Proof of Theorem 2.1

Throughout, we implicitly use the topological Poincaré conjecture to identify the topological manifold underlying a homotopy sphere with the standard sphere.

We first assume $d\le 6$ . For $d\le 3$ or $d=5,6$ , there is nothing to show since there are no nontrivial homotopy spheres. For $d=4$ there may be, but we show that $E_{M\sharp \Sigma }\simeq E_M$ in $\mathrm {PSh}(\mathscr {D}\mathrm {isc}^{+}_d)$ for all oriented homotopy d-spheres $\Sigma $ (and hence also in $\mathrm {PSh}(\mathscr {D}\mathrm {isc}^{+}_{d,\leq k})$ for any k, by restriction). This is closely related to [Reference Knudsen and KupersKK24a, Theorem B]. By smoothing theory for embedding calculus [Reference Krannich and KupersKK26, Theorem 5.21 (ii)], it suffices to show that the two lifts of the topological tangent bundle $M\rightarrow \mathrm {BSTop}(4)$ along $\mathrm {BSO}(4)\rightarrow \mathrm {BSTop}(4)$ induced by the smooth structure of M and of $M\sharp \Sigma $ are homotopic as lifts. For that it is enough to show that the two lifts of $D^4\rightarrow \mathrm {BSTop}(4)$ induced by the smooth structure of $D^4$ and of $D^4\sharp \Sigma $ are homotopic as lifts, relative to the given lift on $\partial D^4$ induced by the standard smooth structure. But the obstruction for the existence of such a homotopy of lifts lies in $\pi _4(\mathrm {Top}(4)/\mathrm {O}(4))$ and this group vanishes, since $\pi _4(\mathrm {Top}(4)/\mathrm {O}(4))\cong \pi _4(\mathrm {Top}/\mathrm {O})$ by [Reference Freedman and QuinnFQ90, Theorem 8.7A] and $\pi _4(\mathrm {Top}/\mathrm {O})$ vanishes by [Reference Kirby and SiebenmannKS77, V.5.0 (5)].

Figure 1

Constructing $M \# \Sigma $ by modifying the gluing diffeomorphism in a splitting $M=M_0\cup _PM_1$ along a codimension zero submanifold P.

Figure 1 Long description

Top left panel shows a rounded rectangle split by a green curve labeled P, with M zero on the left and M one on the right. Top right panel shows the same regions separated, with a green arrow labeled i d sub P pointing right and an isomorphism symbol above the arrow. Bottom left panel shows the same split, but with a dashed circle labeled Sigma intersecting the green curve. Bottom right panel shows the separated regions again, with a green arrow labeled e x t sub P open parenthesis t w open parenthesis Sigma close parenthesis close parenthesis pointing right and an isomorphism symbol above. The sequence of panels illustrates the process of modifying the gluing diffeomorphism by introducing a twist along Sigma within P.

Turning to the case $d\ge 7$ , we begin with a general observation: Given a decomposition $M=M_0\cup M_1$ of an oriented d-manifold M into two codimension $0$ submanifolds $M_0,M_1\subset N$ that intersect in their common boundary $P{:=} \partial M_0=\partial M_1$ , then for any embedded disc $D^{d-1}\subset P$ compatible with the orientation, there is an orientation-preserving diffeomorphism $M\sharp \Sigma \cong M_0\cup _{\mathrm {ext}_P(\mathrm {tw}(\Sigma ))}M_1$ where $\mathrm {tw} \colon \Theta _d \to \pi _0\,{\mathrm {Diff}}_\partial (D^{d-1})$ is the inverse of the isomorphism from (3) (see Fig. 1). Combining this with Theorem 2.2, when given oriented homotopy d-spheres $\Sigma _0$ and $\Sigma _1\in \Theta _d$ , to show $E_{M\sharp \Sigma _0}\simeq E_{M\sharp \Sigma _1}$ in $\mathrm {PSh}(\mathscr {D}\mathrm {isc}_{d,\le k}^{+})$ for some fixed $1\le k\le \infty $ it suffices to show that $\smash {\mathrm {tw}(\Sigma _0\sharp \overline {\Sigma _1})=\mathrm {tw}(\Sigma _0)\circ \mathrm {tw}(\Sigma _1)^{-1}}$ lies in the kernel of the composition of $\mathrm {ext}_P$ with the morphism $\smash {E\colon \pi _0\,{\mathrm {Diff}}^{+}(P)\rightarrow \pi _0\,\mathrm {Aut}_{\mathrm {PSh}(\mathscr {D}\mathrm {isc}^{+}_{d-1,\le k})}(E_{P})}$ . Since $\mathrm {bP}_{d+1}=0$ for d even by [Reference Kervaire and MilnorKM63, Theorem 5.1, Lemma 2.3] and we assumed $d\not \equiv 1\ (\mathrm {mod}\ 4)$ , this shows that in order to prove Theorem 2.1, it suffices to find for $d=2n+1$ with $n\ge 3$ odd a decomposition $M=M_0\cup M_1$ as above, such that the following composition is trivial for $1\le k<\infty $ :

(4) $$ \begin{align}\smash{\mathrm{bP}_{2n+2}\le \smash{\Theta_{2n+1}\overset{({3})}{\cong}\pi_0\,{\mathrm{Diff}}_\partial(D^{2n})\xrightarrow{\mathrm{ext}_P} \pi_0\,{\mathrm{Diff}}^{+}(P)\overset{E}{\longrightarrow} \pi_0\,\mathrm{Aut}_{\mathrm{PSh}(\mathscr{D}\mathrm{isc}^{+}_{d,\le k})}(E_{P})}.} \end{align} $$

The decomposition we use is the following: for any fixed $g\ge 0$ , choose an embedding of the iterated boundary connected sum $V_g{:=} \natural ^g D^{n+1}\times S^n$ into M (such an embedding exists since $V_g$ embeds into a closed disc $D^{2n+1}$ which in turn embeds into any nonempty manifold), set $M_0$ to be the image of the embedding and $M_1$ to be the closure of its complement. The submanifolds $M_0$ and $M_1$ intersect in their common boundary which is diffeomorphic to $W_{g}= \partial (V_g)=\sharp ^g S^n \times S^n$ . It thus suffices to show that (4) is trivial for $P=W_g$ for some value of $g\ge 0$ . This is true for any $g\ge 2$ , since the composition $\mathrm {bP}_{2n+2}\rightarrow \pi _0\,{\mathrm {Diff}}^{+}(W_g)$ lands in the finite residual of $\pi _0\,{\mathrm {Diff}}^{+}(W_g)$ by Theorem 2.3 (it is in fact equal to it, but we will not need this), so as finite residuals are preserved by group homomorphisms, the image of the composition (4) is for $g\ge 2$ contained in the finite residual of $\smash {\pi _0\,\mathrm {Aut}_{\mathrm {PSh}(\mathscr {D}\mathrm {isc}^{+}_{d,\le k})}(E_{W_g})}$ . But the latter finite residual is trivial by [Reference MezherMez24, Theorem 3.16], so the claim follows.

