2.1 Spaces

Let $\mathrm{Top}$ be the Quillen model category of compactly generated weak Hausdorff spaces [Reference Hirschhorn12].Footnote ³ The weak equivalences of $\mathrm{Top}$ are the weak homotopy equivalences, and the fibrations are the Serre fibrations. The cofibrations are defined using the right lifting property with respect to the trivial fibrations. In particular, every object $\mathrm{Top}$ is fibrant. An object is cofibrant whenever it is a retract of a cell complex. We let $\mathrm{Top}_*$ denote the category of based spaces. Then $\mathrm{Top}_*$ inherits a Quillen model structure from $\mathrm{Top}$ by means of the forgetful functor $\mathrm{Top}_\ast \to \mathrm{Top}$.

An object in $\mathrm{Top}$ or $\mathrm{Top}_\ast$ is finite if it is a finite cell complex. It is homotopy finite if it is weakly equivalent to a finite object. A object is $X$ is finitely dominated if it is a retract of a homotopy finite object.

We employ the following notation, most of which is standard. For objects $X,Y\in \mathrm{Top}_\ast$, we let $X\vee Y$ denote the wedge and we let $X\wedge Y$ denote the smash product. The suspension of $X$ is then $S^1 \wedge X$, where $S^1$ is the circle. When we write $X^{[k]}$, we mean the $k$-fold smash product of $X$ with itself. When passing to the stable category of based spaces we sometimes write to indicate a stable map. If $X$ is an unbased space, we write $X_+$ for the based space $X \amalg \ast$ given by taking the disjoint union with a basepoint. For a based space $A$ and an unbased space $B$, our convention will be $A\wedge B_+ := A \wedge (B_+)$ and $B_+ \wedge A := (B_+) \wedge A$.

2.2 Poincaré duality spaces

Recall that an object $M \in \mathrm{Top}$ is a Poincaré duality space of (formal) dimension $d$ if there exists a pair

\[ (\mathscr L,[M]) \]

in which $\mathscr L$ is a rank one local coefficient system (or orientation sheaf) and

\[ [M] \in H_d(M;\mathscr L) \]

is a fundamental class such that for all local coefficient systems $\mathscr B$, the cap product homomorphism

\[{\cap} [M] \colon\! H^\ast(M;\mathscr B) \to H_{d-{\ast}}(M;\mathscr L\otimes\mathscr B) \]

is an isomorphism in all degrees (cf. [Reference Klein19, Reference Wall30]). The Poincaré spaces considered in this paper are assumed to be connected, finitely dominated and cofibrant. Note that if the pair $(\mathscr L, [M])$ exists, then it is determined up to unique isomorphism by the above property. Also note that closed manifolds are homotopy finite Poincaré duality spaces.

More generally, one has the notion of a Poincaré pair $(N,\partial N)$ with fundamental class $[N] \in H_d(N,\partial N; \mathscr L)$. In this case one assumes that the cap product

\[{\cap} [N] \colon\! H^\ast(N;\mathscr B) \to H_{d-{\ast}}(N,\partial N;\mathscr L\otimes\mathscr B) \]

is an isomorphism and in addition the image of $[N]$ with respect to the boundary homomorphism $H_d(N,\partial N; \mathscr L) \to H_{d-1}(\partial N; \mathscr L_{|\partial L})$ equips $\partial N$ with the structure of a Poincaré space of dimension $d-1$.

Given a spherical fibration $\xi \colon \! S(\xi ) \to M$, we write $D(\xi ) \to M$ for the mapping cylinder. The Thom space is the quotient

\[ M^\xi := D(\xi)/S(\xi). \]

Note that the first Stiefel-Whitney class of $\xi$ defines an orientation sheaf $\mathscr L^\xi$. Moreover, by the Thom isomorphism, $H_\ast (M^\xi ) \cong \tilde H_{\ast -k}(M)$, where in the latter, the coefficients are taken with respect to the orientation sheaf.

A Spivak normal fibration for a Poincaré duality space $M$ of dimension $d$ consists of a pair $(\nu,c)$ in which $\nu$ is a $(k-1)$-spherical fibration for some $k$ and

\[ c\colon\! S^{d+k} \to M^\nu \]

is a degree one map in the sense that the image $c_\ast ([S^{d+k}]) \in H_{d+k}(M^\xi ) \cong H_d(M;\mathscr L^\xi )$ is a fundamental class for $M$. The pair $(\nu,c)$ always exists and is unique up to stable fibre homotopy equivalence (see e.g., [Reference Klein15]). The map $c$ is sometimes called the normal invariant.

