Introduction

A number of cohomology theories refining $K$ -theory and motivic cohomology have been introduced and studied over the past 20 years. Well-known examples include hermitian $K$ -theory [Reference Karoubi and Villamayor21, Reference Schlichting36], the theory of the Chow–Witt groups [Reference Asok and Fasel4, Reference Barge and Morel7, Reference Fasel15, Reference Fasel and Srinivas17], Milnor–Witt motivic cohomology [Reference Bachmann and Fasel6, Reference Calmès and Fasel10, Reference Déglise and Fasel12–Reference Druzhinin14], Witt theory and the cohomology of the Witt sheaves [Reference Ananyevskiy3]. All these theories have the property of being SL-oriented, that is, there are Thom classes for vector bundles endowed with a trivialization of the determinant line bundle. Witt theory and the cohomology of the Witt sheaves have the additional property of being $\unicode[STIX]{x1D702}$ -invertible, that is multiplication by the algebraic Hopf map $\unicode[STIX]{x1D702}$ is an isomorphism on the theory.

The $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible theories ${\mathcal{E}}$ have been studied by Ananyevskiy [Reference Ananyevskiy3]. He computes the ${\mathcal{E}}$ -cohomology of $\operatorname{BSL}_{n}$ and proves a fundamental $\operatorname{SL}_{2}$ -splitting principle, replacing the classical $\mathbb{G}_{m}$ -splitting principle which plays a central role in the calculus of characteristic classes in $\operatorname{GL}$ -oriented theories such as $K$ -theory, motivic cohomology or $\operatorname{MGL}$ -theory.

Works by Hoyois [Reference Hoyois18] and Kass–Wickelgren [Reference Kass and Wickelgren22, Reference Kass and Wickelgren23], as well as our papers [Reference Levine24, Reference Levine25], have initiated a study of Euler characteristics and Euler classes with values in the Grothendieck–Witt ring or in the Chow–Witt groups. In this paper we continue this study, looking at Euler characteristics and characteristic classes with values in Witt cohomology and other $\operatorname{SL}$ -oriented $\unicode[STIX]{x1D702}$ -invertible theories.

Using the six-functor formalism on the motivic stable homotopy category, we define a motivic version of the Becker–Gottlieb transfer. This includes the case of a Nisnevich locally trivial fiber bundle $f:E\rightarrow B$ with fiber $F$ , such that the suspension spectrum of $F$ is a strongly dualizable object in the appropriate motivic stable homotopy category. Just as for the classical Becker–Gottlieb transfer, the motivic transfer defines a splitting to the pullback map $f^{\ast }$ on ${\mathcal{E}}$ -cohomology for a motivic spectrum ${\mathcal{E}}$ , as long as the categorical Euler characteristic of $F$ is invertible in the coefficient ring of ${\mathcal{E}}$ . This is discussed in Section 1.

In Section 2, we consider a conjecture of Morel that for $N$ the normalizer of a maximal torus in a split reductive group $G$ over a perfect field $k$ , the quotient scheme $N\backslash G$ has Euler characteristic one in the Grothendieck–Witt ring of $k$ . This generalizes the fact that, for $N$ the normalizer of a maximal torus in a reductive complex Lie group $G$ , $N\backslash G$ has topological Euler characteristic one. We verify a weak form of this conjecture for $\operatorname{GL}_{n}$ and $\operatorname{SL}_{n}$ over a characteristic zero field, as well as the strong form for $\operatorname{GL}_{2}$ and $\operatorname{SL}_{2}$ over an arbitrary perfect field.

We recall the results of Ananyevskiy that we will be using here in Section 3. In Section 4, we reduce the computation of characteristic classes in an $\unicode[STIX]{x1D702}$ -invertible, $\operatorname{SL}$ -oriented theory (satisfying an additional technical hypothesis, see Theorem 4.1 for the precise statement) from the general case to that of bundles with trivialized determinant.

Ananyevskiy’s splitting principle for $\operatorname{SL}_{n}$ -bundles reduces the computation of characteristic classes for such bundles in $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible theories ${\mathcal{E}}$ to the case of $\operatorname{SL}_{2}$ -bundles. In Sections 5 and 6 we look more closely at $\operatorname{SL}_{2}$ , refining Ananyevskiy’s splitting principle to reduce further to bundles with group the normalizer $N_{\operatorname{SL}_{2}}(T)$ of the standard torus $T$ in $\operatorname{SL}_{2}$ . This makes computations much easier, as an irreducible representation of $N_{\operatorname{SL}_{2}}(T)$ has dimension either one or two, and the representation theory of $N_{\operatorname{SL}_{2}}(T)$ is formally (almost) the same as that of real $O(2)$ -bundles. In Section 6 we introduce the rank two bundles $\tilde{O} (m)$ over $BN_{\operatorname{SL}_{2}}(T)$ , as analogs of the classical bundles $O(m)$ on $B\mathbb{G}_{m}$ , and show that the analogy gives a method for computing of the Euler class in Witt cohomology. We apply these considerations in Section 7 to carry out our computation of the Euler class and Pontryagin class of $\tilde{O} (m)$ in Theorem 7.1. This is used in Section 8 to compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, see Theorem 8.1.

We compute the characteristic classes in Witt cohomology for tensor products in Section 9, which together with Ananyevskiy’s splitting principle and our computation of the characteristic classes of symmetric powers of a rank two bundle gives a complete calculus for the characteristic classes in Witt cohomology. We conclude with an example, computing the “quadratic count” of lines on a smooth hypersurface of degree $2d-1$ in $\mathbb{P}^{d+1}$ . This recovers results of Kass–Wickelgren for cubic surfaces [Reference Kass and Wickelgren23] and gives a purely algebraic treatment of some results of Okonek–Teleman [Reference Okonek and Teleman33].

We would like to thank Eva Bayer, Jesse Kass and Kirsten Wickelgren for their comments and suggestions. Eva Bayer drew our attention to Serre’s theorem on computing Hasse–Witt invariants of trace forms, and pointed out several other facts about quadratic forms, such as her paper with Suarez [Reference Bayer-Fluckiger and Suarez8], which were all very important for computing the Euler characteristic of the bundles $\tilde{O} (m)$ , especially our Lemma 7.3. We also thank the referee for making a number of very helpful comments and suggestions.

1 Becker–Gottlieb transfers

We fix a Noetherian base scheme $S$ and let $\mathbf{Sch}/S$ denote the category of finite type separated $S$ -schemes.

We have the motivic stable homotopy category

$$\begin{eqnarray}\operatorname{SH}(-):\mathbf{Sch}/S^{\text{}op}\rightarrow \mathbf{Tr}\end{eqnarray}$$

where $\mathbf{Tr}$ is the 2-category of symmetric monoidal triangulated categories. Following [Reference Ayoub5, Reference Cisinski and Déglise11, Reference Hoyois19], the functor $\operatorname{SH}(-)$ admits the Grothendieck six operations; we briefly review the aspects of this structure that we will be needing here. We note that the theory in [Reference Ayoub5, Reference Hoyois19] is for quasi-projective $S$ -schemes; the extension to separated $S$ -schemes of finite type is accomplished in [Reference Cisinski and Déglise11, Theorem 2.4.50].

For each $X$ , $\operatorname{SH}(X)$ is the homotopy category of a closed symmetric stable model category [Reference Jardine20], which makes $\operatorname{SH}(X)$ into a closed symmetric monoidal category. We denote the symmetric monoidal product in $\operatorname{SH}(X)$ by $\wedge _{X}$ and the adjoint internal Hom by ${\mathcal{H}}om_{X}(-,-)$ . The unit $1_{X}$ is the motivic sphere spectrum over $X$ , $\mathbb{S}_{X}$ . We have for each morphism $f:X\rightarrow Y$ in $\mathbf{Sch}/S$ the exact symmetric monoidal functor $f^{\ast }:\operatorname{SH}(Y)\rightarrow \operatorname{SH}(X)$ with right adjoint $f_{\ast }:\operatorname{SH}(X)\rightarrow \operatorname{SH}(Y)$ , and exact functors $f^{!}:\operatorname{SH}(Y)\rightarrow \operatorname{SH}(X)$ , $f_{!}:\operatorname{SH}(X)\rightarrow \operatorname{SH}(Y)$ , with $f_{!}$ left adjoint to $f^{!}$ .

We refer to a commutative monoid ${\mathcal{E}}$ in the symmetric monoidal category $\operatorname{SH}(X)$ as a motivic ring spectrum, that is, ${\mathcal{E}}$ is endowed with a multiplication map $\unicode[STIX]{x1D707}:{\mathcal{E}}\wedge _{X}{\mathcal{E}}\rightarrow {\mathcal{E}}$ and a unit map $e:1_{X}\rightarrow {\mathcal{E}}$ such that $({\mathcal{E}},\unicode[STIX]{x1D707},e)$ is a commutative monoid in $(\operatorname{SH}(X),\wedge _{X},1_{X})$ .

The adjoint pair $f_{!}\dashv f^{!}$ satisfies a projection formula and the adjunctions extend to internal adjunctions: for $f:X\rightarrow Y$ , there are natural isomorphisms

$$\begin{eqnarray}\displaystyle & & \displaystyle f_{!}(f^{\ast }y\wedge _{X}x)\cong y\wedge _{Y}f_{!}x\nonumber\\ \displaystyle & & \displaystyle f_{\ast }{\mathcal{H}}om_{X}(x,f^{!}x)\cong {\mathcal{H}}om_{Y}(f_{!}x,y).\nonumber\end{eqnarray}$$

There is a natural transformation $\unicode[STIX]{x1D702}_{!\ast }^{f}:f_{!}\rightarrow f_{\ast }$ , which is an isomorphism if $f$ is proper (see e.g. [Reference Cisinski and Déglise11, Section 2.2.6]). Furthermore, for $f$ smooth, the functor $f^{\ast }$ admits a left adjoint $f_{\#}$ .

If $i:X\rightarrow Y$ is a closed immersion with open complement $j:U\rightarrow Y$ , there are localization distinguished triangles

$$\begin{eqnarray}j_{!}j^{!}\rightarrow \text{Id}_{\operatorname{ SH}(Y)}\rightarrow i_{\ast }i^{\ast }\end{eqnarray}$$

and

$$\begin{eqnarray}i_{!}i^{!}\rightarrow \text{Id}_{\operatorname{ SH}(Y)}\rightarrow j_{\ast }j^{\ast }.\end{eqnarray}$$

In addition, $j_{!}=j_{\#}$ , $j^{!}=j^{\ast }$ and $i_{!}=i_{\ast }$ , the counit $i^{\ast }i_{\ast }\rightarrow \text{Id}$ is an isomorphism and the pair $(i^{\ast },j^{\ast })$ is conservative. This gives the following weak Nisnevich descent property: let $\{g_{i}:U_{i}\rightarrow Y\}$ be a Nisnevich cover. A morphism $f:a\rightarrow b$ in $\operatorname{SH}(Y)$ is an isomorphism if and only if $g_{i}^{\ast }f:g_{i}^{\ast }a\rightarrow g_{i}^{\ast }b$ is an isomorphism in $\operatorname{SH}(U_{i})$ for all $i$ (see [Reference Cisinski and Déglise11, Proposition 3.2.8, Corollary 3.3.5]).

If $p:V\rightarrow X$ is a vector bundle with 0-section $s:X\rightarrow V$ , we have the endomorphism $\unicode[STIX]{x1D6F4}^{V}:\operatorname{SH}(X)\rightarrow \operatorname{SH}(X)$ defined as $\unicode[STIX]{x1D6F4}^{V}:=p_{\#}s_{\ast }$ . This is an auto-equivalence, with quasi-inverse $\unicode[STIX]{x1D6F4}^{-V}:=s^{!}p^{\ast }$ . For $f:X\rightarrow Y$ smooth, there are canonical isomorphisms

$$\begin{eqnarray}f_{!}\cong f_{\#}\circ \unicode[STIX]{x1D6F4}^{-T_{f}};\quad f^{!}\cong \unicode[STIX]{x1D6F4}^{T_{f}}\circ f^{\ast }\end{eqnarray}$$

where $T_{f}$ is the relative tangent bundle of $f$ .

For $\unicode[STIX]{x1D70B}_{Y}:Y\rightarrow X$ in $\mathbf{Sm}/X$ , we let $Y/X\in \operatorname{SH}(X)$ denote the object $\unicode[STIX]{x1D70B}_{Y\#}(1_{Y})$ and for $\unicode[STIX]{x1D70B}_{Y}:Y\rightarrow X$ in $\mathbf{Sch}/X$ , we let $Y_{\operatorname{B-M}}/X\in \operatorname{SH}(X)$ denote the object $\unicode[STIX]{x1D70B}_{Y!}(1_{Y})$ . Sending $Y\in \mathbf{Sm}/X$ to $Y/X$ defines a functor

$$\begin{eqnarray}-/X:\mathbf{Sm}/X\rightarrow \operatorname{SH}(X)\end{eqnarray}$$

and sending $Y\in \mathbf{Sch}/X$ to $Y_{\operatorname{B-M}}/X$ defines a functor

$$\begin{eqnarray}-_{\operatorname{B-M}}/X:\mathbf{Sch}^{\text{pr}}/X^{\text{}op}\rightarrow \operatorname{SH}(X),\end{eqnarray}$$

where $\mathbf{Sch}^{\text{pr}}/X$ is the subcategory of proper morphisms in $\mathbf{Sch}/X$ . For a morphism $g:Y\rightarrow Z$ in $\mathbf{Sm}/X$ , the map $g/X:Y/X\rightarrow Z/X$ is induced by the counit of the adjunction $e_{g}:g_{!}g^{!}\rightarrow \text{Id}_{\operatorname{SH}(Z)}$ as the composition

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{Y\#}(1_{Y}) & = & \displaystyle \unicode[STIX]{x1D70B}_{Y\#}\unicode[STIX]{x1D70B}_{Y}^{\ast }(1_{X})\cong \unicode[STIX]{x1D70B}_{Y!}\unicode[STIX]{x1D70B}_{Y}^{!}(1_{X})\cong \unicode[STIX]{x1D70B}_{Z!}g_{!}g^{!}\unicode[STIX]{x1D70B}_{Z}^{!}(1_{X})\nonumber\\ \displaystyle & & \displaystyle \xrightarrow[{}]{e_{g}}\unicode[STIX]{x1D70B}_{Z!}\unicode[STIX]{x1D70B}_{Z}^{!}(1_{X})\cong \unicode[STIX]{x1D70B}_{Z\#}\unicode[STIX]{x1D70B}_{Z}^{\ast }(1_{X})=\unicode[STIX]{x1D70B}_{Z\#}(1_{Z}).\nonumber\end{eqnarray}$$

For a proper morphism $g:Z\rightarrow Y$ in $\mathbf{Sch}/X$ the map $g/X_{\operatorname{B-M}}:Y/X_{\operatorname{B-M}}\rightarrow Z/X_{\operatorname{B-M}}$ is the composition

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{Y!}(1_{Y})\xrightarrow[{}]{u_{g}}\unicode[STIX]{x1D70B}_{Y!}g_{\ast }g^{\ast }(1_{Y})\xrightarrow[{}]{(\unicode[STIX]{x1D702}_{!\ast }^{g})^{-1}}\unicode[STIX]{x1D70B}_{Y!}g_{!}g^{\ast }(1_{Y})\cong \unicode[STIX]{x1D70B}_{Z!}(1_{Z}),\end{eqnarray}$$

where $u_{g}:\text{Id}_{\operatorname{SH}(Y)}\rightarrow g_{\ast }g^{\ast }$ is the unit of the adjunction.