Proof of Theorem 2.5

We first recall the definition of the element $[\Sigma ]\in \mathrm {coker}(J)_d$ associated to an oriented homotopy d-sphere (cf. [Reference Kervaire and MilnorKM63, Section 4]): Given an oriented homotopy d-sphere $\Sigma $ , consider its stable collapse map $\mathbf {S}\rightarrow {\mathrm {Th}}(-T\Sigma )$ . Choosing a stable framing of $\Sigma $ gives an equivalence ${\mathrm {Th}}(-T\Sigma )\simeq \mathbf {S}^{-d}$ , so the stable collapse map gives an element in $\pi _0(\mathbf {S}^{-d})\cong \pi _d(\mathbf {S})$ . Its image in $\mathrm {coker}(J)_d$ turns out to be independent of the choice of stable framing as long as the latter is chosen to be compatible with the given orientation of $\Sigma $ ; this defines the homomorphism $\Theta _d\rightarrow \mathrm {coker}(J)_d$ . In particular, we see that the class $[\Sigma ]\in \mathrm {coker}(J)_d$ depends only on (a) a classifier of the stable oriented tangent bundle $T\Sigma \colon \Sigma \rightarrow \mathrm {BSO}$ and (b) the homotopy class of the stable collapse map $\mathbf {S}\rightarrow {\mathrm {Th}}(-T\Sigma )$ . By the discussion above, in particular Theorem 2.6, both of these can be recovered from the equivalence class of the $2$ -truncated presheaf $E_\Sigma \in \mathrm {PSh}(\mathscr {D}\mathrm {isc}^{+}_{d,\le 2})$ , so the claim follows.

Proof. We begin with a few preliminary observations:

1. (a) Using the adjunctions $\kappa _i\vdash \partial _i$ and that mapping spaces in pullbacks of categories are given by the pullbacks of mapping spaces, one sees that the two projections $\mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1\rightarrow \mathscr {D}_i$ induce for $d_i,d_i'\in \mathscr {D}_i$ equivalences

   $$ \begin{align*}\mathrm{Map}_{\mathscr{D}_0\times_{\mathscr{D}_{01}}\mathscr{D}_1}((d_0,\kappa_1\partial_0(d_0)),(d_0',d_1'))&\overset{\simeq}\longrightarrow \mathrm{Map}_{\mathscr{D}_0}(d_0,d_0'), \\[-0.1cm] \mathrm{Map}_{\mathscr{D}_0\times_{\mathscr{D}_{01}}\mathscr{D}_1}((\kappa_0\partial_1(d_1),d_1),(d_0',d_1'))&\overset{\simeq}\longrightarrow \mathrm{Map}_{\mathscr{D}_1}(d_1,d_1').\end{align*} $$

2. (b) The functor $\iota $ in 1 is induced by the two functors $(\mathrm {id},\kappa _1\partial _0)\colon \mathscr {D}_0\rightarrow \mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1$ and $(\kappa _0\partial _1,\mathrm {id})\colon \mathscr {D}_1\rightarrow \mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1$ which are both fully faithful as a result of (a), and agree on $\mathscr {D}_{01}$ since $\partial _1 \kappa _1 \simeq \mathrm {id} \simeq \partial _0\kappa _0$ as the $\kappa _i$ were assumed to be fully faithful.

3. (c) Since fully faithful functors are preserved by taking pushouts [Reference Haine, Ramzi and SteinebrunnerHRS25, Theorem 0.1], the inclusion functors $\mathscr {D}_i\rightarrow \mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1$ are fully faithful for $i=0,1$ .

Combining (b) with (c) and observing that $\mathscr {D}_i \to \mathscr {D}_0 \cup _{\mathscr {D}_{01}} \mathscr {D}_1$ for $i=0,1$ are jointly essentially surjective, to show the claim in 1 it suffices to show that for $d_i\in \mathscr {D}_i$ , the functor induces an equivalence on the mapping space from $d_0$ to $d_1$ and on that from $d_1$ to $d_0$ . By symmetry, it suffices to check the former. By Theorem 0.1 loc.cit., the map

(14)

induced by the inclusions of $\mathscr {D}_0,\mathscr {D}_1,\mathscr {D}_{01}$ into $\mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1$ is an equivalence (for the equivalence in (14), see Remark 3.4 loc.cit.). Here $\lvert -\rvert \colon \mathscr {C}\mathrm {at}_\infty \rightarrow \mathscr {S}$ denotes the left adjoint to the inclusion $\mathscr {S}\subset \mathscr {C}\mathrm {at}_\infty $ , and $(-)_{(-)/}$ and $(-)_{/(-)}$ indicate under- and overcategories. Now consider the commutative diagram of pullbacks of categories

(15)

We claim that the bottom left horizontal map is the inclusion of a terminal object. To show this, we fix an object $(c_{01},f\colon \kappa _1(c_{01})\rightarrow d_1)\in \mathscr {D}_{01}\times _{\mathscr {D}_1}(\mathscr {D}_1)_{/d_1}$ and consider the commutative diagram of mapping spaces

Both squares are pullbacks, by the description of mapping spaces in pullbacks and overcategories as pullbacks and fibres, respectively. Hence the outer square is also a pullback. But the bottom composition is an equivalence by the adjunction $\kappa _1\vdash \partial _1$ , hence the top composition is an equivalence as well, so the top-left corner is contractible which confirms that $(\partial _1(d_1),\epsilon _1)$ is a terminal object. Now note that the rightmost vertical map in (15) is a cocartesian fibration (it classifies the functor $\mathrm {Map}_{\mathscr {D}_0}(d_0,-)$ ), so as cocartesian fibrations are closed under pullback [Reference LurieLur09, 2.4.2.3], $\mathrm {pr}_2$ in (15) is a cocartesian fibration as well and is thus by 4.1.2.15 loc.cit.smooth in the sense of 4.1.2.9 loc.cit.. Since we saw that the bottom left horizontal map is the inclusion of a terminal object and therefore cofinal, it follows from 4.1.2.10 loc.cit. that the upper left horizontal map in (15) is also cofinal and thus an equivalence after applying $\lvert -\rvert $ by 4.1.1.3 (3) loc.cit.. Combining this with (14), we conclude that the composition

induced by the inclusion $\mathscr {D}_0\rightarrow \mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1$ and postcompostion with the counit $\kappa _1\partial _1(d_1)\rightarrow d_1$ , is an equivalence. This implies the claim in 1, since postcomposition of this composition with the map induced by $\iota $ is equivalent to the identity on ${\mathrm {Map}_{\mathscr {D}_0}(d_0,\kappa _0\partial _1(d_1))}$ as a result of (b).