2.3 Poincaré embeddings of the diagonal

Suppose $M$ is a connected finitely dominated Poincaré duality space of dimension $d$. If $(\mathscr L,[M])$ is a choice of orientation and fundamental class for $M$, then $(\mathscr L^{\times 2},[M]^{\times 2})$ is one for $M\times M$. In particular $M\times M$ is a Poincaré duality space of dimension $2d$.

A diagonal Poincaré embedding of $M$ consists of a map of (finitely dominated) spaces $S(\xi ) \to C$ that fits into a commutative homotopy cocartesian square

in which

• • $\Delta$ is the diagonal map,

• • $\xi$ is a $(d-1)$-spherical fibration,

• • $(C,S(\xi ))$ is a Poincaré pair of dimension $2d$ with orientation sheaf given by the restriction $\mathscr L^{\times 2}_{|C}$ and fundamental class $[C]$ given by taking the image of $[M]^{\times 2}$ under the homomorphism

  \[ H_{2d}(M\times M; \mathscr L^{{\times} 2}) \to H_{2d}(M\times M, M; \mathscr L^{{\times} 2}) \cong H_{2d}(C,S(\xi);\mathscr L^{{\times} 2}_{|C}), \]
  where the displayed isomorphism is given by excision.

It follows that the local system associated to $\xi$ is $\mathscr L^{-1}$. We refer the reader to [Reference Klein19, §5] or [Reference Klein17, §2] for the definition of Poincaré embeddings in full generality.

2.4 $G$-spaces

A topological group object $G$ of $\mathrm{Top}$ is said to be cofibrant if its underlying space is. Let $\mathrm{Top}(G)$ and $\mathrm{Top}_\ast (G)$ denote the category of left $G$-spaces and based left $G$-spaces. We remark here that a right $G$-space can always be converted to a left $G$-space using the involution of $G$ defined by $g \mapsto g^{-1}$.

The categories $\mathrm{Top}(G)$ and $\mathrm{Top}_\ast (G)$ inherit a model structure using the forgetful functor to $\mathrm{Top}$. In both instances, all objects are fibrant. An object of $\mathrm{Top}(G)$ is cofibrant if it is a retract of a free $G$-cell complex, i.e., a space built up from the empty space by free $G$-cell attachments, where a free $G$-cell is defined to be $D^n \times G$ for some $n \ge 0$. Similarly, an object of $\mathrm{Top}_\ast (G)$ is cofibrant if it is built up from a point by based free $G$-cell attachments. In $\mathrm{Top}(G)$ and $\mathrm{Top}_\ast (G)$ one can speak of finite, homotopy finite, and finitely dominated objects.

According to Milnor [Reference Milnor23], given a connected based simplicial complex $X$, one may associate a cofibrant topological group $G$ and a universal $G$-principal fibre bundle $\tilde X \to X$. As $\tilde X$ is contractible, it follows that $X$ is identified with $BG$ up to homotopy. On the other hand, any connected space $Y$ is weakly equivalent to a simplicial complex $X$. It follows that $Y$ is weakly equivalent to $BG$.

If $X, Y\in \mathrm{Top}_\ast (G)$ are objects, we write

\[ [X,Y]_G \]

for the set of homotopy classes of based $G$-maps $X\to Y$. Similarly, we write

\[ \{X,Y\}_G \]

for the abelian group of homotopy classes of stable $G$-maps .

2.5 Naive $G$-spectra

We will also be considering the category $\mathrm{Sp}(G)$ of (naive) $G$-spectra formed from objects of $\mathrm{Top}_\ast (G)$. A $G$-spectrum $E$ in this sense consists of based $G$-spaces $E_n$ for $n \ge 0$ together with based $G$-maps $\Sigma E_n \to E_{n+1}$ (structure maps), where $G$ acts trivially on the suspension coordinate of $\Sigma E_n$. A morphism $f\colon \! E\to E'$ of $G$-spectra consists of based $G$-maps of $f_n \colon \! E_n \to E_n'$ for all $n$ which are compatible with the structure maps.

Then $\mathrm{Sp}(G)$ forms a model category in which the fibrant objects are the $\Omega$-spectra and the weak equivalences are the maps which induce isomorphisms on homotopy groups after applying fibrant replacement [Reference Schwede25].