For $\unicode[STIX]{x1D70B}_{Y}:Y\rightarrow X$ in $\mathbf{Sm}/X$ , there is a morphism

$$\begin{eqnarray}ev_{Y}:Y_{\operatorname{B-M}}/X\wedge _{X}Y/X\rightarrow 1_{X}\end{eqnarray}$$

defined as follows: we consider $Y\times _{X}Y$ as a $Y$ -scheme via $p_{2}$ . We have the canonical isomorphisms

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{Y\#}(p_{2!}(1_{Y\times _{X}Y})) & \cong & \displaystyle \unicode[STIX]{x1D70B}_{Y\times _{X}Y\#}(\unicode[STIX]{x1D6F4}^{-T_{p_{2}}}(1_{Y\times _{X}Y}))\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D70B}_{Y\#}(\unicode[STIX]{x1D6F4}^{-T_{Y/X}}(1_{Y}))\wedge _{X}\unicode[STIX]{x1D70B}_{Y\#}(1_{Y})\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D70B}_{Y!}(1_{Y})\wedge _{X}\unicode[STIX]{x1D70B}_{Y\#}(1_{Y})\cong Y_{\operatorname{B-M}}/X\wedge _{X}Y/X.\nonumber\end{eqnarray}$$

The diagonal section $s_{\unicode[STIX]{x1D6FF}}:Y\rightarrow Y\times _{X}Y$ to $p_{2}$ is a proper morphism in $\mathbf{Sch}/Y$ , giving the map

$$\begin{eqnarray}s_{\unicode[STIX]{x1D6FF}}^{\ast }:p_{2!}(1_{Y\times _{X}Y})\rightarrow 1_{Y}\end{eqnarray}$$

in $\operatorname{SH}(Y)$ . Applying $\unicode[STIX]{x1D70B}_{Y\#}$ and using the above-mentioned isomorphism gives the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{Y\#}(s_{\unicode[STIX]{x1D6FF}}^{\ast }):Y_{\operatorname{ B-M}}/X\wedge _{X}Y/X\rightarrow Y/X.\end{eqnarray}$$

The map $ev_{Y}$ is then defined as the composition

$$\begin{eqnarray}Y_{\operatorname{B-M}}/X\wedge _{X}Y/X\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{Y\#}(s_{\unicode[STIX]{x1D6FF}}^{\ast })}Y/X\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{Y}/X}X/X=1_{X}.\end{eqnarray}$$

For $\unicode[STIX]{x1D70B}_{Y}:Y\rightarrow X$ a proper morphism in $\mathbf{Sch}/X$ , we have the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{Y\operatorname{B-M}}/X:1_{X}\rightarrow Y_{\operatorname{B-M}}/X.\end{eqnarray}$$

If $Y$ is smooth over $X$ , we consider $Y\times _{X}Y$ as a smooth $Y$ -scheme via $p_{1}$ , giving the map

$$\begin{eqnarray}s_{\unicode[STIX]{x1D6FF}}/Y:1_{Y}=Y/Y\rightarrow Y\times _{X}Y/Y=p_{1\#}(1_{Y\times _{X}Y}).\end{eqnarray}$$

Applying $\unicode[STIX]{x1D70B}_{Y!}$ and using the canonical isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{Y!}(p_{1\#}(1_{Y\times _{X}Y}))\cong Y/X\wedge _{X}Y_{\operatorname{B-M}}/X\end{eqnarray}$$

constructed as above gives the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{Y!}(s_{\unicode[STIX]{x1D6FF}}/Y):Y_{\operatorname{B-M}}/X\rightarrow Y/X\wedge _{X}Y_{\operatorname{B-M}}/X.\end{eqnarray}$$

For $Y$ smooth and proper over $X$ , define the map $\unicode[STIX]{x1D6FF}_{Y}:1_{X}\rightarrow Y/X\wedge _{X}Y_{\operatorname{B-M}}/X$ as the composition

$$\begin{eqnarray}1_{X}\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{Y\operatorname{B-M}}/X}Y_{\operatorname{B-M}}/X\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{Y!}(s_{\unicode[STIX]{x1D6FF}}/Y)}Y/X\wedge _{X}Y_{\operatorname{B-M}}/X.\end{eqnarray}$$

Recall that an object $x\in \operatorname{SH}(X)$ has the dual $x^{\vee }:={\mathcal{H}}om_{X}(x,1_{X})$ , giving the evaluation map

$$\begin{eqnarray}ev:x^{\vee }\wedge _{X}x\rightarrow 1_{X}\end{eqnarray}$$

adjoint to the identity on $x^{\vee }$ . The map $ev$ in turn induces maps $can_{a}:x^{\vee }\wedge _{X}a\rightarrow {\mathcal{H}}om_{X}(x,a)$ and $can_{a}^{\prime }:a\wedge _{X}x\rightarrow {\mathcal{H}}om_{X}(x^{\vee },a)$ ; $x$ is called strongly dualizable if the morphism $can_{x}:x^{\vee }\wedge _{X}x\rightarrow {\mathcal{H}}om_{X}(x,x)$ is an isomorphism. This is equivalent to the existence of an object $y\in \operatorname{SH}(X)$ and morphisms $\unicode[STIX]{x1D6FF}_{x}:1_{X}\rightarrow x\wedge _{X}y$ , $ev_{x}:y\wedge _{X}x\rightarrow 1_{X}$ , such that

(1.1) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}(\text{Id}_{x}\wedge _{X}ev_{x})\circ (\unicode[STIX]{x1D6FF}_{x}\wedge _{X}\text{Id}_{x})=\text{Id}_{x};\\ (ev_{x}\wedge _{X}\text{Id}_{y})\circ (\text{Id}_{y}\wedge _{X}\unicode[STIX]{x1D6FF}_{x})=\text{Id}_{y},\end{array} & & \displaystyle\end{eqnarray}$$

(see [Reference May28, Theorem 2.6]). In this case, there is a canonical isomorphism $y\cong x^{\vee }$ with $ev_{x}$ going over to $ev$ , and $\unicode[STIX]{x1D6FF}_{x}$ going over to the map

$$\begin{eqnarray}1_{X}\rightarrow x\wedge x^{\vee }\end{eqnarray}$$

given as the image of $\text{Id}_{x}$ under the sequence of isomorphisms

$$\begin{eqnarray}\displaystyle \text{Hom}_{\operatorname{SH}(X)}(x,x) & & \displaystyle \cong \text{Hom}_{\operatorname{SH}(X)}(1_{X},{\mathcal{H}}om_{X}(x,x))\nonumber\\ \displaystyle & & \displaystyle \xleftarrow[{}]{can_{x\ast }}\text{Hom}_{\operatorname{SH}(X)}(1_{X},x^{\vee }\wedge _{X}x)\nonumber\\ \displaystyle & & \displaystyle \xrightarrow[{}]{\unicode[STIX]{x1D70F}_{x^{\vee },x\ast }}\text{Hom}_{\operatorname{SH}(X)}(1_{X},x\wedge _{X}x^{\vee }).\nonumber\end{eqnarray}$$

Definition 1.3. Let $x\in \operatorname{SH}(X)$ be a strongly dualizable object. The Euler characteristic $\unicode[STIX]{x1D712}(x/X)\in \text{End}_{\operatorname{SH}(X)}(1_{X})$ is the composition

$$\begin{eqnarray}1_{X}\xrightarrow[{}]{\unicode[STIX]{x1D6FF}_{x}}x\wedge _{X}x^{\vee }\xrightarrow[{}]{\unicode[STIX]{x1D70F}_{x,x^{\vee }}}x^{\vee }\wedge _{X}x\xrightarrow[{}]{ev_{x}}1_{X}.\end{eqnarray}$$

Definition 1.5. 1. Let $f:E\rightarrow B$ be in $\mathbf{Sm}/B$ with $E/B$ a strongly dualizable object in $\operatorname{SH}(B)$ . The relative transfer $\text{Tr}_{f/B}:1_{B}\rightarrow E/B$ is defined as follows: consider the diagonal morphism

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{E}:E\rightarrow E\times _{B}E.\end{eqnarray}$$

Applying $f_{\#}$ gives the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{E}/B:E/B\rightarrow E\times _{B}E/B\cong E/B\wedge _{B}E/B.\end{eqnarray}$$

By duality, we have the isomorphism

$$\begin{eqnarray}(-)^{\text{tr}}:\text{Hom}_{\operatorname{SH}(B)}(E/B,E/B\wedge _{B}E/B)\cong \text{Hom}_{\operatorname{SH}(B)}(E/B^{\vee }\wedge _{B}E/B,E/B),\end{eqnarray}$$

defined by sending a morphism $g:E/B\rightarrow E/B\wedge _{B}E/B$ to the composition

$$\begin{eqnarray}E/B^{\vee }\wedge _{B}E/B\xrightarrow[{}]{\text{Id}\wedge g}E/B^{\vee }\wedge _{B}E/B\wedge _{B}E/B\xrightarrow[{}]{ev_{E/B}\wedge \text{Id}}1_{B}\wedge _{B}E/B\cong E/B.\end{eqnarray}$$

We set

$$\begin{eqnarray}\text{Tr}_{f/B}:=\unicode[STIX]{x1D6E5}_{E}/B^{\text{tr}}\circ \unicode[STIX]{x1D70F}_{E/B,E/B^{\vee }}\circ \unicode[STIX]{x1D6FF}_{E/B}.\end{eqnarray}$$

2. Suppose we have $f:E\rightarrow B$ in $\mathbf{Sm}/B$ as in (1), with $\unicode[STIX]{x1D70B}_{B}:B\rightarrow S$ in $\mathbf{Sm}/S$ , giving the structure morphism $\unicode[STIX]{x1D70B}_{E}:E\rightarrow S$ , $\unicode[STIX]{x1D70B}_{E}:=\unicode[STIX]{x1D70B}_{B}\circ f$ . Define the transfer $\text{Tr}(f/S):B/S\rightarrow E/S$ by

$$\begin{eqnarray}\text{Tr}(f/S):=\unicode[STIX]{x1D70B}_{B\#}(\text{Tr}_{f/B}),\end{eqnarray}$$

where we use the canonical isomorphism

$$\begin{eqnarray}E/S:=\unicode[STIX]{x1D70B}_{E\#}(1_{E})\cong \unicode[STIX]{x1D70B}_{B\#}(f_{\#}(1_{E}))=\unicode[STIX]{x1D70B}_{B\#}(E/B).\end{eqnarray}$$

Lemma 1.6. Given $f:E\rightarrow B$ as in Definition 1.5(1), let $g:B^{\prime }\rightarrow B$ be a morphism in $\mathbf{Sch}/S$ and let $f^{\prime }:E^{\prime }\rightarrow B^{\prime }$ be the pullback $E^{\prime }:=E\times _{B}B^{\prime }$ , $f^{\prime }=p_{2}$ . Then $f^{\prime }:E^{\prime }\rightarrow B^{\prime }$ is strongly dualizable in $\operatorname{SH}(B^{\prime })$ , there are canonical isomorphisms $E^{\prime }/B^{\prime }\cong g^{\ast }(E/B)$ , $1_{B^{\prime }}\cong g^{\ast }(1_{B})$ and via these isomorphisms, we have

$$\begin{eqnarray}\text{Tr}(f^{\prime }/B^{\prime })=g^{\ast }(\text{Tr}(f/B)).\end{eqnarray}$$

Moreover, if $\unicode[STIX]{x1D70B}_{B}:B\rightarrow S$ is in $\mathbf{Sm}/S$ , if $g_{0}:S^{\prime }\rightarrow S$ is a morphism in $\mathbf{Sch}/S$ , if $B^{\prime }=S^{\prime }\times _{S}B$ and $g:B^{\prime }\rightarrow B$ is the projection, then there is a canonical isomorphism $B^{\prime }/S^{\prime }\cong g_{0}^{\ast }(B/S)$ and via this isomorphism

$$\begin{eqnarray}\text{Tr}(f^{\prime }/S^{\prime })=g_{0}^{\ast }(\text{Tr}(f/S)).\end{eqnarray}$$

Proof. For the first assertion, we have the exchange isomorphism (see [Reference Cisinski and Déglise11, § 1.1.6] or [Reference Ayoub5, § 1.4.5])

$$\begin{eqnarray}Ex_{\#}^{\ast }:p_{1\#}\circ p_{2}^{\ast }\rightarrow g^{\ast }\circ f_{\#}~\end{eqnarray}$$

associated to the Cartesian diagram

which gives the isomorphism

$$\begin{eqnarray}g^{\ast }(E/B)=g^{\ast }\circ f_{\#}(1_{E})\xrightarrow[{}]{(Ex_{\#}^{\ast })^{-1}}p_{1\#}\circ p_{2}^{\ast }(1_{E})\cong p_{1\#}(1_{E^{\prime }})=E^{\prime }/B^{\prime }.\end{eqnarray}$$

We have the canonical isomorphism $1_{B^{\prime }}\cong g^{\ast }(1_{B})$ as well. Since $g^{\ast }$ is a symmetric monoidal functor, the assumption that $E/B$ is strongly dualizable in $\operatorname{SH}(B)$ implies that $E^{\prime }/B^{\prime }$ is strongly dualizable in $\operatorname{SH}(B^{\prime })$ with ${E^{\prime }/B^{\prime }}^{\vee }$ canonically isomorphic to $g^{\ast }(E/B^{\vee })$ . Moreover, via these isomorphisms, we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{E^{\prime }/B^{\prime }}=g^{\ast }(\unicode[STIX]{x1D6FF}_{E/B}),~ev_{E^{\prime }/B^{\prime }}=g^{\ast }(ev_{E/B}).\end{eqnarray}$$

Finally, since the exchange isomorphism is a natural transformation, we have

$$\begin{eqnarray}g^{\ast }({\unicode[STIX]{x1D6E5}_{E}/B}^{\text{tr}})={\unicode[STIX]{x1D6E5}_{E^{\prime }}/B^{\prime }}^{\text{tr}}\end{eqnarray}$$

and thus

$$\begin{eqnarray}\displaystyle \text{Tr}(f^{\prime }/B) & := & \displaystyle {\unicode[STIX]{x1D6E5}_{E^{\prime }}/B^{\prime }}^{\text{tr}}\circ \unicode[STIX]{x1D70F}_{E^{\prime }/B^{\prime },{E^{\prime }/B^{\prime }}^{\vee }}\circ \unicode[STIX]{x1D6FF}_{E^{\prime }/B^{\prime }}\nonumber\\ \displaystyle & = & \displaystyle g^{\ast }({\unicode[STIX]{x1D6E5}_{E}/B}^{\text{tr}})\circ g^{\ast }(\unicode[STIX]{x1D70F}_{E/B,E/B^{\vee }})\circ g^{\ast }(\unicode[STIX]{x1D6FF}_{E/B})\nonumber\\ \displaystyle & = & \displaystyle g^{\ast }(\text{Tr}(f/B)).\nonumber\end{eqnarray}$$

The proof that $\text{Tr}(f^{\prime }/S^{\prime })=g_{0}^{\ast }(\text{Tr}(f/S))$ is similar and is left to the reader.

Lemma 1.7. Let $f:E\rightarrow B$ be in $\mathbf{Sm}/B$ , with $E/B$ strongly dualizable in $\operatorname{SH}(B)$ . Then

$$\begin{eqnarray}f/B\circ \text{Tr}(f/B):1_{B}\rightarrow 1_{B}\end{eqnarray}$$

is equal to $\unicode[STIX]{x1D712}(E/B)$ .

Proof. $f/B\circ \text{Tr}(f/B)$ is the composition

$$\begin{eqnarray}1_{B}\xrightarrow[{}]{\unicode[STIX]{x1D6FF}_{E/B}}E/B\wedge _{B}E/B^{\vee }\xrightarrow[{}]{\unicode[STIX]{x1D70F}_{E/B,E/B^{\vee }}}E/B^{\vee }\wedge _{B}E/B\xrightarrow[{}]{\unicode[STIX]{x1D6E5}_{E}/B^{\text{tr}}}E/B\xrightarrow[{}]{f/B}1_{B}.\end{eqnarray}$$

The map $\unicode[STIX]{x1D6E5}_{E}/B^{\text{tr}}$ is the composition

$$\begin{eqnarray}\displaystyle E/B^{\vee }\wedge _{B}E/B\xrightarrow[{}]{\text{Id}\wedge \unicode[STIX]{x1D6E5}_{E}/B}E/B^{\vee }\wedge _{B}E/B\wedge _{B}E/B & & \displaystyle \nonumber\\ \displaystyle \xrightarrow[{}]{ev_{E/B}\wedge \text{Id}}1_{B}\wedge _{B}E/B\cong E/B. & & \displaystyle \nonumber\end{eqnarray}$$

We have the commutative diagram

Thus

$$\begin{eqnarray}f/B\circ \text{Tr}(f/B)=ev_{E/B}\circ \unicode[STIX]{x1D70F}_{E/B,E/B^{\vee }}\circ \unicode[STIX]{x1D6FF}_{E/B}=\unicode[STIX]{x1D712}(E/B).\end{eqnarray}$$

Proof. Under our assumptions, we have $E/B=\unicode[STIX]{x1D70B}_{B}^{\ast }(F/S)$ . Since $\unicode[STIX]{x1D70B}_{B}^{\ast }$ is a symmetric monoidal functor, it follows that $E/B^{\vee }=\unicode[STIX]{x1D70B}_{B}^{\ast }(F/S^{\vee })$ and $\text{Tr}(E/B)=\unicode[STIX]{x1D70B}_{B}^{\ast }(\text{Tr}(F/S))$ . Thus

$$\begin{eqnarray}\displaystyle f/S\circ \text{Tr}(E/S) & = & \displaystyle \unicode[STIX]{x1D70B}_{B\#}(f/B\circ \unicode[STIX]{x1D70B}_{B}^{\ast }(\text{Tr}(F/S))),\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70B}_{B\#}(\unicode[STIX]{x1D70B}_{B}^{\ast }(\unicode[STIX]{x1D70B}_{F}/S)\circ \unicode[STIX]{x1D70B}_{B}^{\ast }(\text{Tr}(F/S)))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70B}_{B\#}(\unicode[STIX]{x1D70B}_{B}^{\ast }(\unicode[STIX]{x1D70B}_{F}/S\circ \text{Tr}(F/S)))\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70B}_{B\#}(\unicode[STIX]{x1D70B}_{B}^{\ast }(\unicode[STIX]{x1D712}(F/S)))\nonumber\end{eqnarray}$$

which is multiplication by $\unicode[STIX]{x1D712}(F/S)$ .