We now turn to proving 2. By the limit-formula for right Kan extensions [Reference LurieLur09, 4.3.2.13], the unit map in the claim is naturally equivalent to the map $X(d_0,d_1)\rightarrow \lim _{(\iota (d)\rightarrow (d_0,d_1))}X(\iota (d))$ where the limit is over $\smash {(\mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1)_{/(d_0,d_1)}{:=} (\mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1)\times _{(\mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1)}(\mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1)_{/(d_0,d_1)}}$ . The latter category is the value of at $\iota \colon \mathscr {D}_0\cup _{\mathscr {D}_{01}}\mathscr {D}_1\rightarrow \mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1$ of the functor

(16) $$ \begin{align}(-)_{/(d_0,d_1)}\colon (\mathscr{C}\mathrm{at}_\infty)_{/\mathscr{D}_0\times_{\mathscr{D}_{01}}\mathscr{D}_1}\rightarrow (\mathscr{C}\mathrm{at}_\infty)_{/((\mathscr{D}_0\times_{\mathscr{D}_{01}}\mathscr{D}_1)_{/(d_0,d_1)})}\end{align} $$

which is given by taking pullback along the forgetful functor $\pi \colon (\mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1)_{/(d_0,d_1)}\rightarrow \mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1$ . Since the latter forgetful functor is a cartesian fibration (it classifies the functor $\smash {\mathrm {Map}_{\mathscr {D}_0\times _{\mathscr {D}_{01}}\mathscr {D}_1}(-,(d_0,d_1))}$ ), the pullback functor (16) is a left adjoint by [Reference Ayala and FrancisAF20, Definition 1.1.6 and Lemma 3.2.1], so it preserves colimits. As colimits in overcategories are computed in the underlying categories, this implies that the commutative square

(17)

of fully faithful inclusions is again a pushout. Therefore the limit $\lim _{(\iota (d)\rightarrow (d_0,d_1))}X(\iota (d))$ is the pullback of the limits of the restriction of the diagram to the full subcategories in (17). The latter all have terminal objects induced by the units of the $(\kappa _i\dashv \partial _i)$ -adjunction: $(\epsilon _0,\mathrm {id})\colon (\kappa _0\partial _0(d_0),d_1)\rightarrow (d_0,d_1)$ is terminal in $(\mathscr {D}_1)_{/(d_0,d_1)}$ , $(\mathrm {id},\epsilon _1)\colon (d_0,d_1)\rightarrow (d_0,\kappa _1\partial _1(d_1)$ is terminal in $(\mathscr {D}_0)_{/(d_0,d_1)}$ , and $(\epsilon _0,\epsilon _1)\colon (d_0,d_1)\rightarrow (\kappa _0\partial _0(d_0),\kappa _1\partial _1(d_1))$ is terminal in $(\mathscr {D}_0)_{/(d_0,d_1)}$ . Using that limits over categories with terminal objects are given by the value at that terminal object, the claim 2 follows.

We prove 3 more generally as stated: assume we have maps of adjunctions (maps of bicartesian fibrations over $\Delta ^1$ [Reference LurieLur17, 4.7.4], e.g., obtained by restricting an adjunction to full subcategories as in 2)

(18)

where the $\kappa _i$ and $\kappa ^{\prime }_i$ are fully faithful. In addition to commutativity of the four squares obtained from (18) by forgetting the $\kappa _i$ and $\kappa _i'$ or the $\partial _i$ and $\partial _i'$ , maps of adjunctions give compatibilities between the unit transformations, which yields a map of commutative diagrams

This yields an equivalence $\iota \phi _\sqcup \simeq \phi _\times \iota '$ where $\phi _\sqcup $ and $\phi _\times $ are the induced functors between the pushouts of the top rows or the pullbacks of the bottom rows, respectively. The more general version of 3 we show is that the diagram

commutes, that is, the Beck–Chevalley transformation $\phi _\times ^{\ast }\iota _{\ast }\rightarrow \iota ^{\prime }_{\ast }\phi _\sqcup ^{\ast }$ is an equivalence. It suffices to prove this after precomposition with $\iota ^{\ast }$ since the latter is essentially surjective as a result of 1. Then the transformation has the form $\phi _\times ^{\ast }\iota _{\ast }\iota ^{\ast }\rightarrow \iota ^{\prime }_{\ast }\phi _\sqcup ^{\ast }\iota ^{\ast }\simeq \iota ^{\prime }_{\ast }(\iota ')^{\ast }\phi _\times $ and the claim that it is an equivalence follows from the description of $\iota ^{\prime }_{\ast }(\iota ')^{\ast }$ and $\iota _{\ast }\iota ^{\ast }$ from 2.

Finally, 4 follows by combining 2, observation (a), and the fact that mapping spaces in pullbacks of categories are the pullbacks of the mapping spaces.

Proof. The claim regarding the first square follows from the same argument as for [Reference Krannich and KupersKK26, Lemma 4.4] by replacing the role of $\mathrm {Env}(\mathscr {O})$ and $\mathrm {Env}(\mathscr {P})$ in loc.cit.by the categories $\mathscr {D}\mathrm {isc}_{d}^{{\shortparallel }}$ and $\mathscr {D}\mathrm {isc}_{d}$ respectively and using that any embedding between manifolds factors uniquely as the composition of an embedding which is surjective on path-components with an inclusion of path-components. Regarding the second square, we first recall that $\smash {\mathscr {D}\mathrm {isc}^{{\shortparallel }}_{d,\le n}}$ was defined as a full subcategory of $\smash {\mathscr {D}\mathrm {isc}^{{\shortparallel }}_{d}=\mathscr {D}\mathrm {isc}^\partial _d\times _{\mathscr {D}\mathrm {isc}_{d-1}}\mathscr {D}\mathrm {isc}^\partial _d}$ , namely the preimage of $\mathscr {D}\mathrm {isc}_{d,\le n}$ under $\gamma $ . This is contained in the full subcategory $\smash {\mathscr {D}\mathrm {isc}^\partial _{d,\le n}\times _{\mathscr {D}\mathrm {isc}_{d-1,\le n}}\mathscr {D}\mathrm {isc}^\partial _{d,\le n}\subset \mathscr {D}\mathrm {isc}^{{\shortparallel }}_{d}}$ , but is smaller. However, as right Kan extension along a fully faithful functor is fully faithful, it suffices to prove commutativity of the second square when replacing $\mathscr {D}\mathrm {isc}^{{\shortparallel }}_{d,\le n}$ with $\smash {\mathscr {D}\mathrm {isc}^\partial _{d,\le n}\times _{\mathscr {D}\mathrm {isc}_{d-1,\le n}}\mathscr {D}\mathrm {isc}^\partial _{d,\le n}}$ for $n=k$ and l respectively, and then the claim becomes an instance of Lemma 3.6 3.

Proof. By an application of the triangle identities for adjunctions, one sees that it suffices to show the claim after replacing the tower $\mathscr {D}\mathrm {isc}^{{\shortparallel }}_{d,\le \bullet }$ by the tower $\smash {\mathscr {D}\mathrm {isc}^\partial _{d,\le \bullet }\times _{\mathscr {D}\mathrm {isc}_{d-1,\le \bullet }}\mathscr {D}\mathrm {isc}^\partial _{d,\le \bullet }}$ in which case it is an instance of Lemma 3.6 4.