Let $S$ denote the sphere spectrum. Given any $G$-space $X$ its suspension spectrum $\Sigma ^\infty (X_+):= S \wedge X_+$, is naturally a $G$-spectrum. We will sometimes denote it by

\[ S[X]. \]

Given a based $G$-space $X$ and a $G$-spectrum $E$, we may form the spectrum of equivariant maps from $X$ to $E$:

\[ F_G(X,E). \]

The $n$-th space of $F_G(X,E)$ is given by the function space $F_G(X,E_n)$ consisting of based $G$-maps $X\to E_n$. For $F_G(X,E)$ to have a sensible homotopy type, one should assume that $X$ is cofibrant and that $E$ is fibrant. Similarly, one may form

\[ X \wedge_G E, \]

which is the spectrum having $n$-space $X\wedge _G E_n$, i.e., the orbits of $G$ acting diagonally on the smash product $X\wedge E_n$. For this construction to have a sensible homotopy type, we should assume that $X$ and $E$ are cofibrant. We will implicitly apply fibrant/cofibrant replacement to arrange that the above constructions are always homotopy invariant.

As a special case of the above, let $EG$ be a cofibrant free contractible $G$-space. Then the homotopy fixed points and homotopy orbits of $G$ acting on $E$ are given by

\[ E^{hG} := F_G(EG_+,E) \quad \text{and} \quad E_{hG} := E\wedge_G EG_+. \]

2.5.1 Function spectra and smash products

By resorting to one of the standard enriched model categories of spectra [Reference Elmendorf, Kriz, Mandell and May8, Reference Hovey, Shipley and Smith13], we can form internal hom objects

\[ \hom(X,Y) \]

for naive spectra $X$ and $Y$.

Note that $X$ should be cofibrant and $Y$ should be fibrant for the function spectrum to be homotopy invariant. If $X$ and $Y$ are naive $G$-spectra in one of these categories, then $\hom (X,Y)$ is equipped with a $(G\times G)$-action and restricting to the $G$-equivariant functions gives the spectrum

\[ \hom_G(X,Y). \]

When $X = \Sigma ^\infty Z$ is a suspension spectrum of a (cofibrant) $G$-space, then one has an identification

\[ \hom_G(X,Y) \simeq F_G(Z,Y). \]

Similarly, the smash product $X\wedge Y$ has the structure of a $(G\times G)$-spectrum and the orbits under the diagonal action of $G$ defines a spectrum $X\wedge _G Y$. Moreover, $X\wedge _G Y \simeq Z\wedge _G Y$ when $X= \Sigma ^\infty Z$.

2.6 The dualizing spectrum

The main reference for this subsection is [Reference Klein15]. Let $G$ be a cofibrant topological group. Let $G^{\ell r}$ denote $G$ with the left action of $G\times G$ given by $(g,h)\cdot x = gxh^{-1}$.

Remark 2.5 The $(G \times G)$-spectrum $S[G^{\ell r}]$ has the following useful property: given any $G$-spectrum $E$, let $E^\ell$ and $E^r$ denote $E$ as a $G \times G$-spectrum with trivial right/left action, respectively. Then one has a natural isomorphism of $(G \times G)$-spectra

\[ E^\ell \wedge S[G^{\ell r}] \cong E^r \wedge S[G^{\ell r}]. \]

If we write $G_\ell$ in place of $G\times 1$ and $G_r$ in place of $1\times G$, then we immediately infer that

\[ (E^\ell \wedge S[G^{\ell r}])_{hG_\ell} \simeq E \simeq (E^r \wedge S[G^{\ell r}])_{hG_r} \]

as $G$-spectra and compatible with the above identification. Similarly, if $E$ is a $(G \times G)$-spectrum, we obtain a $G$-spectrum $E^{\mathrm{ad}}$ by restricting to the diagonal action. We then obtain natural equivalences

(2.1)\begin{equation} (E \wedge S[G^{\ell r}])_{hG_\ell} \simeq E^{\mathrm{ad}} \simeq (E \wedge S[G^{\ell r}])_{hG_r}. \end{equation}

Example 2.6 Let $G^{\mathrm{ad}}$ denote $G$ equipped with its conjugation action $g\cdot x= gxg^{-1}$. We set

\[ S[G^\mathrm{ad}] := S \wedge (G^\mathrm{ad}_+). \]

Then $S[G^\mathrm {ad}] = S[G^{{\ell r}}]^{\mathrm {ad}}$.

The dualizing spectrum of $G$ is the $G$-spectrum

(2.2)\begin{equation} \mathscr D_G := S[G^{\ell r}]^{hG_\ell} = F_{G_\ell}(EG_+,S[G^{\ell r}]), \end{equation}

in which $G$ acts as $G_r$.