Let $f:E\rightarrow B$ be a finite type morphism, with $B\in \mathbf{Sch}/S$ and let $F$ be in $\mathbf{Sch}/S$ . We say that $f$ is a Nisnevich locally trivial bundle with fiber $F$ if there is a covering family $\{g_{i}:U_{i}\rightarrow B,i\in I\}$ for the Nisnevich topology on $B$ and isomorphisms of $U_{i}$ -schemes $E\times _{B}U_{i}\cong F\times _{S}U_{i}$ for each $i\in I$ .

Theorem 1.10. Let $F$ and $B$ be in $\mathbf{Sm}/S$ and let $f:E\rightarrow B$ in $\mathbf{Sm}/B$ be a Nisnevich locally trivial bundle with fiber $F$ .

1. If $F/S$ is strongly dualizable in $\operatorname{SH}(S)$ , then $E/B$ is strongly dualizable in $\operatorname{SH}(B)$ .

2. Suppose that $F/S$ is strongly dualizable in $\operatorname{SH}(S)$ . Let ${\mathcal{E}}\in \operatorname{SH}(S)$ be a motivic ring spectrum such that the map $\unicode[STIX]{x1D712}(F/S):1_{S}\rightarrow 1_{S}$ induces an isomorphism on ${\mathcal{E}}^{0,0}(S)$ . Then the map

$$\begin{eqnarray}f/S^{\ast }:{\mathcal{E}}^{\ast \ast }(B/S)\rightarrow {\mathcal{E}}^{\ast \ast }(E/S)\end{eqnarray}$$

is split injective.

Proof. The assertion (1) follows from the isomorphism $g_{i}^{\ast }(E/B)\cong F\times _{S}U_{i}/U_{i}$ , and Proposition 1.2(2, 4).

For (2), we have

$$\begin{eqnarray}\displaystyle {\mathcal{E}}^{a,b}(E/S) & = & \displaystyle \text{Hom}_{\operatorname{SH}(S)}(\unicode[STIX]{x1D70B}_{B\#}(E/B),\unicode[STIX]{x1D6F4}^{a,b}{\mathcal{E}})\nonumber\\ \displaystyle & = & \displaystyle \text{Hom}_{\operatorname{SH}(B)}(E/B,\unicode[STIX]{x1D6F4}^{a,b}\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}})=\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(E/B);\nonumber\end{eqnarray}$$

similarly ${\mathcal{E}}^{a,b}(B/S)=\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{B})$ and via these isomorphisms, the map $f/S^{\ast }:{\mathcal{E}}^{a,b}(B/S)\rightarrow {\mathcal{E}}^{a,b}(E/S)$ is the map $f/B^{\ast }:\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{B})\rightarrow \unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(E/B)$ . We claim that the endomorphism

$$\begin{eqnarray}\text{Tr}(E/B)^{\ast }\circ f/B^{\ast }:\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{B})\rightarrow \unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{B})\end{eqnarray}$$

is an isomorphism; assuming this the case, the map $\text{Tr}(E/B)^{\ast }$ gives a left quasi-inverse to $f/S^{\ast }$ .

By weak Nisnevich descent, it suffices to show that the pullback

$$\begin{eqnarray}g_{i}^{\ast }(\text{Tr}(E/B)^{\ast }\circ f/B^{\ast }):g_{i}^{\ast }\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{U_{i}})\rightarrow g_{i}^{\ast }\unicode[STIX]{x1D70B}_{B}^{\ast }{\mathcal{E}}^{a,b}(1_{U_{i}})\end{eqnarray}$$

is an isomorphism for all $i$ . But

$$\begin{eqnarray}g_{i}^{\ast }(\text{Tr}(E/B)^{\ast }\circ f/B^{\ast })=\text{Tr}(F\times _{S}U_{i}/U_{i})^{\ast }\circ p_{U_{i}}^{\ast }\end{eqnarray}$$

and $p_{U_{i}}\circ \text{Tr}(F\times _{S}U_{i}/U_{i})$ is multiplication by $\unicode[STIX]{x1D70B}_{U_{i}}^{\ast }\unicode[STIX]{x1D712}(F/S)$ . Using the multiplication $\unicode[STIX]{x1D707}$ in ${\mathcal{E}}$ , we have the commutative diagram

and so $g_{i}^{\ast }(\text{Tr}(E/B)^{\ast }\circ f/B^{\ast })$ is an isomorphism by our assumption on ${\mathcal{E}}$ .

2 The case of split reductive groups: a conjecture

Let $G$ be a split reductive group scheme over a base scheme $S$ , with Borel subgroup $B$ and maximal split torus $T\subset B$ . Let $N_{T}\supset T$ be the normalizer of $T$ in $G$ . We take the model $N_{T}\backslash EG$ for $BN_{T}$ (see the end of this section for details on our model for $EG$ ) and consider the $N_{T}\backslash G$ -bundle $BN_{T}\rightarrow BG$ . For special $G$ , that is, for $G=\operatorname{GL}_{n},\operatorname{SL}_{n},\operatorname{Sp}_{2n}$ , $BN_{T}\rightarrow BG$ is a Nisnevich locally trivial bundle, and so we can apply the motivic Becker–Gottlieb theorem developed in the previous section; for this, we need to compute the Euler characteristic $\unicode[STIX]{x1D712}(N_{T}\backslash G/S)$ .

Following a suggestion of Fabien Morel, who formulated the strong form, we have the following conjecture.

The weak form suffices to apply the motivic Becker–Gottlieb transfer to the map $\unicode[STIX]{x1D70B}_{G}:BN_{T}(G)\rightarrow BG$ in case the $N_{T}(G)\backslash G$ -bundle is Nisnevich locally trivial; see Theorem 2.5 below for a precise statement in case $G=\operatorname{GL}_{n},\operatorname{SL}_{n}$ .

For $S=\text{Spec }\mathbb{C}$ it is well-known that $N_{T}(\mathbb{C})\backslash G(\mathbb{C})$ has (topological) Euler characteristic 1. This is also easy to prove: $T(\mathbb{C})\backslash G(\mathbb{C})$ is homotopy equivalent to $B(\mathbb{C})\backslash G(\mathbb{C})$ , which has homology the free abelian group on the closure of the Schubert cells $B(\mathbb{C})wB(\mathbb{C})$ , $w\in W(G):=N_{T}(\mathbb{C})/T(\mathbb{C})$ ; as these are all of even dimension, we have $\unicode[STIX]{x1D712}^{\text{top}}(G(\mathbb{C})/T(\mathbb{C}))=\#W(G)$ . As the covering space $T(\mathbb{C})\backslash G(\mathbb{C})\rightarrow N_{T}(\mathbb{C})\backslash G(\mathbb{C})$ has degree $\#W(G)$ , it follows that $\unicode[STIX]{x1D712}^{\text{top}}(N_{T}(\mathbb{C})\backslash G(\mathbb{C}))=1$ .

We note that $N_{T}(G)(\mathbb{C})\backslash G(\mathbb{C})=(N_{T}(G)\backslash G)(\mathbb{C})$ , since $\mathbb{C}$ is algebraically closed. For $\mathbb{R}$ , this is no longer the case, which makes the analysis of the Euler characteristic $\unicode[STIX]{x1D712}^{\text{top}}((N_{T}(G)\backslash G)(\mathbb{R}))$ a bit more difficult.

Proof. It suffices to consider the case $G=\operatorname{GL}_{n}$ , since the schemes $N_{T}(\operatorname{GL}_{n})\backslash \operatorname{GL}_{n}$ and $N_{T}(\operatorname{SL}_{n})\backslash \operatorname{SL}_{n}$ are isomorphic. We write $N_{T}$ for $N_{T}(\operatorname{GL}_{n})$ .

$N_{T}=\mathbb{G}_{m}^{n}\rtimes \mathfrak{S}_{n}$ , where $\mathfrak{S}_{n}$ is the symmetric group, so we have

$$\begin{eqnarray}N_{T}\backslash \operatorname{GL}_{n}=\mathfrak{S}_{n}\backslash (\mathbb{G}_{m}^{n}\backslash \operatorname{GL}_{n})\end{eqnarray}$$

and $\mathbb{G}_{m}^{n}\backslash \operatorname{GL}_{n}$ is the open subscheme of $(\mathbb{P}^{n-1})^{n}$ parametrizing $n$ -tuples of lines through 0 in $\mathbb{A}^{n}$ which span $\mathbb{A}^{n}$ . This realizes $N_{T}\backslash \operatorname{GL}_{n}$ as an open subscheme of $\operatorname{Sym}^{n}\mathbb{P}^{n-1}$ , and gives a decomposition of the $\mathbb{R}$ -points $x\in (N_{T}\backslash \operatorname{GL}_{n})(\mathbb{R})$ in terms of the residue fields of the corresponding closed points $y_{1},\ldots ,y_{s}$ of $(\mathbb{P}^{n-1})^{n}$ lying over $x$ . This decomposes $(N_{T}\backslash \operatorname{GL}_{n})(\mathbb{R})$ as a disjoint union of open submanifolds

$$\begin{eqnarray}(N_{T}\backslash \operatorname{GL}_{n})(\mathbb{R})=\amalg _{i=0}^{n}U_{i}\end{eqnarray}$$

where $U_{i}$ parametrizes the collection of lines with exactly $i$ real lines. Of course $U_{i}$ is empty if $n-i$ is odd.

We claim that $\unicode[STIX]{x1D712}^{\text{top}}(U_{i})=0$ if $i\geqslant 2$ . Indeed, we can form a finite covering space $\tilde{U} _{i}\rightarrow U_{i}$ by killing the permutation action on the real lines and choosing an orientation on each real line. Then putting a metric on $\mathbb{C}^{n}$ , we see that $\tilde{U} _{i}$ is homotopy equivalent to an $\text{SO}(i)$ -bundle over the space parametrizing the (unordered) collection of the remaining lines. Since $\unicode[STIX]{x1D712}^{\text{top}}(\text{SO}(i))=0$ for $i\geqslant 2$ , we have $\unicode[STIX]{x1D712}^{\text{top}}(\tilde{U} _{i})=0$ and hence $\unicode[STIX]{x1D712}^{\text{top}}(U_{i})=0$ as well.

Now assume $n=2m$ is even. $U_{0}$ parametrizes the unordered collections of $m$ pairs of complex conjugate lines that span $\mathbb{C}^{n}$ , so we may form the degree $m!$ covering space $\tilde{U} _{0}\rightarrow U_{0}$ of ordered $m$ -tuples of such conjugate pairs. Each pair $\ell ,\bar{\ell }$ spans a $\mathbb{A}^{2}\subset \mathbb{A}^{n}$ defined over $\mathbb{R}$ , so we may map $\tilde{U} _{0}$ to the real points of the quotient $(\operatorname{GL}_{2})^{m}\backslash \operatorname{GL}_{n}$ , which is homotopy equivalent to the real points of the partial flag manifold $Fl_{m}:=P_{m}\backslash \operatorname{GL}_{n}$ , where $P_{m}$ is the parabolic with Levi subgroup $(\operatorname{GL}_{2}(\mathbb{C}))^{m}$ . In fact, the map $\tilde{U} _{0}\rightarrow Fl_{m}(\mathbb{R})$ is also a homotopy equivalence: given a two-plane $\unicode[STIX]{x1D70B}\subset \mathbb{A}^{n}$ defined over $\mathbb{R}$ , the pairs of conjugate lines in $\unicode[STIX]{x1D70B}$ are parametrized by the upper half-plane in the Riemann sphere $\mathbb{P}(\unicode[STIX]{x1D70B})$ , which is contractible.

We claim that $\unicode[STIX]{x1D712}^{\text{top}}(Fl_{m}(\mathbb{R}))=m!$ for all $m\geqslant 2$ . To compute the Euler characteristic, we use the following fact:

If $X\in \mathbf{Sm}/\mathbb{R}$ is cellular, that is, $X$ admits a finite stratification by locally closed subschemes $X_{i}\cong \mathbb{A}^{n_{i}}$ , then $\unicode[STIX]{x1D712}^{\text{top}}(X(\mathbb{R}))=\operatorname{rnk}\,\text{CH}^{\text{even}}(X)-\operatorname{rnk}\,\text{CH}^{\text{odd}}(X)$ .

This follows from [Reference Levine25, Remark 1.11, Proposition 1.14], using the $\operatorname{GW}(\mathbb{R})$ -valued Euler characteristic $\unicode[STIX]{x1D712}(X/\mathbb{R})$ :

(2.1) $$\begin{eqnarray}\begin{array}{@{}l@{}}\text{i}.\text{ For }X\in \mathbf{Sm}/\mathbb{R},\unicode[STIX]{x1D712}^{\text{top}}(X(\mathbb{R}))\text{ is equal to the signature of }\unicode[STIX]{x1D712}(X/\mathbb{R})\\ \text{[24, Remark 1.11]}.\\ \text{i}\text{i}.\text{ For }X\in \mathbf{Sm}/k\text{ cellular},\unicode[STIX]{x1D712}(X/k)=\operatorname{rnk}\,\text{CH}^{\text{even}}\cdot \langle 1\rangle +\,\operatorname{rnk}\,\text{CH}^{\text{odd}}\cdot \langle -1\rangle \\ \text{[24, Proposition 1.14]}.\end{array}\end{eqnarray}$$

Alternatively, one can apply the Lefschetz trace formula to complex conjugation $c$ acting on $H^{\ast }(X(\mathbb{C}),\mathbb{Q})$ , and use the fact that for $X$ cellular, $H^{\text{odd}}(X(\mathbb{C}),\mathbb{Q})=0$ , $H^{2n}(X(\mathbb{C}),\mathbb{Q})$ has dimension $\operatorname{rnk}\,\text{CH}^{n}(X)$ and $c$ acts by $(-1)^{n}$ , since the cycle class map $\text{cl}^{n}$ is a $c$ -equivariant isomorphism from the $c$ -invariant $\mathbb{Q}$ vector space $\text{CH}^{n}(X)\otimes \mathbb{Q}$ to $H^{2n}(X(\mathbb{C}),\mathbb{Q}(2\unicode[STIX]{x1D70B}i)^{n})=\mathbb{Q}\cdot \text{cl}^{n}(\text{CH}^{n}(X))$ .

For $m=2$ , $Fl_{2}=\operatorname{Gr}(2,4)$ , which has 4 Schubert cells of even dimension and 2 of odd dimension. Thus $\unicode[STIX]{x1D712}(Fl_{2}/\mathbb{R})=4\langle 1\rangle +2\langle -1\rangle$ , which has signature 2, and hence $\unicode[STIX]{x1D712}^{\text{top}}(Fl_{2}(\mathbb{R}))=2$ .

In general, we have the fibration $Fl_{m}\rightarrow Fl_{m-1}$ with fiber $\operatorname{Gr}(2,2m)$ . We can compare the Schubert cells for $\operatorname{Gr}(2,2m)$ with those for $\operatorname{Gr}(2,2m-2)$ : the “new” ones come from the partitions $(2m-3,i)$ , $i=0,\ldots ,2m-3$ and $(2m-2,i)$ , $i=0,\ldots ,2m-2$ . This adds the quadratic form $(2m-1)\langle 1\rangle +(2m-2)\langle -1\rangle$ of signature +1 to $\unicode[STIX]{x1D712}(\operatorname{Gr}(2,2m-2))$ , which implies that $\unicode[STIX]{x1D712}^{\text{top}}(\operatorname{Gr}(2,2m)(\mathbb{R}))=\unicode[STIX]{x1D712}^{\text{top}}(\operatorname{Gr}(2,2m-2))+1$ . By induction, this says that $\unicode[STIX]{x1D712}^{\text{top}}(\operatorname{Gr}(2,2m)(\mathbb{R}))=m$ , and similarly by induction we find $\unicode[STIX]{x1D712}^{\text{top}}(Fl_{m})=m\cdot \unicode[STIX]{x1D712}^{\text{top}}(Fl_{m-1})=m!$ . Thus $\unicode[STIX]{x1D712}^{\text{top}}(\tilde{U} _{0})=m!$ and hence $\unicode[STIX]{x1D712}^{\text{top}}(U_{0})=1$ .