Proof. Since we have already seen in (20) that $\gamma _!$ is a map of towers, it suffices to prove the statement for $\bullet =\infty $ . Both sides of the Beck–Chevalley transformation $\delta \colon \gamma _!(\iota ^{{\shortparallel }})^{\ast }\rightarrow \iota ^{\ast }\gamma _!$ commute with colimits, so since presheaf-categories are generated under colimits by representables, it suffices in view of Remark 3.1 to show that for each d-manifold M with no boundary and codimension $0$ submanifolds $M_\ell ,M_r\subset M$ that intersect in their common boundary $P{:=} \partial M_{\ell }=\partial M_r$ , the transformation $\delta $ is an equivalence on ${\mathrm {Map}_{\mathscr {M}\mathrm {fd}_d^{\shortparallel }}(-,M^{\shortparallel })\in \mathrm {PSh}(\mathscr {M}\mathrm {fd}_d^{\shortparallel })}$ for $M^{\shortparallel }{:=} (M_\ell ,M_r,\mathrm {id})\in \mathscr {M}\mathrm {fd}_d^{\shortparallel }$ . To do so, we consider the poset $\mathscr {O}^{{\shortparallel }}_{\infty }(M)$ of open subsets $O\subset M$ such that the manifolds with boundary $O\cap M_\ell $ and $O\cap M_r$ are both contained in $\mathscr {D}\mathrm {isc}^\partial _d$ . Inclusion gives a functor $\mathscr {O}^{{\shortparallel }}_{\infty }(M)\rightarrow (\mathscr {M}\mathrm {fd}^{{\shortparallel }}_d)_{/M^{\shortparallel }}$ that sends $O\in \mathscr {O}^{{\shortparallel }}_{\infty }(M)$ to $O^{{\shortparallel }}{:=} (O\cap M_\ell ,O\cap M_r,\mathrm {id}_{O\cap P})\subset M^{\shortparallel }$ . Now consider the commutative square in $\mathrm {PSh}(\mathscr {D}\mathrm {isc}_d)$

We will show that ①-③ are equivalences, which implies that ④ is one as well, so the claim will follow. To do this, we repeatedly use the facts that restriction and left Kan extension preserve colimits and that left Kan extension preserves representables. From these facts, together with the observation that $\smash {O^{\shortparallel }\in \mathscr {D}\mathrm {isc}_d^{\shortparallel }}$ , one sees that ① is equivalent to the identity on $\smash {\mathrm {colim}_{O\in \mathscr {O}_\infty ^{\shortparallel }(M)}\mathrm {Map}_{\mathscr {D}\mathrm {isc}_d}(-,O)}$ , so in particular an equivalence. Using the above facts again, the map ③ is equivalent to the map $\smash {\mathrm {colim}_{O\in \mathscr {O}^{\shortparallel }_\infty (M)}\mathrm {Map}_{\mathscr {M}\mathrm {fd}_d}(\iota (-),O)\rightarrow \mathrm {Map}_{\mathscr {M}\mathrm {fd}_d}(\iota (-),M)}$ . Since $\smash {\{O\}_{O\in \mathscr {O}_{\infty }^{{\shortparallel }}(M)}}$ is a complete Weiss $\infty $ -cover of M in the sense of [Reference Knudsen and KupersKK24a, Definition 6.3], the proof of Lemma 6.4 loc.cit. shows that this map is indeed an equivalence. Since it will be relevant later, recall that the key step in the proof of this result in loc.cit. is an application of [Reference Dugger and IsaksenDI04, Proposition 4.6 (c)] to the open cover $\{F_n(O)\}_{O\in \mathscr {O}_{\infty }^{{\shortparallel }}(M)}$ of the space $F_n(M)$ of ordered configurations of n points in M, for $n\ge 0$ , using that this is a complete cover in the sense of Definition 4.5 loc.cit.. We will now show by a similar argument that the map $\smash {\mathrm {colim}_{O\in \mathscr {O}^{\shortparallel }_\infty (M)}\mathrm {Map}_{\mathscr {M}\mathrm {fd}^{\shortparallel }_d}(\iota ^{\shortparallel }(-),O^{\shortparallel })\rightarrow \mathrm {Map}_{\mathscr {M}\mathrm {fd}^{\shortparallel }_d}(\iota ^{\shortparallel }(-),M^{\shortparallel })}$ is an equivalence, which will imply that $\gamma _!(-)$ of it – which is precisely ② – is an equivalence as well, so the claim will follow. Adapting the argument in the proof of [Reference Knudsen and KupersKK24a, Lemma 6.4] to this map reduces the claim to showing that for $r,s,t\ge 0$ the open cover $\smash {\{F_{r,s,t}(O)\}_{O\in \mathscr {O}_{\infty }^{{\shortparallel }}(M)}}$ of the space $F_{r,s,t}(M)$ of ordered configurations of $r+s+t$ points in M where the first r points lie in $\mathrm {int}(M_\ell )$ , the second s points in P, and the final t points in $\mathrm {int}(M_r)$ , is a complete cover. But we have $F_{r,s,t}(O)=F_{r+s+t}(O)\cap F_{r,s,t}(M)$ , so the claim follows from the corresponding fact for $\{F_{r+s+t}(O)\}_{O\in \mathscr {O}_{\infty }^{{\shortparallel }}(M)}$ we have already used above by observing that complete covers are preserved by taking intersections with a fixed subspace.

Proof of Theorem 3.2

By the discussion above the diagram (21), it suffices to show that the outermost two compositions in it agree. As a result of Lemma 3.8, it suffices to show that the two leftmost squares in the diagram commute. For the second square, this is Lemma 3.9 and for the first square it is an instance of the naturality of the Yoneda embedding.

Proof of Theorem 2.2

For manifolds $M_i$ and diffeomorphism $\varphi _i$ as in the statement, we have that $E_{M_0\cup _{\varphi _i}M_1}$ in $\mathrm {PSh}(\mathscr {D}\mathrm {isc}_{d,\le k})$ for $i=0,1$ are the values of $(M_0,M_1,\varphi _i)\in \mathscr {M}\mathrm {fd}_d^{{\shortparallel }}$ under the counterclockwise composition in the square of Theorem 3.2 for $\bullet =k$ , so by commutativity they are also the values under the clockwise composition. But the assumption implies that their images $(E^\partial _{M_0},E^\partial _{M_0},E_{\varphi _i})$ under the top horizontal map $E^{\shortparallel }$ agree, so the claim follows. The addendum follows in the same way, using a variant of Theorem 3.2 for oriented manifolds which is proved in the same way as the nonoriented version (This is also a special case of the version for general tangential structures as alluded to in Remark 3.3 1).

Construction 4.1. Given a space B, we view $B\times B$ as a space over B via the projection to the first coordinate. Under the straightening equivalence, this corresponds to the constant functor $\mathrm {const}_B\in \mathrm {Fun}(B,\mathscr {S})$ , so the result $\Sigma ^\infty _B((B\times B)_{+,B})\in \mathrm {Sp}_B$ of adding a fibrewise disjoint basepoint and then taking fibrewise suspension spectra corresponds to the constant functor $\mathrm {const}_{\Sigma _+^\infty B}\in \mathrm {Fun}(B,\mathrm {Sp})$ . The latter is also the straightening of $p^{\ast }(\Sigma _+^\infty B)\in \mathrm {Sp}_B$ where $p\colon B\rightarrow \ast $ is the constant map, so we have a canonical equivalence $p^{\ast }(\Sigma ^\infty _+B)\simeq \Sigma ^\infty _B ((B\times B)_{+,B})$ . Now given $V\in \mathscr {S}_{/B}$ and a map $\varphi \colon V\rightarrow B\times B$ in $\mathscr {S}_{/B}$ , we may apply $\Sigma ^\infty _B(-)$ to the map $(B\times B)_{+,B}\rightarrow C_B(\varphi )$ from (d) to arrive at a map $p^{\ast }(\Sigma ^\infty _+B)\simeq \Sigma ^\infty _B ((B\times B)_{+,B})\rightarrow \Sigma ^\infty _BC_B(\varphi )$ . Combining this with the fibrewise evaluation map $D_B(\Sigma ^\infty _BC_B(\varphi ))\otimes _B \Sigma ^\infty _BC_B(\varphi ) \rightarrow p^{\ast }(\mathbf {S})$ , we obtain a map $D_B(\Sigma ^\infty _BC_B(\varphi ))\otimes _B p^{\ast }(\Sigma ^\infty _+B)\rightarrow p^{\ast }(\mathbf {S})$ which yields using the tensor-hom adjunction a map $D_B(\Sigma ^\infty _BC_B(\varphi ))\rightarrow {\mathrm {map}}_B(p^{\ast }(\Sigma ^\infty _+B), p^{\ast }(\mathbf {S}))\simeq p^{\ast }{\mathrm {map}}(\Sigma ^\infty _+B, \mathbf {S})=p^{\ast }D(\Sigma ^\infty _+B)$ which in turn yields via the $(p_!\dashv p^{\ast })$ -adjunction a map of spectra of the form