For any $G$-spectrum $E$, one has a norm map

(2.3)\begin{equation} \eta = \eta_E \colon\! \mathscr D_G \wedge_{hG} E \to E^{hG} \end{equation}

which is natural in $E$. The map $\eta$ arises by noting that composition of functions gives rise to a pairing

(2.4)\begin{equation} F_{G_\ell}(EG_+,S[G^{\ell r}]) \wedge \hom_{G_\ell}(S[G^{\ell r}],E) \to F_{G_\ell}(EG_+,E) \end{equation}

which is $G$-invariant, and therefore factors through homotopy orbits. The norm map is then obtained using the evident weak equivalence $E \simeq \hom _{G_\ell }(S[G^{\ell r}],E)$. We recall the following.

Theorem 2.7 (Theorem D in [Reference Klein15])

If $BG$ is finitely dominated, then the norm map

\[ \eta_E \colon\! \mathscr D_G \wedge_{hG} E = (\mathscr D_G \wedge E)_{hG} \to E^{hG} \]

is an equivalence for any $G$-spectrum $E$.

Note that the norm map is adjoint to a map of $G$-spectra

\[ \mathscr D_G \wedge_{hG} E \wedge EG_+ \to E, \]

This map can be obtained from the evaluation map

(2.5)\begin{equation} \epsilon\colon\! \mathscr D_G\wedge EG_+ \to S[G^{\ell r}] \end{equation}

which is $(G\times G)$-equivariant. The map $\epsilon$ is defined so that it is adjoint to the identity map of $\mathscr D_G$, i.e., it is given by the evaluating stable maps $EG_+ \to S[G^{\ell r}]$ at points of $EG_+$.

Example 2.8 In the case $E = S$, with $G$ acting trivially, the norm equivalence is

\[ \eta \colon\! \left( \mathscr D_G \right)_{hG} \xrightarrow{\simeq} S^{hG} = F(BG_+,S). \]

Hence, the unit $BG_+ \to S$ corresponds to a map $S \to ( \mathscr D_G )_{hG}$.

This relates to the classical normal invariant of the Spivak normal fibration as follows: Suppose that $M$ is a (connected, finitely dominated) Poincaré duality space of dimension $d$. Let $\tilde M \to M$ be a universal principal bundle over $M$ with topological structure group $G$. Then $\tilde M$ is a free, contractible $G$-space. We may assume that $G$ is cofibrant. Then $\tilde M$ models $EG$ and $M$ models $BG$. In this case $\mathscr D_G$ is unequivariantly weakly equivalent to a sphere of dimension $-d$, and we write

\[ S^{-\tau} := \mathscr D_G \]

in this instance. Then the stable spherical fibration

\[ S^{-\tau} \times_G \tilde M \to M \]

is the Spivak normal fibration of $M$.Footnote ⁴ In this context, the Thom spectrum $M^{-\tau }$ is identified with $S^{-\tau }_{hG} = S^{-\tau } \wedge _G (\tilde M_+)$, and the norm equivalence in this notation is just Atiyah duality [Reference Atiyah1] for Poincaré duality spaces:

\[ M^{-\tau} \simeq F(M_+,S). \]

Summarizing, with respect to $E = S$, the norm equivalence specializes to Atiyah duality and the stable normal invariant $S \to M^{-\tau }$ corresponds to the unit $1\colon \! M_+ \to S$.

In what follows we set

\[ S^\tau := \hom(S^{-\tau},S)\wedge EG_+. \]

Then $G=G_\ell$ acts on $S^\tau$ and

\[ S^{\tau} \times_G \tilde M \to M \]

is the stable Spivak tangent fibration of $M$. In what follows we extend the action of $G = G_\ell$ on $S^\tau$ to $G\times G$ by letting $G_r$ act trivially.

2.7 The umkehr map $\Delta _!$

To the diagonal map $\Delta \colon \! M \to M\times M$ we associate a $(G\times G$)-equivariant umkehr map

(2.6)\begin{equation} \Delta_!\colon\! EG_+{\wedge} EG_+ \to S^\tau[G^{\ell r}], \end{equation}

where $S^\tau [G^{\ell r}] := S^\tau \wedge G^{\ell r}_+$. The umkehr map is defined as the evaluation map

\[ EG_+{\wedge} F_{G_\ell}(EG_+,S^\tau[G^{\ell r}]) \to S^\tau[G^{\ell r}], \]

in conjunction with the observation that the norm map with respect to $E= S^\tau [G^{\ell r}]$ defines an equivalence

\[ EG_+{\simeq} \mathscr D_G \wedge_{hG} S^\tau[G^{\ell r}] \mathrel{\mathop{\rightarrow}^{\eta}_{\sim}} F_{G_\ell}(EG_+,S^\tau[G^{\ell r}]) . \]

We can now rephrase the norm equivalence in terms of $\Delta _!$. Given any $G$-spectrum $E$ we obtain a map of $G$-spectra

\[ EG_+{\wedge} E_{hG} \cong (EG_+{\wedge} EG_+{\wedge} E^r)_{G_r} \mathrel{\mathop{\rightarrow}^{\Delta_! \wedge \operatorname{id}_E}} (S^\tau[G^{\ell r}] \wedge E^r)_{G_r} \cong S^\tau \wedge E , \]

The proof of the following is straightforward, but tedious. We leave its verification to the reader.