In case $n=2m+1$ is odd, we can again choose a metric and fiber $U_{1}$ over the real projective space $\mathbb{R}\mathbb{P}^{2m}$ , with fiber over $[\ell ]\in \mathbb{R}\mathbb{P}^{2m}$ homotopy equivalent to the $U_{0}$ for the hyperplane perpendicular to the $\mathbb{C}$ -span of $\ell$ . As $\unicode[STIX]{x1D712}^{\text{top}}(\mathbb{R}\mathbb{P}^{2m})=1$ , this yields $\unicode[STIX]{x1D712}^{\text{top}}(U_{1})=1$ .

We take the opportunity here to make explicit our model for $BG$ , $G$ a reductive group scheme over a perfect field $k$ . We follow Morel–Voevodsky [Reference Morel and Voevodsky32, Section 4.2.2], by using the Totaro–Graham–Edidin model for the Morel–Voevodsky $B_{\acute{\text{e}}\text{t}}G$ as our $BG$ .

Fixing integers $n>0$ , $m\geqslant 0$ , let $M_{n\times m+n}\cong \mathbb{A}^{n(m+n)}$ be the affine space of $m\times (m+n)$ matrices and let $E_{m}\operatorname{GL}_{n}\subset M_{n\times m+n}$ be the open subscheme of the matrices of maximal rank $n$ . We give $E_{m}\operatorname{GL}_{n}$ the base-point $x_{0}=(I_{n},0_{n},\ldots ,0_{n})$ where $I_{n}\in M_{n\times n}$ is the identity matrix and $0_{n}\in M_{n\times 1}$ is the 0-matrix. We let $E\operatorname{GL}_{n}$ be the presheaf on $\mathbf{Sm}/k$

$$\begin{eqnarray}E\operatorname{GL}_{n}=colim_{m}E_{m}\operatorname{GL}_{n}\end{eqnarray}$$

which we may consider as a pointed presheaf with base-point $x_{0}$ if we wish. We note that $\operatorname{GL}_{n}$ acts freely on $E_{m}\operatorname{GL}_{n}$ for each $m\geqslant 0$ . Fixing an embedding $G\subset \operatorname{GL}_{n}$ for some $n$ , we have the (pointed) quotient scheme

$$\begin{eqnarray}B_{m}G:=G\backslash E_{m}\operatorname{GL}_{n}\end{eqnarray}$$

and the (pointed) presheaf on $\mathbf{Sm}/k$

$$\begin{eqnarray}BG:=colim_{m}B_{m}G.\end{eqnarray}$$

For $X\in \mathbf{Sm}/k$ , we have

$$\begin{eqnarray}BG(X)=colim_{m}B_{m}G(X)\end{eqnarray}$$

where $B_{m}G(X)$ denotes the morphisms from $X$ to $B_{m}G$ in $\mathbf{Sm}_{k}$ and $BG(X)$ is the set of morphisms from the representable presheaf $X$ to $BG$ . Thus we can speak of the $k$ -points of $BG$ .

Although $B_{m}G$ is almost never the presheaf quotient of $E_{m}\operatorname{GL}_{n}$ by $G$ , $B_{m}G$ is the quotient $G\backslash E_{m}\operatorname{GL}_{n}$ as étale sheaf, and if $G$ is special, $B_{m}G$ is the quotient $G\backslash E_{m}\operatorname{GL}_{n}$ as Zariski sheaf. These facts pass to the morphism $E\operatorname{GL}_{n}\rightarrow BG$ . Although $BG$ is not in general the presheaf quotient, we nonetheless often write $BG=G\backslash E\operatorname{GL}_{n}$ . We note that the $\mathbb{A}^{1}$ -homotopy type of $BG$ is independent of the choice of embedding $G\subset \operatorname{GL}_{n}$ and does in fact give a model for the Morel–Voevodsky $B_{\acute{\text{e}}\text{t}}G$ in the unstable motivic homotopy category [Reference Morel and Voevodsky32, Remark 4.27].

For each $m\geqslant 0$ , we have the principal $G$ -bundle

$$\begin{eqnarray}G\rightarrow E_{m}\operatorname{GL}_{n}\rightarrow B_{m}G\end{eqnarray}$$

which is étale locally trivial if $G$ is smooth over $k$ and is Zariski locally trivial if $G$ is special. We refer to the sequence

$$\begin{eqnarray}G\rightarrow E\operatorname{GL}_{n}\rightarrow BG\end{eqnarray}$$

as a principal $G$ -bundle.

For any $G$ defined as a closed subgroup scheme of $\operatorname{GL}_{n}$ or of $\operatorname{SL}_{n}$ , we always use the embedding $G\subset \operatorname{GL}_{n}$ in our definition of $B_{m}G$ and $BG$ , unless we explicitly say otherwise.

For $G=\operatorname{GL}_{n}$ , we recall that $B_{m}\operatorname{GL}_{n}$ is just the Grassmann scheme $\operatorname{Gr}(n,n+m)$ . For $G=N_{T}:=N_{T}(\operatorname{GL}_{n})\subset \operatorname{GL}_{n}$ we have the fiber bundle

$$\begin{eqnarray}B_{m}N_{T}\rightarrow B_{m}\operatorname{GL}_{n}.\end{eqnarray}$$

Since $\operatorname{GL}_{n}$ is special the bundle $E_{m}\operatorname{GL}_{n}\rightarrow B_{m}\operatorname{GL}_{n}$ is a Zariski locally trivial principal $\operatorname{GL}_{n}$ -bundle; this gives the description of $B_{m}N_{T}\rightarrow B_{m}\operatorname{GL}_{n}$ as

$$\begin{eqnarray}B_{m}N=(N_{T}\backslash \operatorname{GL}_{n})\times ^{\operatorname{GL}_{n}}E_{m}\operatorname{GL}_{n}\end{eqnarray}$$

that is $\unicode[STIX]{x1D70B}_{\operatorname{GL}_{n},m}:B_{m}N_{T}\rightarrow B_{m}\operatorname{GL}_{n}$ is a Zariski locally trivial fiber bundle with fiber $F=N_{T}\backslash \operatorname{GL}_{n}$ .

Applying this construction to $N_{T}(\operatorname{SL}_{n})\subset \operatorname{SL}_{n}\subset \operatorname{GL}_{n}$ , gives us the Zariski locally trivial fiber bundle

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\operatorname{SL}_{n},m}:B_{m}N_{T}(\operatorname{SL}_{n})\rightarrow B_{m}\operatorname{SL}_{n}\end{eqnarray}$$

with fiber $N_{T}\backslash \operatorname{GL}_{n}$ , noting that the inclusion $\operatorname{SL}_{n}\rightarrow \operatorname{GL}_{n}$ induces an isomorphism of schemes

$$\begin{eqnarray}N_{T}(\operatorname{SL}_{n})\backslash \operatorname{SL}_{n}\rightarrow N_{T}(\operatorname{GL}_{n})\backslash \operatorname{GL}_{n}\!.\end{eqnarray}$$

Theorem 2.5. Let $k$ be a field of characteristic zero and let ${\mathcal{E}}\in \operatorname{SH}(k)$ be a motivic ring spectrum. Then the maps $\unicode[STIX]{x1D70B}_{\operatorname{GL}_{n}}:B_{m}N_{T}(\operatorname{GL}_{n})\rightarrow B_{m}\operatorname{GL}_{n}$ and $\unicode[STIX]{x1D70B}_{\operatorname{SL}_{n}}:B_{m}N_{T}(\operatorname{SL}_{n})\rightarrow B_{m}\operatorname{SL}_{n}$ induce split injections

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{\operatorname{GL}_{n}}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(B_{m}\operatorname{GL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(B_{m}N_{T}(\operatorname{GL}_{n})) & & \displaystyle \nonumber\\ \displaystyle \unicode[STIX]{x1D70B}_{\operatorname{SL}_{n}}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(B_{m}\operatorname{SL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(B_{m}N_{T}(\operatorname{GL}_{n})), & & \displaystyle \nonumber\end{eqnarray}$$

for each $m\geqslant 0$ , and injections

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{\operatorname{GL}_{n}}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(\operatorname{BGL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(BN_{T}(\operatorname{GL}_{n})), & & \displaystyle \nonumber\\ \displaystyle \unicode[STIX]{x1D70B}_{\operatorname{SL}_{n}}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(\operatorname{BSL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(BN_{T}(\operatorname{SL}_{n})). & & \displaystyle \nonumber\end{eqnarray}$$

Proof. We give the proof for $\operatorname{GL}_{n}$ , the proof for $\operatorname{SL}_{n}$ is the same; we write $N_{T}$ for $N_{T}(\operatorname{GL}_{n})$ .

Since $k$ has characteristic zero and $N_{T}\backslash \operatorname{GL}_{n}$ is smooth over $k$ , $N_{T}\backslash \operatorname{GL}_{n}$ is strongly dualizable. By our computations above verifying the weak form of Conjecture 2.1, the Euler characteristic $\unicode[STIX]{x1D712}(N_{T}\backslash \operatorname{GL}_{n}/k)$ is invertible in $\operatorname{GW}(k)$ , thus by Theorem 1.10, the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{m}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(B_{m}\operatorname{GL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(B_{m}N_{T})\end{eqnarray}$$

is split injective, with splitting natural in $m$ .

This gives split injections

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:R^{i}\lim _{m}{\mathcal{E}}^{\ast ,\ast }(B_{m}\operatorname{GL}_{n})\rightarrow R^{i}\lim _{m}{\mathcal{E}}^{\ast ,\ast }(B_{m}N_{T})\end{eqnarray}$$

for $i=0,1$ , and by looking at the Milnor sequences

we see that $\unicode[STIX]{x1D70B}_{\operatorname{GL}_{n}}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(\operatorname{BGL}_{n})\rightarrow {\mathcal{E}}^{\ast ,\ast }(BN_{T_{\operatorname{GL}_{n}}})$ is injective.

We conclude this section with a proof of the strong form of the conjecture for $\operatorname{GL}_{2}$ (or $\operatorname{SL}_{2}$ ). We argue as above: $(\mathbb{G}_{m})^{2}\backslash \operatorname{GL}_{2}=\mathbb{P}^{1}\times \mathbb{P}^{1}-\unicode[STIX]{x1D6E5}$ , where $\unicode[STIX]{x1D6E5}\cong \mathbb{P}^{1}$ is the diagonal, and thus $N_{T}\backslash \operatorname{GL}_{2}=\mathfrak{S}_{2}\backslash (\mathbb{P}^{1}\times \mathbb{P}^{1}-\unicode[STIX]{x1D6E5})$ . We have $\mathfrak{S}_{2}\backslash \mathbb{P}^{1}\times \mathbb{P}^{1}=\operatorname{Sym}^{2}\mathbb{P}^{1}\cong \mathbb{P}^{2}$ . Via this isomorphism the quotient map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\mathbb{P}^{1}\times \mathbb{P}^{1}\rightarrow \mathbb{P}^{2}=\operatorname{Sym}^{2}\mathbb{P}^{1}\end{eqnarray}$$

is given by symmetric functions,

$$\begin{eqnarray}\unicode[STIX]{x1D70B}([x_{0}:x_{1}],[y_{0}:y_{1}])=[x_{0}y_{0}:x_{0}y_{1}+x_{1}y_{0}:x_{1}y_{1}].\end{eqnarray}$$

Thus the restriction of $p$ to $\unicode[STIX]{x1D6E5}$ is the map $\bar{\unicode[STIX]{x1D70B}}([x_{0}:x_{1}])=[x_{0}^{2}:2x_{0}x_{1}:x_{1}^{2}]$ with image the conic $C\subset \mathbb{P}^{2}$ defined by $Q:=T_{1}^{2}-4T_{0}T_{2}$ . Furthermore, this identifies the double cover $\mathbb{P}^{1}\times \mathbb{P}^{1}\rightarrow \mathbb{P}^{2}$ with $\text{Spec }{\mathcal{O}}_{\mathbb{P}^{2}}(\sqrt{Q})$ , that is, the closed subscheme of $O_{\mathbb{P}^{2}}(1)$ defined by pulling back the section of $O_{\mathbb{P}^{2}}(2)$ defined by $Q$ via the squaring map $O_{\mathbb{P}^{2}}(1)\rightarrow O_{\mathbb{P}^{2}}(2)$ .

For $k$ of characteristic $\neq 2$ , the map $\bar{\unicode[STIX]{x1D70B}}:\mathbb{P}^{1}\rightarrow C$ is an isomorphism. If $\text{char}k=2$ , then $C$ is just the line $T_{1}=0$ and $N_{T}\backslash \operatorname{GL}_{2}\cong \mathbb{A}^{2}$ .

We have the $\mathbb{A}^{1}$ - cofiber sequence $\mathbb{P}^{2}-C{\hookrightarrow}\mathbb{P}^{2}\rightarrow \operatorname{Th}(N_{C/\mathbb{P}^{2}})$ . The identity $\unicode[STIX]{x1D712}(Th(N_{C/\mathbb{P}^{2}})/k)=\langle -1\rangle \unicode[STIX]{x1D712}(C/k)$ [Reference Levine25, Proposition 1.10(2)] together with the additivity of $\unicode[STIX]{x1D712}(-/k)$ in distinguished triangles gives

$$\begin{eqnarray}\unicode[STIX]{x1D712}(N_{T}\backslash \operatorname{GL}_{2}/k)=\unicode[STIX]{x1D712}(\mathbb{P}^{2}-C/k)=\unicode[STIX]{x1D712}(\mathbb{P}^{2}/k)-\langle -1\rangle \unicode[STIX]{x1D712}(\mathbb{P}^{1}/k).\end{eqnarray}$$

Applying (2.1)(ii), this gives

$$\begin{eqnarray}\unicode[STIX]{x1D712}((N_{T}\backslash \operatorname{GL}_{2})/k)=2\langle 1\rangle +\langle -1\rangle -\langle -1\rangle \cdot (\langle 1\rangle +\langle -1\rangle )=\langle 1\rangle ,\end{eqnarray}$$

verifying the strong form of the conjecture in this case.

3 Some background on Euler and Pontryagin classes

We now turn to a more detailed study of Euler classes and Pontryagin classes. In this section we recall some necessary background on characteristic classes, which we use throughout the rest of the paper. This material is all taken from [Reference Ananyevskiy3].

We work over a fixed perfect base field $k$ , giving us the motivic stable homotopy category $\operatorname{SH}(k)$ as the setting for our constructions.

We have the algebraic Hopf map $\unicode[STIX]{x1D702}_{2}:\mathbb{A}^{2}-\{0\}\rightarrow \mathbb{P}^{1}$ , giving the stable Hopf map $\unicode[STIX]{x1D702}:\unicode[STIX]{x1D6F4}_{\mathbb{G}_{m}}\mathbb{S}_{k}\rightarrow \mathbb{S}_{k}$ , that is $\unicode[STIX]{x1D702}\in \unicode[STIX]{x1D70B}_{1,1}(\mathbb{S}_{k})(k)$ . For ${\mathcal{E}}$ a motivic ring spectrum, the unit map gives us the element $\unicode[STIX]{x1D702}_{{\mathcal{E}}}\in {\mathcal{E}}^{-1,-1}(k)$ . ${\mathcal{E}}$ is said to be an $\unicode[STIX]{x1D702}$ -invertible theory if multiplication by $\unicode[STIX]{x1D702}_{{\mathcal{E}}}$ is invertible on ${\mathcal{E}}^{\ast \ast }(k)$ , that is, there is an element $\unicode[STIX]{x1D702}_{{\mathcal{E}}}^{-1}\in {\mathcal{E}}^{1,1}(k)$ with $\unicode[STIX]{x1D702}_{{\mathcal{E}}}\cdot \unicode[STIX]{x1D702}_{{\mathcal{E}}}^{-1}=1_{{\mathcal{E}}}$ . This implies that multiplication by $\unicode[STIX]{x1D702}_{{\mathcal{E}}}$ is an isomorphism on ${\mathcal{E}}^{\ast \ast }(F)$ for all $F\in \operatorname{SH}(k)$ .