(23) $$ \begin{align}p_!D_B(\Sigma^\infty_BC_B(\varphi))\longrightarrow D(\Sigma^\infty_+B).\end{align} $$

Construction 4.2. Suppose we are given a d-dimensional vector bundle $\xi \colon B\rightarrow \mathrm {BO}(d)$ over a space B and a commutative square in $\mathscr {S}$ of the form

(24)

where $S(\xi )$ is the $(d-1)$ -dimensional spherical fibration induced by $\xi $ , the bottom horizontal map is the diagonal, and i is any map. Viewing this square as a square in $\mathscr {S}_{/B}$ via the projection of $B\times B$ onto the first coordinate, the naturality in (d) yields a map $C_B(S(\xi ))\rightarrow C_B(i)$ and thus by applying $p_!D_B(\Sigma _B^\infty (-))$ a map $p_!D_B(\Sigma _B^\infty C_B(i))\rightarrow p_!D_B(\Sigma _B^\infty C_B(S(\xi )))\simeq {\mathrm {Th}}(-\xi )$ (see (d) for the final equivalence). This features in a natural zig-zag of spectra

(25) $$ \begin{align} \mathbf{S}\simeq D(\Sigma_+^\infty \ast)\longrightarrow D(\Sigma_+^\infty B) \longleftarrow p_!D_B(\Sigma^\infty_B C_B(i))\longrightarrow {\mathrm{Th}}(-\xi)\end{align} $$

where the first map results from applying $D(\Sigma _+^\infty (-))$ to the map $p\colon B\rightarrow \ast $ , the second map is an instance of (23), and the final map is the one we just described.

Proof. To see this, one first uses that evaluation at the centre induces an equivalence between the map of pairs (28) for $X=E_M=\mathrm {Emb}(-,M)$ and the map of pairs

(29)

induced by composition of embeddings. It thus suffices to construct an equivalence of pairs

$$\begin{align*}(TM,TM\backslash 0_M)\xrightarrow{\simeq} \Big(\mathrm{Emb}(\ast,\mathbf{R}^d)^{\times 2}\times_{{\mathrm{Diff}}(\mathbf{R}^d)}\mathrm{Emb}(\mathbf{R}^d,M),\mathrm{Emb}(\ast\sqcup \ast,\mathbf{R}^d)\times_{{\mathrm{Diff}}(\mathbf{R}^d)}\mathrm{Emb}(\mathbf{R}^d,M)\Big)\end{align*}$$

such that its postcomposition with (29) agrees with the map $(TM,TM\backslash 0_M)\rightarrow (M\times M,M\times M\backslash \Delta )$ used in Example 4.3 (involving the choice of a tubular neighbourhood). By the uniqueness of tubular neighbourhoods, we may assume that the tubular neighbourhood $TM\hookrightarrow M\times M$ involved is on the first coordinate given by the projection and on the second coordinate given by a smooth retraction $e\colon TM\rightarrow M$ of the $0$ -section that is an embedding when restricted to each individual tangent space (to obtain e, fix a Riemannian metric and define e to first fibrewise scale vectors to be shorter than the injectivity radius and then apply the exponential map; see, e.g., [Reference Bröcker and JänichBJ82, Section 12]). Such a map e yields an $\mathrm {GL}_d(\mathbf {R})$ -equivariant equivalence $\smash {\mathrm {Fr}(TM)\xrightarrow {\simeq } \mathrm {Emb}(\mathbf {R}^d,M)}$ out of the frame bundle, which together with the $\mathrm {O}(d)$ -equivariant equivalence of pairs $\smash {(\mathbf {R}^d,\mathbf {R}^{d}\backslash \{0\})\xrightarrow {\simeq } (\mathrm {Emb}(*,\mathbf {R}^d)^{\times 2},\mathrm {Emb}(*\sqcup *,\mathbf {R}^d))}$ given by $\mathbf {R}^d\ni x\mapsto (0,x)\in \mathrm {Emb}(*,\mathbf {R}^d)^{\times 2}$ yields an equivalence of pairs of the form

Going through the construction, the precomposition of this equivalence with the standard identification of its source with $(TM,TM\backslash 0_M)$ has the desired properties.

Proof of Theorem 2.6

Specialising (25) to Example 4.4 yields a natural zig-zag

(30) $$ \begin{align}\mathbf{S}\longrightarrow D(\Sigma_+^\infty\lvert X\rvert)\longleftarrow p_!D_B(\Sigma^\infty C_B(i))\longrightarrow {\mathrm{Th}}(-\xi_X),\end{align} $$

as in the claim. The asserted property of it in the case $X=E_M$ follows by combining Lemma 4.5 with the discussion at the end of Example 4.3.

Proof. We start by recalling the definition of the morphism $\smash {\chi ^2\colon \mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\rightarrow \mathbf {Z}}$ from [Reference KrannichKra20, Section 3.5]. Any cohomology class in $\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$ can be represented by a continuous map $\smash {f\colon S\rightarrow B(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))}$ from an oriented closed connected surface S. The composition $\smash {f_2{:=} (B(\mathrm {pr}_2)\circ f)\colon S\rightarrow \mathrm {BSp}_{2g}(\mathbf {Z})}$ gives rise to a symmetric bilinear form $\smash {\langle -,-\rangle _{f_2}\colon \mathrm {H}^1(S;f_2^{\ast }\mathbf {Z}^{2g})^{\otimes 2}\rightarrow \mathbf {Z}}$ (see Section 3.4 loc.cit.) and $f_1{:=} \smash {(\mathrm {pr}_1\circ \pi _1(f))\colon \pi _1(S)\rightarrow \mathbf {Z}^{2g}}$ defines a $1$ -cocycle, so a class $\smash {[f_1]\in \mathrm {H}^1(S;f_2^{\ast }\mathbf {Z}^{2g})}$ . In these terms, the morphism $\chi ^2\colon \mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\rightarrow \mathbf {Z}$ is defined to send a class represented by $\smash {f\colon S\rightarrow B(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))}$ to $\smash {\langle [f_1],[f_1]\rangle _{f_2}\in \mathbf {Z}}$ (see Section 3.5 loc.cit.). Using this, we compute

$$\begin{align*}\chi^2((k,\mathrm{id})_{\ast}([f]))=\chi^2([B(k,\mathrm{id})\circ f])=\langle [(B(k,\mathrm{id})\circ f)_1],[(B(k,\mathrm{id})\circ f)_1]\rangle_{f_2}=\langle k\cdot [f_1],k\cdot [f_1]\rangle_{f_2}\end{align*}$$

for any class $\smash {[f\colon S\rightarrow B(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))]\in \mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})}$ . Here the first equality holds by the definition of $\chi ^2$ as just recalled, the second equality holds by observing that $(B(k,\mathrm {id})\circ f)_2=f_2$ since $\mathrm {pr}_2\circ (k,\mathrm {id})=\mathrm {pr}_2$ , and the third equality holds since $(B(k,\mathrm {id})\circ f)_1=\mathrm {pr}_1\circ \pi _1(B(k,\mathrm {id})\circ f)=\mathrm {pr}_1\circ (k,\mathrm {id})\circ \pi _1(f)=(k\cdot -)\circ \mathrm {pr}_1\circ \pi _1(f)=k\cdot f_1$ . Since $\langle -,-\rangle _{f_2}$ is bilinear, we have $\chi ^2((k,\mathrm {id})_{\ast }([f]))=\langle k\cdot [f_1],k\cdot [f_1]\rangle _{f_2}=k^2\cdot \langle [f_1],[f_1]\rangle _{f_2}=k^2\cdot \chi ^2([f])$ , which proves the relation in 1. To establish the relation 2, we first recall from Definition 3.20 (i) loc.cit. the lift of the morphism $\smash {(\chi ^2\circ \mathrm {inc}_{\ast })/2\in \mathrm {Hom}(\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z}),\mathbf {Z})}$ to a cohomology class $\smash {\frac {\chi ^2}{2}\in \mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})}$ . For $g\ge 3$ , there is a certain morphism $\smash {a\colon \mathbf {Z}/4\mathbf {Z}\rightarrow \mathrm {Sp}_{2g}^q(\mathbf {Z})\le \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z})}$ defined in Section 3.4 loc.cit.which features in a splitting of the form (see p.81 loc.cit.)

(35) $$ \begin{align}\smash{a^{\ast}\oplus h\colon \mathrm{H}^2(\mathbf{Z}^{2g}\rtimes\mathrm{Sp}_{2g}^q(\mathbf{Z});\mathbf{Z})\overset{\cong}{\longrightarrow} \mathrm{H}^2(\mathbf{Z}/4\mathbf{Z})\oplus\mathrm{Hom}(\mathrm{H}_2(\mathbf{Z}^{2g}\rtimes\mathrm{Sp}_{2g}^q(\mathbf{Z})),\mathbf{Z})}\end{align} $$

where h is the map from the universal coefficient theorem. For $g\ge 3$ , the lift $\smash {\frac {\chi ^2}{2}}$ is defined as the inverse of $(0,\chi ^2/2)$ under this isomorphism, and for $g\le 2$ it is defined by pulling back $\smash {\frac {\chi ^2}{2}}$ along the inclusion $\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z})\le \mathbf {Z}^{2g'}\rtimes \mathrm {Sp}_{2g'}^q(\mathbf {Z})$ for $g'\gg 0$ . Since $\smash {a\colon \mathbf {Z}/4\rightarrow \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z})}$ factors through the inclusion $\smash {\mathrm {Sp}_{2g}(\mathbf {Z})\le \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ , it commutes with the morphism $\smash {(k,\mathrm {id})\colon \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})\rightarrow \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ , so the induced map is with respect to the splitting (35) given by the identity on the first summand and by precomposition with $(k,id)_{\ast }$ on the second. Hence, since $\chi ^2\circ (k,\mathrm {id})_{\ast }=k^2$ by the first relation 1, we can deduce the claimed relation $\smash {(k,\mathrm {id})^{\ast }\frac {\chi ^2}{2}=k^2\cdot \frac {\chi ^2}{2}}$ .

Proof. For $n\equiv 1(4)$ or $n=3,7$ , we have $\Gamma _{g,1}=E(\mu \cdot \Sigma _P)$ by (36), so there is nothing to show. We thus assume $n\equiv 3(4)$ and $n\neq 3,7$ . Fixing an isomorphism $\pi _n(\mathrm {O})\cong \mathbf {Z}$ , we define a self-monomorphism $\kappa \colon \Gamma _{g,{\unicode{x00BD}}}\hookrightarrow \Gamma _{g,{\unicode{x00BD}}}$ by the following commutative square whose horizontal arrows are isomorphisms by (c) and whose left vertical map is induced by multiplication by k in $\mathbf {Z}^{2g}$ for some fixed number $k\neq 0$

(37)

We now consider the map of extensions

(38)

obtained by pulling back the extension (31) along $\kappa $ . From Lemma 5.1 2, we obtain $\mathrm {fr}(\Gamma _{g,1})\cap \Theta _{2n+1}=\mathrm {fr}(\kappa ^{\ast }\Gamma _{g,1})\cap \Theta _{2n+1}$ , so it suffices to show that there exists $k\neq 0$ such that the top extension in (38) is classified by $\mu =\frac {\sigma }{8}\cdot \Sigma _P$ . Since the bottom extension is classified by $\smash {\frac {\sigma }{8}+\frac {\chi ^2}{2}\in \mathrm {H}^2(\Gamma _{g,{\unicode{x00BD}}};\Theta _{2n+1})}$ by (36), this is equivalent to showing that

(39) $$ \begin{align}\smash{\textstyle{\kappa^{\ast}(\frac{\sigma}{8})\cdot \Sigma_P+\kappa^{\ast}(\frac{\chi^2}{2})\cdot \Sigma_Q=\frac{\sigma}{8} \in\mathrm{H}^2(\Gamma_{g,{\unicode{x00BD}}};\Theta_{2n+1})}}.\end{align} $$

Since the endomorphism $(k,\mathrm {id})$ in (37) commutes with the projection to $\smash {\mathrm {Sp}_{2g}^q(\mathbf {Z})}$ , the same applies to $\kappa $ in (37), so as $\smash {\frac {\sigma }{8}}\in \mathrm {H}^2(\Gamma _{g,{\unicode{x00BD}}};\Theta _{2n+1})$ is pulled back along $p\colon \Gamma _{g,{\unicode{x00BD}}} \rightarrow \mathrm {Sp}_{2g}^q(\mathbf {Z})$ , this implies that $\smash {\kappa ^{\ast }(\frac {\sigma }{8})=\frac {\sigma }{8}}$ , so it remains to show that $\smash {\kappa ^{\ast }(\frac {\chi ^2}{2})\cdot \Sigma _Q=0}$ in $\mathrm {H}^2(\Gamma _{g,{\unicode{x00BD}}};\Theta _{2n+1})$ for some $k\neq 0$ . Since the class $\smash {\frac {\chi ^2}{2}}$ is pulled back along the isomorphism $(s_F,p)$ in (37), it suffices to show that $\smash {(k,\mathrm {id})^{\ast }(\frac {\chi ^2}{2})\cdot \Sigma _Q=0}$ in $\mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}^q_{2g}(\mathbf {Z});\Theta _{2n+1})$ . But in view of Lemma 5.3, we have $\smash {(k,\mathrm {id})^{\ast }(\frac {\chi ^2}{2})=k^2\cdot \frac {\chi ^2}{2}}$ , so if we choose k to be any multiple of the order of $\Sigma _Q$ in $\Theta _{2n+1}$ , then we can conclude that $\smash {(k,\mathrm {id})^{\ast }(\frac {\chi ^2}{2})\cdot \Sigma _Q= \frac {\chi ^2}{2}\cdot k^2\cdot \Sigma _Q=0}$ holds as claimed.