We take the definition of an $\operatorname{SL}$ -orientation on ${\mathcal{E}}$ from [Reference Ananyevskiy3, Definition 4], for us an $\operatorname{SL}$ -orientation is always normalized in the sense of loc. cit.

Following [Reference Ananyevskiy3, Definition 1], an $\operatorname{SL}_{n}$ -bundle on some $X\in \mathbf{Sm}/k$ is a vector bundle $E\rightarrow X$ together with an isomorphism $\unicode[STIX]{x1D703}:\operatorname{det}E\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{X}$ . We use as well the definition of the Euler class $e(E,\unicode[STIX]{x1D703})\in {\mathcal{E}}^{2n,n}(X)$ of an $\operatorname{SL}_{n}$ -bundle in an $\operatorname{SL}$ -oriented theory ${\mathcal{E}}$ [Reference Ananyevskiy3, Definition 4], and the Pontryagin classes $p_{m}(E)\in {\mathcal{E}}^{4m,2m}(X)$ of an arbitrary vector bundle $E\rightarrow X$ , for an $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible theory ${\mathcal{E}}$ [Reference Ananyevskiy3, Definition 19].

For an $\operatorname{SL}$ -oriented theory ${\mathcal{E}}$ and a line bundle $L\rightarrow X$ , $X\in \mathbf{Sm}/k$ , one has the twisted cohomology

$$\begin{eqnarray}{\mathcal{E}}^{a,b}(X;L):={\mathcal{E}}_{0_{L}}^{a+2,b+1}(L).\end{eqnarray}$$

The Euler class for $\operatorname{SL}_{n}$ bundles extends to a theory of Euler classes

$$\begin{eqnarray}e(E)\in {\mathcal{E}}^{2n,n}(X;\operatorname{det}^{-1}E)\end{eqnarray}$$

for arbitrary rank $n$ vector bundles $E\rightarrow X$ , $X\in \mathbf{Sm}/k$ as follows: let $L=\operatorname{det}E\rightarrow X$ . One has a canonical trivialization of $\operatorname{det}(E\oplus L^{-1})$ , giving the Thom class $\operatorname{th}(E\oplus L^{-1},can)\in {\mathcal{E}}_{0_{E\oplus L^{-1}}}^{2n+2,n+1}(E\oplus L^{-1})$ . Pulling back by the inclusion $i:L^{-1}=0_{X}\oplus L^{-1}\rightarrow E\oplus L^{-1}$ gives the class

$$\begin{eqnarray}e(E):=i^{\ast }\operatorname{th}(E\oplus L^{-1},can)\in {\mathcal{E}}_{0_{L^{-1}}}^{2n+2,n+1}(L^{-1})={\mathcal{E}}^{2n,n}(X;\operatorname{det}^{-1}E).\end{eqnarray}$$

For details and properties of the twisted cohomology for $\operatorname{SL}$ -oriented theories, we refer the reader to [Reference Levine and Raksit26, § 3] and [Reference Ananyevskiy1, § 3].

For an $\text{SL}_{n}$ -bundle $E\rightarrow X$ , we have the total Pontryagin class $p(E)=1+\sum _{i\geqslant 1}p_{i}(E)\in {\mathcal{E}}^{4\ast ,2\ast }(X)$ , which satisfies the Whitney formula for $\operatorname{SL}$ -bundles: if

$$\begin{eqnarray}0\rightarrow E^{\prime }\rightarrow E\rightarrow E^{\prime \prime }\rightarrow 0\end{eqnarray}$$

is an exact sequence of $\operatorname{SL}$ -vector bundles on $X\in \mathbf{Sm}/k$ , then

$$\begin{eqnarray}p(E)=p(E^{\prime })\cdot p(E^{\prime \prime }).\end{eqnarray}$$

In addition, for an $\operatorname{SL}$ bundle $E$ of rank $n$ , $p_{i}(E)=0$ for $2i>n$ . These facts follow from [Reference Ananyevskiy3, Corollary 3, Lemma 15]. For theories ${\mathcal{E}}$ which are $\operatorname{SL}$ -oriented and $\unicode[STIX]{x1D702}$ -invertible, these facts hold for arbitrary vector bundles and arbitrary short exact sequences of vector bundles [Reference Ananyevskiy1, Corollary 7.9].

We recall the following results of Ananyevskiy.

Theorem 3.3. [Reference Ananyevskiy3, Theorem 10]

For ${\mathcal{E}}$ an $\unicode[STIX]{x1D702}$ -invertible, $\operatorname{SL}$ -oriented theory, one has

$$\begin{eqnarray}{\mathcal{E}}^{\ast ,\ast }(\operatorname{BSL}_{n})=\left\{\begin{array}{@{}ll@{}}{\mathcal{E}}^{\ast \ast }(k)[p_{1},\ldots ,p_{m-1},e]\quad & \text{ for }n=2m,\\ {\mathcal{E}}^{\ast \ast }(k)[p_{1},\ldots ,p_{m}]\quad & \text{ for }n=2m+1.\end{array}\right.\end{eqnarray}$$

Here $p_{i}=p_{i}(E_{n})$ , $e=e(E_{n})$ , where $E_{n}\rightarrow \operatorname{BSL}_{n}$ is the universal rank $n$ bundle. Moreover, $p_{m}(E_{2m})=e(E_{2m})^{2}$ .

and

Theorem 3.4. ( $\operatorname{SL}_{2}$ -splitting principle)

Let $i_{m}:(\operatorname{SL}_{2})^{m}\rightarrow \operatorname{SL}_{2m}$ be the block-diagonal inclusion and let ${\mathcal{E}}$ be an $\unicode[STIX]{x1D702}$ -invertible, $\operatorname{SL}$ -oriented theory. Then

$$\begin{eqnarray}i_{m}^{\ast }:{\mathcal{E}}^{\ast ,\ast }(\operatorname{BSL}_{2m})\rightarrow {\mathcal{E}}^{\ast ,\ast }((\operatorname{BSL}_{2})^{m})\end{eqnarray}$$

is injective. Moreover,

$$\begin{eqnarray}{\mathcal{E}}^{\ast ,\ast }((\operatorname{BSL}_{2})^{m})={\mathcal{E}}^{\ast ,\ast }(\operatorname{BSL}_{2})^{\otimes _{{\mathcal{E}}^{\ast \ast }(k)}m}={\mathcal{E}}^{\ast \ast }(k)[e_{1},\ldots ,e_{m}]\end{eqnarray}$$

where $e_{i}$ is the pullback of $e\in {\mathcal{E}}^{2,1}(\operatorname{BSL}_{2})$ via the $i$ th projection, and

$$\begin{eqnarray}i_{m}^{\ast }e=e_{1}\cdot \ldots \cdot e_{m};\quad i_{m}^{\ast }\biggl(1+\mathop{\sum }_{i}p_{i}\biggr)=\mathop{\prod }_{i=1}^{r}(1+e_{i}^{2}).\end{eqnarray}$$

This latter result is a consequence of [Reference Ananyevskiy3, Theorems 6, 10].

4 $\operatorname{BGL}_{n}$ and $\operatorname{BSL}_{n}$

Let $\unicode[STIX]{x1D70B}_{n}:\operatorname{BSL}_{n}\rightarrow \operatorname{BGL}_{n}$ be the canonical map, $E_{n}\rightarrow \operatorname{BGL}_{n}$ the universal rank $n$ vector bundle. For ${\mathcal{E}}$ an $\operatorname{SL}$ -oriented motivic ring spectrum, we have the pullback map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})\rightarrow {\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n}).\end{eqnarray}$$

In addition, the bundle $\unicode[STIX]{x1D70B}^{\ast }E_{n}$ is canonically isomorphic to the universal bundle $\tilde{E}_{n}\rightarrow \operatorname{BSL}_{n}$ , hence there is a canonical trivialization $\unicode[STIX]{x1D703}:\unicode[STIX]{x1D70B}^{\ast }\operatorname{det}^{-1}E_{n}\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{\operatorname{BSL}_{n}}$ . We may therefore define

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};\operatorname{det}^{-1}E_{n})\rightarrow {\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})\end{eqnarray}$$

as the composition

$$\begin{eqnarray}{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};\operatorname{det}^{-1}E_{n})\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\ast }}{\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n}(\unicode[STIX]{x1D70B}_{n}^{\ast }\operatorname{det}^{-1}E_{n}))\xrightarrow[{}]{\unicode[STIX]{x1D703}_{\ast }}{\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n}).\end{eqnarray}$$

Our main theorem of this section is

Theorem 4.1. Let ${\mathcal{E}}\in \operatorname{SH}(k)$ be an $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible motivic ring spectrum. Suppose that either ${\mathcal{E}}^{0,0}$ is a Zariski sheaf on $\mathbf{Sm}/k$ or, that the unit map $\mathbb{S}_{k}\rightarrow {\mathcal{E}}$ makes ${\mathcal{E}}^{0,0}$ a ${\mathcal{W}}$ -module. Then for $n=2m$ , $\unicode[STIX]{x1D70B}^{\ast }$ induces an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})\oplus {\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};\operatorname{det}E_{n})\rightarrow {\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n}).\end{eqnarray}$$

More precisely,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}^{\ast }({\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})) & = & \displaystyle {\mathcal{E}}^{\ast \ast }(k)[p_{1},\ldots ,p_{m-1},e^{2}]\nonumber\\ \displaystyle \unicode[STIX]{x1D70B}^{\ast }({\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n},\operatorname{det}E_{n})) & = & \displaystyle e\cdot {\mathcal{E}}^{\ast \ast }(k)[p_{1},\ldots ,p_{m-1},e^{2}].\nonumber\end{eqnarray}$$

For $n=2m+1$ , ${\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};\operatorname{det}E_{n})\!)=0$ and

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})\rightarrow {\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})\end{eqnarray}$$

is an isomorphism.

Proof. Write $\bar{\unicode[STIX]{x1D70B}}:L\rightarrow \operatorname{BGL}_{n}$ for the line bundle $\operatorname{det}E_{n}$ . We may identify the morphism $\unicode[STIX]{x1D70B}:\operatorname{BSL}_{n}\rightarrow \operatorname{BGL}_{n}$ with the principal $\mathbb{G}_{m}$ -bundle $L-0_{L}$ associated to $L\rightarrow \operatorname{BGL}_{n}$ . Identifying ${\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})$ with ${\mathcal{E}}^{\ast \ast }(L)$ via $\bar{\unicode[STIX]{x1D70B}}^{\ast }$ and using homotopy invariance gives us the localization sequences

$$\begin{eqnarray}\displaystyle \cdots \rightarrow {\mathcal{E}}^{a,b}(\operatorname{BGL}_{n}) & \xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\ast }} & \displaystyle {\mathcal{E}}^{a,b}(\operatorname{BSL}_{n})\nonumber\\ \displaystyle & \rightarrow & \displaystyle {\mathcal{E}}^{a-1,b-1}(\operatorname{BGL}_{n};L)\xrightarrow[{}]{\unicode[STIX]{x1D719}_{a-1}}{\mathcal{E}}^{a+1,b}(\operatorname{BGL}_{n})\rightarrow \cdots \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{a-1}$ is cup product with the Euler class $e(L)$ .

By Lemma 4.3 or [Reference Ananyevskiy2, Theorem 2(Ia)], $e(L)=0$ and thus the localization sequence splits into short exact sequences

$$\begin{eqnarray}0\rightarrow {\mathcal{E}}^{a,b}(\operatorname{BGL}_{n})\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{n}^{\ast }}{\mathcal{E}}^{a,b}(\operatorname{BSL}_{n})\xrightarrow[{}]{\unicode[STIX]{x2202}_{a,b}}{\mathcal{E}}^{a-1,b-1}(\operatorname{BGL}_{n};L)\rightarrow 0.\end{eqnarray}$$

Next, we define a splitting to $\unicode[STIX]{x2202}_{a,b}$ . Let $t\in H^{0}(L,\bar{\unicode[STIX]{x1D70B}}^{\ast }L)$ be the tautological section, giving a generator of $\bar{\unicode[STIX]{x1D70B}}^{\ast }L$ over $L-0_{L}$ . Let $t^{\vee }\in H^{0}(L-0_{L},\bar{\unicode[STIX]{x1D70B}}^{\ast }L^{-1})$ be the dual of $t$ . This defines the $\bar{\unicode[STIX]{x1D70B}}^{\ast }L^{-1}$ -valued quadratic form $q(s)=s^{2}\cdot t^{\vee }$ , giving the section $\langle t^{\vee }\rangle$ of the sheaf of twisted Witt groups $W(L^{-1})$ over $L-\{0_{L}\}$ .

Suppose the unit map $\mathbb{S}_{k}\rightarrow {\mathcal{E}}$ makes ${\mathcal{E}}^{\ast \ast }(-)$ into a ${\mathcal{W}}(-)$ module. This gives us the section $\langle t^{\vee }\rangle _{{\mathcal{E}}}:=\langle t^{\vee }\rangle \cdot 1_{{\mathcal{E}}}\in {\mathcal{E}}^{0,0}(L-\{0_{\operatorname{BGL}_{n}}\};L^{-1})$ , that is $\langle t^{\vee }\rangle _{{\mathcal{E}}}\in {\mathcal{E}}^{0,0}(\operatorname{BSL}_{n},L^{-1})$ . One finds

$$\begin{eqnarray}\unicode[STIX]{x2202}\langle t^{\vee }\rangle _{{\mathcal{E}}}=\unicode[STIX]{x1D702}\cdot 1_{{\mathcal{E}}}\in {\mathcal{E}}^{-1,-1}(\operatorname{BGL}_{n})\end{eqnarray}$$

by making a computation in local coordinates. Indeed, we may first make the computation for ${\mathcal{E}}=\operatorname{EM}({\mathcal{W}})$ , $\operatorname{EM}({\mathcal{W}})^{a,b}(-)=H^{a-b}(-,{\mathcal{W}})$ , in which case ${\mathcal{E}}^{-1,-1}(-)=H^{0}(-,{\mathcal{W}})$ , justifying the local computation. In general, we have $\langle t^{\vee }\rangle _{{\mathcal{E}}}=\langle t^{\vee }\rangle \cdot 1_{{\mathcal{E}}}$ and so $\unicode[STIX]{x2202}\langle t^{\vee }\rangle _{{\mathcal{E}}}=\unicode[STIX]{x2202}\langle t^{\vee }\rangle \cdot 1_{{\mathcal{E}}}=\unicode[STIX]{x1D702}\cdot 1_{{\mathcal{E}}}$ . In addition, the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{a,b}(\operatorname{BGL}_{n};L)\rightarrow {\mathcal{E}}^{a,b}(\operatorname{BSL}_{n})\end{eqnarray}$$

is equal to the composition

$$\begin{eqnarray}{\mathcal{E}}^{a,b}(\operatorname{BGL}_{n};L)\xrightarrow[{}]{\unicode[STIX]{x1D70B}^{\ast }}{\mathcal{E}}^{a,b}(\operatorname{BSL}_{n};L)\xrightarrow[{}]{\langle t^{\vee }\rangle _{{\mathcal{E}}}\cdot (-)}{\mathcal{E}}^{a,b}(\operatorname{BSL}_{n}),\end{eqnarray}$$

which one sees by noting that multiplication by $t^{\vee }$ is the same as the canonical trivialization $\unicode[STIX]{x1D70B}^{\ast }L\rightarrow {\mathcal{O}}_{\operatorname{BSL}_{n}}$ . Thus

$$\begin{eqnarray}\unicode[STIX]{x2202}\circ \unicode[STIX]{x1D70B}^{\ast }=\unicode[STIX]{x1D702}\cdot (-):{\mathcal{E}}^{a,b}(\operatorname{BGL}_{n};L)\rightarrow {\mathcal{E}}^{a-1,b-1}(\operatorname{BGL}_{n};L)\end{eqnarray}$$

which is an isomorphism, since ${\mathcal{E}}$ is an $\unicode[STIX]{x1D702}$ -invertible theory.

In case ${\mathcal{E}}^{0,0}$ is a Zariski sheaf, as ${\mathcal{E}}$ is $\unicode[STIX]{x1D702}$ -invertible, ${\mathcal{E}}^{-1,-1}$ is also a Zariski sheaf, so the computation of $\unicode[STIX]{x2202}\langle t^{\vee }\rangle _{{\mathcal{E}}}$ may be made locally as above, with the same result.

It remains to compute the image of the two summands. For $n=2m+1$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\ast }:{\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})\rightarrow {\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n}).\end{eqnarray}$$

is surjective: since the Pontryagin classes are defined for arbitrary vector bundles, and are functorial, we have

$$\begin{eqnarray}p_{i}=\unicode[STIX]{x1D70B}^{\ast }(p_{i}(E_{n}));\quad i=1,\ldots ,m.\end{eqnarray}$$

The surjectivity then follows from Ananyevskiy’s description of ${\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})$ , and this implies that the remaining summand ${\mathcal{E}}^{\ast ,\ast }(\operatorname{BGL}_{n};L)$ must be zero.

Now take $n=2m$ . We write $R_{m-1}$ for the subring ${\mathcal{E}}^{\ast \ast }(k)[p_{1},\ldots ,p_{m-1}]$ of ${\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})$ . The same argument as above together with the identity

$$\begin{eqnarray}p_{m}(\tilde{E}_{2m})=e(\tilde{E}_{2m})^{2}\end{eqnarray}$$

implies that $\unicode[STIX]{x1D70B}_{n}^{\ast }({\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n}))$ contains $R_{m-1}[e^{2}]$ , and $e$ is in $\unicode[STIX]{x1D70B}_{n}^{\ast }({\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};L))$ . Since ${\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};L)$ is a module for ${\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n})$ , this shows that the $R_{m-1}[e^{2}]$ -submodule $e\cdot R_{m-1}[e^{2}]$ of ${\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})$ is contained in $\unicode[STIX]{x1D70B}_{n}^{\ast }({\mathcal{E}}^{\ast \ast }(\operatorname{BGL}_{n};L))$ . Since

$$\begin{eqnarray}{\mathcal{E}}^{\ast \ast }(\operatorname{BSL}_{n})=R_{m-1}[e^{2}]\oplus e\cdot R_{m-1}[e^{2}],\end{eqnarray}$$

the result for $n=2m$ follows.

Remark 4.2. Since the Eilenberg–MacLane spectrum $\operatorname{EM}({\mathcal{W}})$ is $\operatorname{SL}$ -oriented and $\unicode[STIX]{x1D702}$ invertible we may apply Theorem 4.1 to give a computation of $H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}})$ and $H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}}(\operatorname{det}E_{n}))$ : for $n=2m$ ,

$$\begin{eqnarray}\displaystyle & H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}})=W(k)[p_{1},\ldots ,p_{m-1},p_{m}], & \displaystyle \nonumber\\ \displaystyle & H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}}(\operatorname{det}E_{n}))=e\cdot W(k)[p_{1},\ldots ,p_{m-1},p_{m}] & \displaystyle \nonumber\end{eqnarray}$$

and for $n=2m+1$ ,

$$\begin{eqnarray}\displaystyle & & \displaystyle H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}})=W(k)[p_{1},\ldots ,p_{m-1},p_{m}],\nonumber\\ \displaystyle & & \displaystyle H^{\ast }(\operatorname{BGL}_{n},{\mathcal{W}}(\operatorname{det}E_{n}))=0.\nonumber\end{eqnarray}$$

We used the following result, a slight extension of a result of Ananyevskiy’s [Reference Ananyevskiy1, Theorem 7.4], in the proof of Theorem 4.1.

Proof. Let $L=\operatorname{det}^{-1}V$ . By definition, ${\mathcal{E}}^{2r,r}(X;L)={\mathcal{E}}_{0_{L}}^{2r+2,r+1}(L)$ . We have the canonical trivialization $\unicode[STIX]{x1D706}:\operatorname{det}(V\oplus L)\rightarrow {\mathcal{O}}_{X}$ , giving the Thom class $\operatorname{th}(V\oplus L,\unicode[STIX]{x1D706})\in {\mathcal{E}}_{0_{V}\times 0_{L}}^{2r+2,r+1}(V\oplus L)$ , and the Euler class $e(V)\in {\mathcal{E}}^{2r,r}(X;L)$ is by definition the pullback $s_{0_{V}}^{\ast }(\operatorname{th}(V\oplus L,\unicode[STIX]{x1D706}))\in {\mathcal{E}}_{0_{L}}^{2r+2,r+1}(L)$ , where $s_{0_{V}}:L\rightarrow V\oplus L$ is the closed immersion $s_{0_{V}}(x)=(0_{V},x)$ .

Let $\unicode[STIX]{x1D70B}:X\times \mathbb{A}^{1}-\{0\}\rightarrow X$ be the projection, let $t$ be the standard unit on $\mathbb{A}^{1}-\{0\}$ and let $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}^{\ast }(V\oplus L)\rightarrow \unicode[STIX]{x1D70B}^{\ast }(V\oplus L)$ be the automorphism $t\cdot \text{Id}_{\unicode[STIX]{x1D70B}^{\ast }V}\oplus \text{Id}_{\unicode[STIX]{x1D70B}^{\ast }L}$ . We also use $\unicode[STIX]{x1D70B}$ to denote the induced projections $\unicode[STIX]{x1D70B}^{\ast }V\rightarrow V$ , $\unicode[STIX]{x1D70B}^{\ast }L\rightarrow L$ , and so on, and let $\unicode[STIX]{x1D70B}^{\ast }s_{0_{V}}:\unicode[STIX]{x1D70B}^{\ast }L\rightarrow \unicode[STIX]{x1D70B}^{\ast }(V\oplus L)$ denote the map induced by $s_{0_{V}}$ . The proof of [Reference Ananyevskiy1, Theorem 7.4] gives the identity

$$\begin{eqnarray}\langle t^{2r+1}\rangle \cdot \operatorname{th}(\unicode[STIX]{x1D70B}^{\ast }(V\oplus L),\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D706})=\unicode[STIX]{x1D70C}^{\ast }\operatorname{th}(\unicode[STIX]{x1D70B}^{\ast }(V\oplus L),\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D706})\end{eqnarray}$$

and since $\unicode[STIX]{x1D70C}\circ \unicode[STIX]{x1D70B}^{\ast }s_{0_{V}}=\unicode[STIX]{x1D70B}^{\ast }s_{0_{V}}$ , this implies

$$\begin{eqnarray}\langle t\rangle \cdot \unicode[STIX]{x1D70B}^{\ast }e(V)=\langle t^{2r+1}\rangle \cdot \unicode[STIX]{x1D70B}^{\ast }e(V)=\unicode[STIX]{x1D70B}^{\ast }e(V)\in {\mathcal{E}}_{0_{L}\times \mathbb{A}^{1}-\{0\}}^{2r+2,r+1}(L\times \mathbb{A}^{1}-\{0\}).\end{eqnarray}$$

Following Ananyevskiy’s argument (where the minus sign disappears since we are using the “alternate” normalization mentioned in [Reference Ananyevskiy1, Remark 6.5]), applying the boundary in the localization sequence

$$\begin{eqnarray}\displaystyle \cdots \rightarrow {\mathcal{E}}^{2r,r}(X\times \mathbb{A}^{1}-\{0\};\operatorname{det}^{-1}V) & \xrightarrow[{}]{\unicode[STIX]{x1D6FF}} & \displaystyle {\mathcal{E}}^{2r-1,r-1}(X;\operatorname{det}^{-1}V)\nonumber\\ \displaystyle & \rightarrow & \displaystyle {\mathcal{E}}^{2r+1,r+1}(X\times \mathbb{A}^{1};\operatorname{det}^{-1}V)\rightarrow \cdots \nonumber\end{eqnarray}$$

gives the identity

$$\begin{eqnarray}0=\unicode[STIX]{x1D6FF}((\langle t\rangle -1)\cdot \unicode[STIX]{x1D70B}^{\ast }e(V))=\unicode[STIX]{x1D702}\cdot e(V)\end{eqnarray}$$

in ${\mathcal{E}}^{2r-1,r-1}(X;\operatorname{det}^{-1}V)$ . Since ${\mathcal{E}}$ is by assumption an $\unicode[STIX]{x1D702}$ -invertible theory, we have $e(V)=0$ .

5 Witt cohomology of $BN_{T}(\operatorname{SL}_{2})$

In this section, for $V\rightarrow X$ a vector bundle, we let $f:\mathbb{P}(V)\rightarrow X$ denote the associated projective space bundle $\operatorname{Proj}_{{\mathcal{O}}_{X}}(\operatorname{Sym}^{\ast }V^{\vee })$ . We have the tautological subbundle $O(-1)\rightarrow \mathbb{P}(V)$ of $f^{\ast }V$ and its dual $O(1)$ .

We have the $\operatorname{SL}$ -oriented theory $\operatorname{EM}({\mathcal{W}})^{a,b}(X):=H^{a-b}(X,{\mathcal{W}})$ on $\mathbf{Sm}/k$ , where ${\mathcal{W}}$ is the sheaf of Witt rings. Our main goal in this section is to compute $H^{\ast }(BN_{T}(\operatorname{SL}_{2}),{\mathcal{W}})$ , where $N_{T}(\operatorname{SL}_{2})\subset \operatorname{SL}_{2}$ is the normalizer of the standard torus

$$\begin{eqnarray}T:=\left\{\left(\begin{array}{@{}cc@{}}t & 0\\ 0 & t^{-1}\end{array}\right)\right\}\subset \operatorname{SL}_{2}.\end{eqnarray}$$

We write $N:=N_{T}(\operatorname{SL}_{2})$ . $N$ is generated by $T$ and an additional element $\unicode[STIX]{x1D70E}=(\!\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix}\!)$ , and sending $\unicode[STIX]{x1D70E}$ to $-1$ gives us the exact sequence

$$\begin{eqnarray}1\rightarrow T\rightarrow N\rightarrow \{\pm 1\}\rightarrow 0.\end{eqnarray}$$

This gives us the representation $\unicode[STIX]{x1D70C}^{-}:N\rightarrow \mathbb{G}_{m}$ sending $T$ to $1$ and $\unicode[STIX]{x1D70E}$ to $-1$ (see §6 below).

We work over a field $k$ of characteristic $\neq 2$ . In this case, the computation of Section 2 gives the isomorphism

$$\begin{eqnarray}N\backslash \operatorname{SL}_{2}\cong N_{T}(\operatorname{GL}_{2})\backslash \operatorname{GL}_{2}\cong \mathbb{P}^{2}-C\end{eqnarray}$$

with $C$ the conic defined by $Q:=T_{1}^{2}-4T_{0}T_{2}$ .

The natural right $\operatorname{SL}_{2}$ -action on $N\backslash \operatorname{SL}_{2}$ can be described in the model $\mathbb{P}^{2}-C$ as follows. Let $F=\mathbb{A}^{2}$ with the standard (right) $\operatorname{SL}_{2}$ -action. We have the canonical map of schemes

$$\begin{eqnarray}sq:\mathbb{P}(F)\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)\end{eqnarray}$$

induced by the squaring map $sq:F\rightarrow \operatorname{Sym}^{2}F$ , where $sq(v)$ is the image of $v\otimes v$ under the canonical surjection $F\otimes F\rightarrow \operatorname{Sym}^{2}F$ . As the right $\operatorname{SL}_{2}$ -action on $\operatorname{Sym}^{2}F$ is induced by the diagonal action on $F\otimes F$ , $(v\otimes w)\cdot g:=v\cdot g\otimes w\cdot g$ , the map $sq$ is $\operatorname{SL}_{2}$ -equivariant.

This identifies $C\subset \mathbb{P}^{2}$ with $sq((\mathbb{P}(F))\subset \mathbb{P}(\operatorname{Sym}^{2}F)$ . Furthermore, the quadratic polynomial $Q=T_{1}^{2}-4T_{0}T_{2}$ gives us a global section $\langle Q\rangle$ of ${\mathcal{O}}_{\mathbb{P}^{2}-C}^{\times }/{\mathcal{O}}_{\mathbb{P}^{2}-C}^{\times ~2}$ , with value $Q/L^{2}$ on $\mathbb{P}^{2}-(C\cup \{L=0\})$ for each linear form $L$ . As the map sending a unit $u$ to the class of the rank one quadratric form $\langle u\rangle$ descends to a map of sheaves ${\mathcal{O}}^{\times }/{\mathcal{O}}^{\times 2}\rightarrow {\mathcal{G}}{\mathcal{W}}^{\times }$ , we have the section $\langle q\rangle$ of ${\mathcal{G}}{\mathcal{W}}$ on $\mathbb{P}^{2}-C$ corresponding to $\langle Q\rangle$ .

Since $C$ is stable under the $\operatorname{SL}_{2}$ action, the form $Q$ is $\operatorname{SL}_{2}$ invariant up to a scalar, but as

$$\begin{eqnarray}g\mapsto \frac{g^{\ast }Q}{Q}\end{eqnarray}$$

defines a character of $\operatorname{SL}_{2}$ , $Q$ is $\operatorname{SL}_{2}$ invariant and thus $\langle Q\rangle \in H^{0}(\mathbb{P}^{2}-C,{\mathcal{O}}^{\times }/{\mathcal{O}}^{\times 2})$ is also $\operatorname{SL}_{2}$ invariant.

We may use the model

$$\begin{eqnarray}(N\backslash \operatorname{SL}_{2})\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\cong N\backslash E\operatorname{SL}_{2}\end{eqnarray}$$

for $BN$ . The $\operatorname{SL}_{2}$ -invariant section $\langle Q\rangle$ of ${\mathcal{O}}_{N\backslash \operatorname{SL}_{2}}^{\times }/{\mathcal{O}}_{N\backslash \operatorname{SL}_{2}}^{\times ~2}$ on $N\backslash \operatorname{SL}_{2}$ thus descends to a global section $\langle \bar{Q}\rangle$ of ${\mathcal{O}}^{\times }/{\mathcal{O}}^{\times ~2}$ on $BN$ , which gives us the global section $\langle \bar{q}\rangle$ of the sheaf of Grothendieck–Witt rings ${\mathcal{G}}{\mathcal{W}}$ on $BN$ . We also denote the image of $\langle \bar{q}\rangle$ in $H^{0}(BN,{\mathcal{W}})$ by $\langle \bar{q}\rangle$ , using the context to make the meaning clear.

Since $\operatorname{SL}_{2}$ is special, the bundle $E\operatorname{SL}_{2}\rightarrow \operatorname{BSL}_{2}$ is Zariski locally trivial, and thus the same holds for $BN\rightarrow \operatorname{BSL}_{2}$ , as explained in Section 2.

Our description of $N\backslash \operatorname{SL}_{2}$ as $\mathbb{P}(\operatorname{Sym}^{2}F)-sq(\mathbb{P}(F))$ realizes $BN$ as an open subscheme of the $\mathbb{P}^{2}$ -bundle $\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \operatorname{BSL}_{2}$ with complement the $\mathbb{P}^{1}$ -bundle $\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \operatorname{BSL}_{2}$ :

The representation $\unicode[STIX]{x1D70C}^{-}:N\rightarrow \mathbb{G}_{m}$ gives us the line bundle $\unicode[STIX]{x1D6FE}$ on $BN$ , generating $\text{Pic}(BN)$ . We have $\text{Pic}(N\backslash \operatorname{SL}_{2})=\mathbb{Z}/2$ with generator the restriction of $O_{\mathbb{P}^{2}}(1)$ , and $\unicode[STIX]{x1D6FE}$ is the same as the bundle induced by the restriction of $p_{1}^{\ast }O_{\mathbb{P}^{2}}(1)$ via the canonical $\operatorname{SL}_{2}$ -linearization of $O_{\mathbb{P}^{2}}(1)$ . Similarly, we have the bundle $p_{1}^{\ast }O(m)$ on $\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ , induced by the canonical $\operatorname{SL}_{2}$ -linearization of $O_{\mathbb{P}^{1}}(1)$ .

Lemma 5.1.

$$\begin{eqnarray}H^{n}(\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})=\left\{\begin{array}{@{}ll@{}}W(k)\quad & \text{ for }n=0,\\ 0\quad & \text{else.}\end{array}\right.\end{eqnarray}$$

Proof. The element $\unicode[STIX]{x1D702}\in K_{-1}^{MW}(k)$ is realized in $\operatorname{SH}(k)$ as the stable Hopf map, the unstable version of which is the map $\unicode[STIX]{x1D702}_{2}:\mathbb{A}^{2}-\{0\}\rightarrow \mathbb{P}^{1}$ , $\unicode[STIX]{x1D702}_{2}(x,y)=[x:y]$ . As ${\mathcal{W}}={\mathcal{K}}_{\ast }^{MW}[\unicode[STIX]{x1D702}^{-1}]$ , the projection

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{2}\times \text{Id}:(\mathbb{A}^{2}-\{0\})\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\end{eqnarray}$$

induces an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{2}^{\ast }:H^{\ast }(\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})\rightarrow H^{\ast }((\mathbb{A}^{2}-\{0\})\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}).\end{eqnarray}$$

Here we give $\mathbb{A}^{2}-\{0\}$ the right $\operatorname{SL}_{2}$ -action induced from the standard right action on $\mathbb{A}^{2}=F$ .