Proof. Since $\mathrm {fr}(\Gamma _{g,{\unicode{x00BD}}})=1$ by (c), we have $\mathrm {fr}(\Gamma _{g,1})\subset \Theta _{2n+1}$ by applying Lemma 5.1 1 to the morphism $\Gamma _{g,1}\rightarrow \Gamma _{g,{\unicode{x00BD}}}$ in (31) with kernel $\Theta _{2n+1}$ . In particular, we have $\mathrm {fr}(\Gamma _{g,1})=\mathrm {fr}(\Gamma _{g,1})\cap \Theta _{2n+1}$ , so together with Claim ①, we get $\mathrm {fr}(\Gamma _{g,1})=\mathrm {fr}(E(\mu \cdot \Sigma _P))\cap \Theta _{2n+1}$ . Now consider the map of central extensions

where the bottom extension is obtained from the top extension by taking quotients by the subgroup $\mathrm {bP}_{2n+2}\le \Theta _{2n+1}$ . This extension is classified by the image of $\mu \cdot \Sigma _P$ under the morphism $\mathrm {H}^2(\Gamma _{g,{\unicode{x00BD}}};\Theta _{2n+1})\rightarrow \mathrm {H}^2(\Gamma _{g,{\unicode{x00BD}}};\Theta _{2n+1}/\mathrm {bP}_{2n+2})$ induced by the quotient map $\Theta _{2n+1}\rightarrow \Theta _{2n+1}/\mathrm {bP}_{2n+2}$ . Since $\Sigma _P$ is contained in $\mathrm {bP}_{2n+2}$ , this image is trivial, so it follows that $E(\mu \cdot \Sigma _P)/\mathrm {bP}_{2n+2}\cong \Theta _{2n+1}\times \Gamma _{g,{\unicode{x00BD}}}$ . Thus, since $\mathrm {fr}(\Gamma _{g,1})=1$ , we deduce from Lemma 5.1 3 that $\mathrm {fr}(E(\mu \cdot \Sigma _P)/\mathrm {bP}_{2n+2})=1$ , so applying Lemma 5.1 1 to $E(\mu \cdot \Sigma _P)\rightarrow E(\mu \cdot \Sigma _P)/\mathrm {bP}_{2n+2}$ we conclude that $\mathrm {fr}(\mu \cdot \Sigma _P)\subset \mathrm {bP}_{2n+2}$ . Combining this with the first part of the argument, we arrive at the claim $\mathrm {fr}(\Gamma _{g,1})=\mathrm {fr}(E(\mu \cdot \Sigma _P))\cap \Theta _{2n+1}\subset \mathrm {bP}_{2n+2}$ .

Proof. For $g=0$ we have $\Gamma _{g,1}=\pi _0{\mathrm {Diff}}_\partial (D^{2n})\cong \Theta _{2n+1}$ , so since $\Theta _{2n+1}$ is a finite abelian group, we have $\mathrm {fr}(\Gamma _{g,1})=1$ by Lemma 5.1 3, as claimed. In the case $g=1$ and $n\neq 3,7$ , we use that $\smash {\frac {\sigma }{8}}$ is trivial for $g=1$ (see 4), which implies $\smash {E(\frac {\sigma }{8}\cdot \Sigma _P)\cong \Gamma _{g,{\unicode{x00BD}}}\times \Theta _{2n+1}}$ and thus $\mathrm {fr}(E(\frac {\sigma }{8}\cdot \Sigma _P))=1$ by Lemma 5.1 3 and the fact from (c) that $\mathrm {fr}(\Gamma _{g,{\unicode{x00BD}}})=1$ . Combining this with Claim ①, we obtain $\mathrm {fr}(\Gamma _{g,1})\cap \Theta _{2n+1}=1$ . But since $\mathrm {fr}(\Gamma _{g,1})\subset \Theta _{2n+1}$ by Claim ②, we arrive at the claim $\mathrm {fr}(\Gamma _{g})=1$ . In the remaining case $n=3,7$ and $g=1$ , we first consider the map of extensions from [Reference KrannichKra20, Equation (2.2)] which involves the monomorphism $(s_F,p)$ from (c)

By Lemma 2.1, the image of the monomorphism $\mathrm {im}(\pi _n(\mathrm {SO}(n)\rightarrow \pi _n(\mathrm {SO}(n+1)))\hookrightarrow \pi _n(\mathrm {SO})\cong \mathbf {Z}$ has index $2$ . By Theorem 2.2, the upper extension is trivial, so we can conclude that there is an isomorphism $\smash {\Gamma _{g,{\unicode{x00BD}}}\cong \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ with respect to which the monomorphism $\smash {(s_F,p)\colon \Gamma _{g,{\unicode{x00BD}}}\hookrightarrow \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ from (c) is given by $\smash {(2,\mathrm {id})\colon \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})\hookrightarrow \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ . We can now define a self-monomorphism $\kappa \colon \Gamma _{g,{\unicode{x00BD}}}\hookrightarrow \Gamma _{g,{\unicode{x00BD}}}$ for any fixed number $k\neq 0$ similar to (37), but using the above isomorphism $\smash {\Gamma _{g,{\unicode{x00BD}}}\cong \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})}$ instead of $(s_F,p)$ . Arguing as in the proof of Claim ① we have $\smash {\mathrm {fr}(\Gamma _{g,1})\cap \Theta _{2n+1}=\mathrm {fr}(\kappa ^{\ast }\Gamma _{g,1})\cap \Theta _{2n+1}}$ which together with Claim ② yields $\mathrm {fr}(\Gamma _{g,1})=\mathrm {fr}(\kappa ^{\ast }\Gamma _{g,1})\cap \Theta _{2n+1}$ , so it would suffice to show $\mathrm {fr}(\kappa ^{\ast }\Gamma _{g,1})=1$ . This follows from an application of Lemma 5.1 5 to the central extension $\smash {0\rightarrow \Theta _{2n+1}\rightarrow \kappa ^{\ast }\Gamma _{g,1}\rightarrow \Gamma _{g,1{\unicode{x00BD}}}\rightarrow 0}$ once we verify the three assumptions that (I) $\Gamma _{g,1{\unicode{x00BD}}}$ has finitely generated abelianisation, (II) $\mathrm {fr}(\Gamma _{g,1{\unicode{x00BD}}})=1$ , and (III) the cohomology class $\smash {\kappa ^{\ast }(-\frac {\chi ^2-\sigma }{8})\cdot \Sigma _P\in \mathrm {H}^2(\Gamma _{g,1{\unicode{x00BD}}};\Theta _{2n+1})}$ which classifies this extension by (36) has trivial image in $\mathrm {Hom}(\mathrm {H}_2(\Gamma _{g,1{\unicode{x00BD}}}),\Theta _{2n+1})$ for some choice of $k\neq 0$ . Condition (I) holds since $\Gamma _{g,1{\unicode{x00BD}}}$ is itself finitely generated, because we saw above that it is isomorphic to $\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})$ . Condition (II) holds by (c). This leaves us with checking condition (III) for which it suffices to show that $ \smash {\kappa ^{\ast }(-\frac {\chi ^2-\sigma }{8})}$ maps under the canonical morphism $h\colon \mathrm {H}^2(\Gamma _{g,1{\unicode{x00BD}}});\mathbf {Z})\rightarrow \mathrm {Hom}\mathrm {H}_2(\Gamma _{g,1{\unicode{x00BD}}}),\mathbf {Z})$ to $\smash {-k^2\cdot (\chi ^2-\sigma )/8\circ (s_F,p)_{\ast }}$ , since then $\smash {\kappa ^{\ast }(-\frac {\chi ^2-\sigma }{8})\cdot \Sigma _P}$ maps to $-k^2\cdot \Sigma _P\cdot ((\chi ^2-\sigma )/8\circ (s_F,p)_{\ast })$ in $\mathrm {Hom}(\mathrm {H}_2(\Gamma _{g,1{\unicode{x00BD}}}),\Theta _{2n+1})$ which is trivial if we choose $k\neq 0$ to divide the order of $\Sigma _P$ . To show this remaining identity $h( \smash {\kappa ^{\ast }(-\frac {\chi ^2-\sigma }{8})})=\smash {-k^2\cdot (\chi ^2-\sigma )/8\circ (s_F,p)_{\ast }}$ , we compute