Giving $\operatorname{SL}_{2}$ the right $\operatorname{SL}_{2}$ -action via right multiplication, we have the $\operatorname{SL}_{2}$ -equivariant map

$$\begin{eqnarray}r:\operatorname{SL}_{2}\rightarrow \mathbb{A}^{2}-\{0\};\qquad r\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)=(a,b)\end{eqnarray}$$

making $\operatorname{SL}_{2}$ an $\mathbb{A}^{1}$ -bundle over $\mathbb{A}^{2}-\{0\}$ . Thus, by homotopy invariance,

$$\begin{eqnarray}\displaystyle & & \displaystyle r^{\ast }:H^{\ast }((\mathbb{A}^{2}-\{0\})\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})\rightarrow H^{\ast }(\operatorname{SL}_{2}\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})\nonumber\\ \displaystyle & & \displaystyle \quad =H^{\ast }(E\operatorname{SL}_{2},{\mathcal{W}})\nonumber\end{eqnarray}$$

is also an isomorphism. Since $E\operatorname{SL}_{2}$ is $\mathbb{A}^{1}$ -contractible, we have

$$\begin{eqnarray}H^{n}(E\operatorname{SL}_{2},{\mathcal{W}})=\left\{\begin{array}{@{}ll@{}}W(k)\quad & \text{ for }n=0,\\ 0\quad & \text{ else,}\end{array}\right.\end{eqnarray}$$

and the lemma is proven.

Let $T\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)$ be the tangent bundle. The $\operatorname{SL}_{2}$ -action on $\operatorname{Sym}^{2}F$ extends canonically to an action on $T$ over its action on $\mathbb{P}(\operatorname{Sym}^{2}F)$ , and $T\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ is the relative tangent bundle of $\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ over $BN$ , which we denote by ${\mathcal{T}}$ .

Proof. (1) We use the Leray spectral sequence

$$\begin{eqnarray}E_{1}^{p,q}=\bigoplus _{x\in \operatorname{BSL}_{2}^{(p)}}H^{q}(\unicode[STIX]{x1D70B}^{-1}(x),{\mathcal{W}})\Rightarrow H^{p+q}(\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}).\end{eqnarray}$$

As $\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \operatorname{BSL}_{2}$ is Zariski locally trivial, it follows that $\unicode[STIX]{x1D70B}^{-1}(x)\cong \mathbb{P}_{k(x)}^{2}$ for all $x\in \operatorname{BSL}_{2}$ . Fasel [Reference Fasel16, Theorem 11.7] computes $H^{\ast }(\mathbb{P}_{k(x)}^{2},{\mathcal{W}})$ as

$$\begin{eqnarray}H^{n}(\mathbb{P}_{k(x)}^{2},{\mathcal{W}})=\left\{\begin{array}{@{}ll@{}}W(k(x))\quad & \text{ for }n=0,\\ 0\quad & \text{ else.}\end{array}\right.\end{eqnarray}$$

Feeding this into the Leray spectral sequence gives the result.

For (2), we do the same, using the Leray spectral sequence converging to $H^{p+q} (\!\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}(p_{1}^{\ast }O(1))$ and the isomorphism

$$\begin{eqnarray}H^{n}(\mathbb{P}_{k(x)}^{2},{\mathcal{W}}(O(1)))=\left\{\begin{array}{@{}ll@{}}W(k(x))\quad & \text{ for }n=2,\\ 0\quad & \text{ else,}\end{array}\right.\end{eqnarray}$$

again by [Reference Fasel16, Theorem 11.7]. The Gysin map $i_{p\ast }:H^{n-2}(\text{Spec }k(x),{\mathcal{W}})\rightarrow (\mathbb{P}_{k(x)}^{2},{\mathcal{W}}(\unicode[STIX]{x1D714}_{\mathbb{P}^{2}}))$ is left inverse to the pushforward $\unicode[STIX]{x1D70B}_{\ast }:H^{n}(\mathbb{P}_{k(x)}^{2},{\mathcal{W}}(\unicode[STIX]{x1D714}_{\mathbb{P}^{2}}))\rightarrow H^{n-2}(\text{Spec }k(x),{\mathcal{W}})$ ; it is not hard to see that for $n=2$ , these are the isomorphism and its inverse, as given by Fasel’s computation. As $\unicode[STIX]{x1D714}_{\mathbb{P}_{k(x)}^{2}/k(x)}=O_{\mathbb{P}^{2}}(-3)$ , we may replace ${\mathcal{W}}(\unicode[STIX]{x1D714}_{\mathbb{P}^{2}})$ with ${\mathcal{W}}(O(1))\!)$ .

The motivic Euler characteristic $\unicode[STIX]{x1D712}(\mathbb{P}_{k(x)}^{2}/k(x))\in \operatorname{GW}(k(x))$ has been computed by Hoyois [Reference Hoyois18, Ex. 1.8] as $2\langle 1\rangle +\langle -1\rangle$ , hence its image in $W(k)$ is $\langle 1\rangle$ . We have the Gauß-Bonnet formula [Reference Levine and Raksit26, Theorem 1.5], which says that $\unicode[STIX]{x1D712}(\mathbb{P}_{k(x)}^{2}/k(x))=\unicode[STIX]{x1D70B}_{\ast }(e(T_{\mathbb{P}_{k(x)}^{2}}))$ . This in turn implies that $e(T_{\mathbb{P}_{k(x)}^{2}})$ is a $W(k)$ -generator for $H^{2}(\mathbb{P}_{k(x)}^{2},{\mathcal{W}}(\unicode[STIX]{x1D714}_{\mathbb{P}^{2}}))$ (one can also show this by computing the local indices associated to a section of $T_{\mathbb{P}^{2}}$ with isolated zeros, we leave the details of this to the reader). This shows that the Euler class of the relative tangent bundle $e({\mathcal{T}})\in H^{2} (\!\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}(p_{1}^{\ast }O(1))$ gives a global class restricting to the generators in the $E_{1}$ -term, and thus gives a generator for $H^{\ast }(\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}(p_{1}^{\ast }O_{\mathbb{P}^{2}}(1)))$ as a free $H^{\ast }(\operatorname{BSL}_{2},{\mathcal{W}})$ -module.

Recall we have the closed immersion

$$\begin{eqnarray}i:\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}\end{eqnarray}$$

with open complement

$$\begin{eqnarray}j:BN\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}.\end{eqnarray}$$

Proof. We recall Ananyevskiy’s computation of $H^{\ast }(\operatorname{BSL}_{2},{\mathcal{W}})$ (Theorem 3.3):

$$\begin{eqnarray}H^{\ast }(\operatorname{BSL}_{2},{\mathcal{W}})=W(k)[e]\end{eqnarray}$$

where $e=e(E_{2})$ is the Euler class of the tautological rank two vector bundle on $\operatorname{BSL}_{2}$ .

Since $\text{Pic}(\operatorname{BSL}_{2})=\{1\}$ , it follows that $\text{Pic}(\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2})=\mathbb{Z}[{\mathcal{O}}(1)]$ , and thus the normal bundle of $i$ is the bundle $p_{1}^{\ast }O_{\mathbb{P}(F)}(2)$ on $\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2} )$ . Moreover, $i^{\ast }p_{1}^{\ast }O_{\mathbb{P}(\operatorname{Sym}^{2}F)}(m)=p_{1}^{\ast }O_{\mathbb{P}(F)}(2m)$ for all $m$ . This gives us the localization sequence

$$\begin{eqnarray}\displaystyle & & \displaystyle \cdots \rightarrow H^{n-1}(\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})\nonumber\\ \displaystyle & & \displaystyle \qquad \xrightarrow[{}]{i_{\ast }}H^{n}(\mathbb{P}(\operatorname{Sym}^{2}F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}}(p_{1}^{\ast }O_{\mathbb{P}(\operatorname{Sym}^{2}F)}(m)))\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \xrightarrow[{}]{j^{\ast }}H^{n}(BN,{\mathcal{W}})(\unicode[STIX]{x1D6FE}^{m})\xrightarrow[{}]{\unicode[STIX]{x1D6FF}}H^{n}(\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})\rightarrow \cdots \,.\nonumber\end{eqnarray}$$

The result for $n\geqslant 1$ and for $n=0$ and $m=1$ follows from this and the Lemmas 5.1 and 5.2. For $n=0$ and $m=0$ , one can check by looking at a single fiber $\unicode[STIX]{x1D70B}^{-1}(x)$ that $\unicode[STIX]{x1D6FF}(\langle \bar{q}\rangle )$ is a $W(k)$ -generator for $H^{0}(\mathbb{P}(F)\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2},{\mathcal{W}})=W(k)$ .

To conclude, we compute the multiplicative structure for $H^{\ast }(BN,{\mathcal{W}})$ and the $H^{\ast }(BN,{\mathcal{W}})$ -module structure for $H^{\ast }(BN,{\mathcal{W}}(\unicode[STIX]{x1D6FE}))$ . As the class $\langle \bar{q}\rangle \in H^{0}(BN,{\mathcal{W}})$ is represented by a 1-dimensional form, we have $\langle \bar{q}\rangle ^{2}=1$ . It remains to compute $\langle \bar{q}\rangle \cdot p^{\ast }e\in H^{2}(BN,{\mathcal{W}})$ and $\langle \bar{q}\rangle \cdot j^{\ast }e({\mathcal{T}})\in H^{2}(BN,{\mathcal{W}}(\unicode[STIX]{x1D6FE}))$ .

For this, we consider the inclusion of the torus $\mathbb{G}_{m}\rightarrow N$ and the associated map of classifying spaces $B\mathbb{G}_{m}\rightarrow BN$ , where we use the model $(\mathbb{G}_{m}\backslash \operatorname{SL}_{2})\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ for $B\mathbb{G}_{m}$ . We recall that $BN=(\mathbb{P}(\operatorname{Sym}^{2}F)-\mathbb{P}(F))\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ . The $\operatorname{SL}_{2}$ -invariant section $Q$ of ${\mathcal{O}}_{\mathbb{P}(\operatorname{Sym}^{2}F)}(2)$ gives us the section $p_{1}^{\ast }Q$ of $p_{1}^{\ast }{\mathcal{O}}(2)$ . We claim that the corresponding double cover $\text{Spec }_{{\mathcal{O}}_{BN}}(\sqrt{p_{1}^{\ast }Q})\rightarrow BN$ is isomorphic to $B\mathbb{G}_{m}\rightarrow BN$ .

To see this, we note the isomorphism of schemes

$$\begin{eqnarray}\mathbb{G}_{m}\backslash \operatorname{SL}_{2}\cong \mathbb{P}^{1}\times \mathbb{P}^{1}-\unicode[STIX]{x1D6E5};\qquad \left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\mapsto ([a:b],[c:d])\end{eqnarray}$$

which is an $\operatorname{SL}_{2}$ -equivariant isomorphism for the right action of $\operatorname{SL}_{2}$ on $\mathbb{P}^{1}\times \mathbb{P}^{1}-\unicode[STIX]{x1D6E5}$ induced by the right action on $F$ . Since the cover $\mathbb{P}^{1}\times \mathbb{P}^{1}-\unicode[STIX]{x1D6E5}\rightarrow \mathbb{P}(\operatorname{Sym}^{2}F)-\mathbb{P}(F)$ is isomorphic to $\text{Spec }_{{\mathcal{O}}_{\mathbb{ P}(\operatorname{Sym}^{2}F)-\mathbb{P}(F)}}(\sqrt{Q})$ , equivariantly for the right $\operatorname{SL}_{2}$ -actions, the assertion follows.

Proof. The relation $(1+\langle \bar{q}\rangle )\cdot p^{\ast }e=0$ implies that $(1+\langle \bar{q}\rangle )\cdot j^{\ast }e({\mathcal{T}})=0$ . Indeed, since $H^{\ast }(BN,{\mathcal{W}}(\unicode[STIX]{x1D6FE}))$ is a free $W(k)[p^{\ast }e]$ -module, there is a unique element $\unicode[STIX]{x1D706}\in W(k)$ with $(1+\langle \bar{q}\rangle )\cdot j^{\ast }e({\mathcal{T}})=\unicode[STIX]{x1D706}\cdot j^{\ast }e({\mathcal{T}})$ . Then $0=(1+\langle \bar{q}\rangle )\cdot p^{\ast }e\cdot j^{\ast }e({\mathcal{T}})=\unicode[STIX]{x1D706}\cdot (p^{\ast }e\cdot j^{\ast }e({\mathcal{T}}))$ , and as $p^{\ast }e\cdot j^{\ast }e({\mathcal{T}})$ is a generator for the free $W(k)$ -module $H^{4} (\!BN,{\mathcal{W}}(\unicode[STIX]{x1D6FE})$ , $\unicode[STIX]{x1D706}=0$ . We now show that $(1+\langle \bar{q}\rangle )\cdot p^{\ast }e=0$ .

We have the tautological rank two bundle $E_{2}\rightarrow \operatorname{BSL}_{2}$ , $E_{2}:=F\times ^{\operatorname{SL}_{2}}E\operatorname{SL}_{2}$ and its pullback $\tilde{E}_{2}\rightarrow BN$ . The class $e\in H^{2}(\operatorname{BSL}_{2},{\mathcal{W}})$ is the Euler class $e(E_{2})$ , so $p^{\ast }e=e(\tilde{E}_{2})$ . We have the cover $\unicode[STIX]{x1D719}:B\mathbb{G}_{m}\rightarrow BN$ with fiber $\mathbb{G}_{m}\backslash N=\mathbb{Z}/2$ corresponding to the inclusion $\mathbb{G}_{m}\rightarrow N$ . The pullback $\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}$ splits as a direct sum

$$\begin{eqnarray}\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}=O(1)\oplus O(-1)\xrightarrow[{}]{\unicode[STIX]{x1D70B}}B\mathbb{G}_{m}\end{eqnarray}$$

corresponding to the decomposition of $F$ into the $t$ and $t^{-1}$ eigenspaces for the torus $\mathbb{G}_{m}\subset \operatorname{SL}_{2}$ . The $\mathbb{Z}/2$ -action on $B\mathbb{G}_{m}$ corresponds to the automorphism $t\mapsto t^{-1}$ of $\mathbb{G}_{m}$ and exchanges the two factors $O(1)$ and $O(-1)$ . By descent, the union of the two “coordinate axes” $O(1)\times 0\cup 0\times O(-1)\subset \unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}$ gives us the cone $\mathfrak{C}\subset \tilde{E}_{2}$ , which on each geometric fiber is the union of two lines through the origin. The normalization $\unicode[STIX]{x1D708}:\mathfrak{C}^{N}\rightarrow \mathfrak{C}$ is thus isomorphic to the total space of the bundle $O(1)$ , that is, we have the commutative diagram

Let ${\mathcal{I}}_{\mathfrak{C}}$ denote the ideal sheaf of $\mathfrak{C}$ in $\tilde{E}_{2}$ , and define ${\mathcal{I}}_{O(1)}$ similarly. The normal bundle $N_{O(1)/\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}}$ of the subscheme $O(1)$ in $\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}$ is $\unicode[STIX]{x1D70B}^{\ast }O(-1)_{|O(1)}$ . The restriction of $\unicode[STIX]{x1D70B}^{\ast }O(1)_{|O(1)}$ to $O(1)-\{0_{O(1)}\}$ is trivialized by the tautological section $s$ of $\unicode[STIX]{x1D70B}^{\ast }O(1)_{|O(1)}$ over $O(1)$ , which induces a nowhere vanishing dual section $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}s$ of the dual $N_{O(1)/\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}}$ over $O(1)-\{0_{O(1)}\}$ . This gives us the section $\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle$ of the twisted Grothendieck–Witt sheaf ${\mathcal{G}}{\mathcal{W}}(O(-1))$ over $O(1)-\{0_{O(1)}\}$ , as the class of the $O(-1)$ -valued quadratic form $q_{s}$ with $q_{s}(1)=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}s$ .