$$\begin{align*}\textstyle{8\cdot h( \smash{\kappa^{\ast}(-\frac{\chi^2-\sigma}{8})})=-(\chi^2-\sigma)\circ (s_F,p)_{\ast}\circ \kappa_{\ast}=-(\chi^2-\sigma)\circ (k,\mathrm{id})_{\ast}\circ (s_F,p)_{\ast}}\end{align*}$$

where the first equality follows from 3 and the second equality holds since $(s_F,p)\circ \kappa =(k,\mathrm {id})\circ (s_F,p)$ by definition of $\kappa $ . We then compute further

$$\begin{align*}-(\chi^2-\sigma)\circ (k,\mathrm{id})_{\ast}\circ (s_F,p)_{\ast}=-\chi^2\circ (k,\mathrm{id})_{\ast}\circ (s_F,p)_{\ast}=-k^2\cdot \chi^2\circ (s_F,p) \end{align*}$$

using the final part of 4 for the first equality and Lemma 5.3 for the second. In total we obtain $8\cdot h( \smash {\kappa ^{\ast }(-\frac {\chi ^2-\sigma }{8})})=-k^2\cdot \chi ^2\circ (s_F,p)$ in $\mathrm {Hom}(\mathrm {H}_2(\Gamma _{g,1{\unicode{x00BD}}}),\mathbf {Z})$ , so the claimed identity follows by dividing this equality by $8$ .

Proof. We first form the pullback involving the monomorphism $(s_F,p)$ from (c)

(40)

and consider the diagram of horizontal of extensions

(41)

obtained from the front bottom row classified by $\mu \in \mathrm {H}^{2}(\Gamma _{g,{\unicode{x00BD}}};\mathbf {Z})$ by pulling it back along the inclusion $\iota _F$ to obtain the front part of the diagram, and then pushing out along $\Sigma _P\colon \mathbf {Z}\rightarrow \Theta _{2n+1}$ to obtain the back part. In a moment, we will show that $\smash {\mathrm {fr}(E(\iota _F^{\ast }\mu ))=\mathbf {Z}}$ . Assuming this for now, the proof concludes as follows: Applying Lemma 5.1 1 to the map $E(\iota _F^{\ast }\mu )\rightarrow E(\mu \cdot \Sigma _P)$ , we deduce $\langle \Sigma _P\rangle \subset \mathrm {fr}(E(\mu \cdot \Sigma _P))$ , so since $\mathrm {bP}_{2n+2}=\langle \Sigma _P\rangle $ , we obtain $\mathrm {bP}_{2n+2}\subset \mathrm {fr}(E(\mu \cdot \Sigma _P))$ which combined with Claim ① implies $\mathrm {bP}_{2n+2}\subset \mathrm {fr}(\Gamma _{g,1})$ as claimed. We are thus left to show that $\smash {\mathrm {fr}(E(\iota _F^{\ast }\mu ))=\mathbf {Z}}$ which we do by verifying the assumptions of Lemma 5.2 for the class $\smash {\iota _F^{\ast }\mu \in \mathrm {H}^{2}(\Gamma _{g,{\unicode{x00BD}}}^F;\mathbf {Z})}$ , viewing $\smash {\Gamma _{g,{\unicode{x00BD}}}^F}$ as a subgroup of $\smash {\mathrm {Sp}_{2g}(\mathbf {Z})}$ via the inclusion $p_F$ in (40). Since the index of the image of $p_F$ is the index of the image of $(s_F,p)$ in (40) which is finite by Item (c), the first condition in Lemma 5.2 holds for $\iota _F^{\ast }\mu $ . To show the second condition, that is, that the abelianisation $\smash {\mathrm {H}_1(\Gamma _{g,{\unicode{x00BD}}}^F)}$ is finite, note that since the right vertical inclusion in (40) is a split injection, the same applies for the left vertical inclusion, so $\smash {\mathrm {H}_1(\Gamma _{g,{\unicode{x00BD}}}^F)}$ is a summand of $\smash {\mathrm {H}_1(\Gamma _{g,{\unicode{x00BD}}})}$ and thus it suffices to prove that $\smash {\mathrm {H}_1(\Gamma _{g,{\unicode{x00BD}}})}$ is finite. This holds by a combination of Corollary 2.4 and Table 2 in loc.cit., using the assumption that $g\ge 2$ . This leaves us with establishing the third condition in Lemma 5.2 for the class $\iota _F^{\ast }\mu $ , that is, that we have $8\cdot \mu \circ (\iota _F)_{\ast }=\sigma \circ (p_F)_{\ast }$ in $\mathrm {Hom}(\mathrm {H}_2(\Gamma _{g,{\unicode{x00BD}}}^F),\mathbf {Z})$ . From the definition of $\mu $ and commutativity of (40), we see that the left-hand side of the claimed equation is given by the composition

$$\begin{align*}\smash{\mathrm{H}_2(\Gamma_{g,{\unicode{x00BD}}}^F)\xrightarrow{(p_F)_{\ast}}\mathrm{H}_2(\mathrm{Sp}_{2g}(\mathbf{Z}))\xrightarrow{\mathrm{inc}_{\ast}}\mathrm{H}_2((\mathbf{Z}^{2g}\otimes\pi_n(\mathrm{O}))\rtimes \mathrm{Sp}_{2g}(\mathbf{Z}))\longrightarrow \mathbf{Z}} \end{align*}$$

where the final arrow is $\smash { \sigma \circ (\mathrm {pr}_2)_{\ast }}$ if $n\neq 3,7$ and $\smash {-\chi ^2+(\sigma \circ (\mathrm {pr}_2)_{\ast })}$ if $n=3,7$ . As $\smash {\chi ^2\circ \mathrm {inc}_{\ast }}$ vanishes by Lemma 5.3, we have $8\cdot \mu \circ (\iota _F)_{\ast }=\sigma \circ (\mathrm {pr}_2)_{\ast }\circ \mathrm {inc}_{\ast }\circ (p_F)_{\ast }$ in both cases. But since $\smash {\mathrm {pr}_2\circ \mathrm {inc}=\mathrm {id}_{\mathrm {Sp}_{2g}(\mathbf {Z})}}$ , this implies $8\cdot \mu \circ (\iota _F)_{\ast }=\sigma \circ (p_F)_{\ast }$ as claimed.