The boundary $\unicode[STIX]{x2202}_{0_{\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}}}(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle )$ lives in $H^{0} (\!0_{\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}},{\mathcal{K}}_{-1}^{MW}(\operatorname{det}(\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}))$ ; the canonical trivialization of $\operatorname{det}E_{2}$ sets this equal to $H^{0}(0_{\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}},{\mathcal{K}}_{-1}^{MW})$ . We claim that a calculation in local coordinates gives the identity

$$\begin{eqnarray}\unicode[STIX]{x2202}_{0_{\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}}}(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle )=\unicode[STIX]{x1D702}\in H^{0}(0_{q^{\ast }\tilde{E}_{2}},{\mathcal{K}}_{-1}^{MW}).\end{eqnarray}$$

Indeed, take a local generator $\unicode[STIX]{x1D706}$ for $O(1)$ over $U\subset B\mathbb{G}_{m}$ , giving the dual local generator $\unicode[STIX]{x1D706}^{\vee }$ for $O(-1)=O(1)^{\vee }$ . This gives a local isomorphism of $\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}$ with

$$\begin{eqnarray}\mathbb{A}^{1}\times \mathbb{A}^{1}\times U=\text{Spec }k[x,y]\times _{k}U,\end{eqnarray}$$

with $O(1)=\mathbb{A}^{1}\times 0\times U$ , $O(-1)=0\times \mathbb{A}^{1}\times U$ . This also identifies $N_{O(1)/\unicode[STIX]{x1D719}^{\ast }\tilde{E}_{2}}$ with the trivial bundle on $\mathbb{A}^{1}\times 0\times U$ and the section $\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle$ goes over to $\langle 1\rangle \otimes x^{-1}=p_{1}^{\ast }\langle x\rangle$ and the canonical generator of the normal bundle of $0_{O(1)}$ in $O(1)$ goes over to $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$ . On $\mathbb{A}^{1}=\text{Spec }k[x]$ , the element $\langle x\rangle \in K_{0}^{MW}(k[x])$ has boundary

$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x}(\langle x\rangle )=\unicode[STIX]{x2202}_{\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x}(1+\unicode[STIX]{x1D702}[x])=\unicode[STIX]{x1D702}\end{eqnarray}$$

since $\unicode[STIX]{x2202}_{\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x}([x])=1$ . Under the canonical identification ${\mathcal{K}}_{-1}^{MW}\cong {\mathcal{W}}$ , $\unicode[STIX]{x1D702}$ goes over to $\langle 1\rangle$ . We refer the reader to [Reference Morel30, § 6] or [Reference Morel31, § 6] for details on the Milnor–Witt sheaves and their relation to ${\mathcal{G}}{\mathcal{W}}$ and ${\mathcal{W}}$ .

Identifying $\mathfrak{C}-\{0_{\tilde{E}_{2}}\}$ with $O(1)-\{0_{O(1)}\}$ via $\unicode[STIX]{x1D719}$ , we have the section $\unicode[STIX]{x1D719}_{\ast }(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle )$ of ${\mathcal{W}}(N_{\mathfrak{C}-\{0_{\tilde{E}_{2}}\}/\tilde{E}_{2}})$ over $\mathfrak{C}-\{0_{\tilde{E}_{2}}\}$ . This has boundary

$$\begin{eqnarray}\unicode[STIX]{x2202}(\tilde{\unicode[STIX]{x1D719}}_{\ast }(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle ))=\tilde{\unicode[STIX]{x1D719}}_{\ast }(\unicode[STIX]{x2202}(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle ))=s_{0\ast }(\unicode[STIX]{x1D719}_{\ast }\langle 1\rangle )\in H^{0}(0_{\tilde{E}_{2}},{\mathcal{W}})\end{eqnarray}$$

where $s_{0}:BN\rightarrow \tilde{E}_{2}$ is the 0-section. Since $\unicode[STIX]{x1D719}:B\mathbb{G}_{m}\rightarrow BN$ is the double cover $\text{Spec }_{{\mathcal{O}}_{BN}}({\mathcal{O}}_{BN}(\sqrt{p_{1}^{\ast }Q}))$ , $\unicode[STIX]{x1D719}_{\ast }\langle 1\rangle$ is just the image in $H^{0}(BN,{\mathcal{W}})$ of the trace form of this double cover, which gives us the element $\langle 2\rangle +\langle 2\bar{q}\rangle =\langle 2\rangle (1+\langle \bar{q}\rangle )$ .

Let $\tilde{E}_{2,m}$ denote the pullback to $B_{m}\operatorname{SL}_{2}\subset \operatorname{BSL}_{2}$ . We recall that $H^{\ast }(\tilde{E}_{2,m},{\mathcal{W}})$ may be computed as the cohomology of the Rost–Schmid complex (see [Reference Morel29, Chapter 5])

$$\begin{eqnarray}\displaystyle {\mathcal{W}}(k(\tilde{E}_{2,m})) & \rightarrow & \displaystyle \bigoplus _{x\in \tilde{E}_{2,m}^{(1)}}{\mathcal{W}}(k(x),\operatorname{det}N_{x/\tilde{E}_{2,m}})\nonumber\\ \displaystyle & \rightarrow & \displaystyle \bigoplus _{x\in \tilde{E}_{2,m}^{(2)}}{\mathcal{W}}(k(x),\operatorname{det}N_{x/\tilde{E}_{2,m}})\rightarrow \cdots \,.\nonumber\end{eqnarray}$$

To simplify the notation, we replace $\tilde{E}_{2,m}$ with $\tilde{E}_{2}$ , with the understanding that all objects should be considered as living on $\tilde{E}_{2,m}$ for arbitrary $m$ .

We note that $s_{0\ast }(1)\in H^{2}(\tilde{E}_{2},{\mathcal{W}})$ is represented in the Rost–Schmid complex by the constant section $1$ on $0_{\tilde{E}_{2}}$ . Computing $H^{2}(\tilde{E}_{2},{\mathcal{W}})$ via the Rost–Schmid complex, we see that

$$\begin{eqnarray}\unicode[STIX]{x2202}(\unicode[STIX]{x1D719}_{\ast }(\langle \unicode[STIX]{x2202}/\unicode[STIX]{x2202}s\rangle ))=\langle 2\rangle (1+\langle \bar{q}\rangle )\in H^{0}(0_{\tilde{E}_{2}},{\mathcal{W}})\end{eqnarray}$$

goes to zero in $H^{2}(\tilde{E},{\mathcal{W}})$ . As $p^{\ast }e=e(\tilde{E}_{2})=s_{0}^{\ast }s_{0\ast }(1)$ , and as $s_{0}^{\ast }:H^{2}(\tilde{E}_{2},{\mathcal{W}})\rightarrow H^{2}(BN,{\mathcal{W}})$ is an isomorphism, we see that $\langle 2\rangle (1+\langle \bar{q}\rangle )\cdot p^{\ast }e=0$ in $H^{2}(BN,{\mathcal{W}})$ ; since $\langle 2\rangle$ is a unit in ${\mathcal{W}}$ , it follows that $(1+\langle \bar{q}\rangle )\cdot p^{\ast }e=0$ .

Proposition 5.5. Let $W(k)[x_{0},x_{2}]$ be the graded polynomial algebra over $W(k)$ on generators $x_{0},x_{2}$ , with $\deg x_{i}=i$ . Then sending $x_{0}$ to $\langle \bar{q}\rangle$ and $x_{2}$ to $p^{\ast }e$ defines $W(k)$ -algebra isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D713}:W(k)[x_{0},x_{2}]/(x_{0}^{2}-1,(1+x_{0})x_{2})\rightarrow H^{\ast }(BN,{\mathcal{W}}).\end{eqnarray}$$

Moreover, $H^{\ast }(BN,{\mathcal{W}}(\unicode[STIX]{x1D6FE}))$ is the quotient of the free $H^{\ast }(BN,{\mathcal{W}})$ -module on the generator $j^{\ast }e({\mathcal{T}})$ modulo the relation $(1+\langle q\rangle )j^{\ast }e({\mathcal{T}})=0$ .

As $Q$ is a section of ${\mathcal{O}}_{\mathbb{P}(\operatorname{Sym}^{2}F)}(2)$ , $p_{1}^{\ast }Q$ has a well-defined value in $k/k^{\times 2}$ at each point of $BN(k)$ . Here we use the definition of $BN$ as an Ind-scheme to define $BN(k)$ .

Lemma 5.6. Let $x\in BN(k)$ be a $k$ -point such that $p_{1}^{\ast }Q(x)=-1\,\text{mod}\,k^{\times 2}$ , and let $\tilde{E}_{2x}$ denote the fiber of $\tilde{E}_{2}$ at $x$ with origin $0_{x}$ . Let

$$\begin{eqnarray}s_{0\ast }:H^{0}(BN,{\mathcal{W}})\rightarrow H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})\end{eqnarray}$$

denote the Thom isomorphism. Then the restriction map

$$\begin{eqnarray}i_{x}^{\ast }:H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})\rightarrow H_{0_{x}}^{2}(\tilde{E}_{2x},{\mathcal{W}}).\end{eqnarray}$$

induces an isomorphism

$$\begin{eqnarray}\bar{i}_{x}^{\ast }:H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})/(s_{0\ast }(1+\langle \bar{q}\rangle ))\rightarrow H_{0_{x}}^{2}(\tilde{E}_{2x},{\mathcal{W}}).\end{eqnarray}$$

Proof. Since $p_{1}^{\ast }Q(x)=-1\,\text{mod}\,k^{\times 2}$ , it follows that $i_{x}^{\ast }(1+\langle \bar{q}\rangle )=1+\langle -1\rangle =0$ in $W(k)$ . Thus

$$\begin{eqnarray}i_{x}^{\ast }(s_{0\ast }(1+\langle \bar{q}\rangle ))=0\in H_{0_{x}}^{2}(\tilde{E}_{2x},{\mathcal{W}}),\end{eqnarray}$$

so we have a well-defined homomorphism

$$\begin{eqnarray}\bar{i}_{x}^{\ast }:H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})/(s_{0\ast }(1+\langle \bar{q}\rangle ))\rightarrow H_{0_{x}}^{2}(\tilde{E}_{2x},{\mathcal{W}}).\end{eqnarray}$$

We have the decomposition

$$\begin{eqnarray}H^{0}(BN,{\mathcal{W}})=W(k)\oplus W(k)\cdot \langle \bar{q}\rangle\end{eqnarray}$$

and thus the restriction of the Thom isomorphism $s_{0\ast }:H^{0}(BN,{\mathcal{W}})\rightarrow H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})$ to the factor $W(k)$ induces an isomorphism

$$\begin{eqnarray}\bar{s}_{0\ast }:W(k)\rightarrow H_{0_{\tilde{E}_{2}}}^{2}(\tilde{E}_{2},{\mathcal{W}})/(s_{0\ast }(1+\langle \bar{q}\rangle )).\end{eqnarray}$$

As the composition $\bar{i}_{x}^{\ast }\circ \bar{s}_{0\ast }$ is just the Thom isomorphism

$$\begin{eqnarray}s_{0x\ast }:W(k(x))=W(k)\rightarrow H_{0_{x}}^{2}(\tilde{E}_{2x},{\mathcal{W}})\end{eqnarray}$$

the result follows.

6 Rank two bundles on $BN_{T}(\operatorname{SL}_{2})$

We pursue an analogy of $N:=N_{T}(\operatorname{SL}_{2})$ with $\text{O}(2)$ . We let

$$\begin{eqnarray}\unicode[STIX]{x1D704}:\mathbb{G}_{m}\rightarrow T(\operatorname{SL}_{2})\end{eqnarray}$$

be the isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D704}(t)=\left(\begin{array}{@{}cc@{}}t & 0\\ 0 & t^{-1}\end{array}\right)\end{eqnarray}$$

and set

$$\begin{eqnarray}\unicode[STIX]{x1D70E}:=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right).\end{eqnarray}$$

For $m\geqslant 1$ an integer, define the representations $(F(m),\unicode[STIX]{x1D70C}_{m})$ of $N$ by $F(m)=\mathbb{A}^{2}$ and

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{m}\left(\begin{array}{@{}cc@{}}t & 0\\ 0 & t^{-1}\end{array}\right) & = & \displaystyle \left(\begin{array}{@{}cc@{}}t^{m} & 0\\ 0 & t^{-m}\end{array}\right)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{m}(\unicode[STIX]{x1D70E}) & = & \displaystyle \left(\begin{array}{@{}cc@{}}0 & 1\\ (-1)^{m} & 0\end{array}\right).\nonumber\end{eqnarray}$$

We let $(\unicode[STIX]{x1D70C}_{0},F(0))$ denote trivial one-dimensional representation.

For $m\geqslant 0$ , we let $\unicode[STIX]{x1D70C}_{m}^{-}:N\rightarrow \operatorname{SL}_{2}$ be the representation given by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{m}^{-}(\unicode[STIX]{x1D704}(t)) & = & \displaystyle \unicode[STIX]{x1D70C}_{m}(\unicode[STIX]{x1D704}(t))\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{m}^{-}(\unicode[STIX]{x1D70E}) & = & \displaystyle (-1)\cdot \unicode[STIX]{x1D70C}_{m}(\unicode[STIX]{x1D70E}).\nonumber\end{eqnarray}$$

We let $p^{(m)}:\tilde{O} (m)\rightarrow BN$ denote the rank two vector bundle

$$\begin{eqnarray}\tilde{O} (m)=F(m)\times ^{N}E\operatorname{SL}_{2}\rightarrow N\backslash E\operatorname{SL}_{2}=BN\end{eqnarray}$$

corresponding to $\unicode[STIX]{x1D70C}_{m}$ , with zero-section $s_{0}^{(m)}:BN\rightarrow \tilde{O} (m)$ and let $\operatorname{th}^{(m)}\in H_{0_{\tilde{O} (m)}}^{2}(\tilde{O} (m)),{\mathcal{W}}(\operatorname{det}^{-1}(\tilde{O} (m)))$ denote the Thom class $s_{0\ast }^{(m)}(1)$ . We define the bundle $p^{(m)-}:\tilde{O} ^{-}(m)\rightarrow BN$ replacing $\unicode[STIX]{x1D70C}_{m}$ with $\unicode[STIX]{x1D70C}_{m}^{-}$ . We note that $\text{Pic}(BN)\cong \mathbb{Z}/2$ , with generator $\unicode[STIX]{x1D6FE}:=\tilde{O} ^{-}(0)$ and that we have canonical isomorphisms $\operatorname{det}\tilde{O} ^{\pm }(m)\cong O_{BN}$ for $m$ odd and $\operatorname{det}\tilde{O} ^{\pm }(m)\cong \unicode[STIX]{x1D6FE}$ for $m>0$ even.

Our goal in this section is to compute the Euler classes $e(\tilde{O} ^{\pm }(m))\in H^{2}(BN,{\mathcal{W}}(\operatorname{det}^{-1}(\tilde{O} ^{\pm }(m))))$ .

Let $R$ be a $k$ -algebra, essentially of finite type over $k$ . We consider an $R$ -valued point $x$ of $B_{n}\operatorname{SL}_{2}$ coming from an $R$ -valued point of $E_{n}\operatorname{SL}_{2}$ , namely, two elements $v_{1},v_{2}\in \mathbb{A}^{n+2}(R)$ such that the resulting matrix $(\!\begin{smallmatrix}v_{1}\\ v_{2}\end{smallmatrix}\!)$ has rank two at each maximal ideal of $R$ . This gives us the corresponding $R$ -valued point $\bar{x}$ of the Grassmannian $\operatorname{Gr}(2,n+2)$ ,

$$\begin{eqnarray}\bar{x}:\text{Spec }R\rightarrow \operatorname{Gr}(2,n+2)\end{eqnarray}$$

together with the isomorphism of $R$ -modules $R\cong \operatorname{det}\bar{x}^{\ast }E_{2}$ sending $1\in R$ to $v_{1}\wedge v_{2}$ .

We have the $m$ th-power map

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{m,a}:F(a)\rightarrow F(ma);~\unicode[STIX]{x1D713}_{m,a}(x,y):=(x^{m},y^{m}),\end{eqnarray}$$

which is a (nonlinear!) $N$ -equivariant morphism. This gives us the flat and finite map of $BN$ -schemes

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{m,a}:\tilde{O} (a)\rightarrow \tilde{O} (ma).\end{eqnarray}$$

For $V\rightarrow X$ a rank $r$ vector bundle with trivialized determinant $\unicode[STIX]{x1D703}:\operatorname{det}V\rightarrow O_{X}$ and 0-section $s_{0}:X\rightarrow V$ , we write $s_{0\ast }:{\mathcal{E}}^{a,b}(X)\rightarrow {\mathcal{E}}_{0_{V}}^{a+2r,b+r}(V;\operatorname{det}^{-1}V)$ for the Thom isomorphism.