Lemma 2.10 The map adjoint to $\Delta _! \wedge \operatorname {id}_E$ defines a weak equivalence

\[ E_{hG} \mathrel{\mathop{\rightarrow}^{\sim}}(S^\tau \wedge E)^{hG} \]

that is natural in $E$.

4.1 The Reidemeister characteristic

The Reidemeister characteristic of $M$ is the homology class

\[ r(M) \in H_0(LM) \]

which is given by the stable homotopy class of the composition of maps of the form

The first map in this composition is the stable normal invariant and the second map is defined as follows: recall that the evaluation map

\[ \epsilon\colon\! S^{-\tau} \wedge EG_+ \to S[G^{\ell r}] \]

is $(G\times G)$-equivariant. We take homotopy orbits of $\epsilon$ with respect to the diagonal subgroup $G \subset G\times G$ to obtain the map

\[ M^{-\tau} \simeq S^{-\tau}_{hG} \to S[G^\mathrm{ad}]_{hG} \simeq S\wedge ((LM)_+). \]

(cf. [Reference Berrick, Chatterji and Mislin4, Reference Berrick and Hesselholt5]).

Remark 4.2 As $M$ is finitely dominated, there is a homotopy finite space $X$ and a factorization of the identity map

\[ M \overset s \to X \overset r \to M. \]

Setting $f = s\circ r\colon \! X \to X$, it follows that $f^{\circ 2} = f$, so $f$ is idempotent. Moreover, if $\pi = \pi _1(M)$, and $u \colon \! M \to B\pi$ classifies the universal cover, then one has the composition

\[ v \colon\! X \overset r \to M \overset u \to B\pi. \]

Consequently, $v \circ f = v$, i.e., $f$ is a map over $B\pi$. Then the generalized Lefschetz trace $L(f) \in \mathbb {Z}\langle \bar \pi \rangle = H_0(LM)$ is defined (cf.[Reference Berrick, Chatterji and Mislin4, §4], [Reference Geoghegan9]). It is not hard to show that

\[ r(M) = L(f). \]

However, we will not make use of this fact.

We next consider the norm equivalence

\[ \eta\colon\! S[G^\mathrm{ad}]_{hG} \overset\sim\to S^\tau[G^\mathrm{ad}]^{hG} := F_G(EG_+, S^\tau[G^\mathrm{ad}]) \]

for the $G$-spectrum $S^\tau [G^\mathrm {ad}]$. Recall that $S^\tau [G^\mathrm {ad}]$ is $S^\tau [G^{{\ell r}}]$ when considered as a $G$-spectrum.

The following result is essentially an unravelling of the definitions. We omit the proof.

Lemma 4.3 With respect to the above norm equivalence, the Reidemeister characteristic $r(M)$ is represented by the $G$-equivariant stable composite

\[ EG_+ \overset \Delta\to EG_+{\wedge} EG_+ \overset{\Delta_!} \to S^\tau[G^{{\ell r}}]. \]

Let $E$ be any $(G\times G)$-spectrum. smashing $\Delta _!$ with $E$ and taking homotopy fixed points, we obtain a map

\[ \Delta_! ^E \colon\! E^{h(G \times G)} \to (E\wedge S^{\tau}[G^{\ell r}])^{h(G \times G)}. \]

In the above, we have implicitly used the fact that the diagonal map $\Delta _{EG_+ \wedge EG_+} \colon EG_+ \wedge EG_+ \to (EG_+ \wedge EG_+)^{[2]}$ is a $(G\times G)$-equivariant weak equivalence.

If we restrict $(G \times G)$-fixed points to $G$-fixed points along the diagonal inclusion $\Delta \colon G \to G \times G$, we obtain a map

\[ \Delta_E^* \colon\! E^{h(G \times G)} \to (E^{\mathrm{ad}})^{hG}. \]

The following lemma in effect says that the maps $\Delta _! ^E$ and $\Delta _E^*$ coincide under the norm equivalence:

Lemma 4.4 Let $E$ be any cofibrant $(G{ \times } G)$-spectrum. Then the following triangle commutes up to homotopy

in which the vertical arrow is given by

\begin{align*} (E^{\mathrm{ad}})^{hG} & \simeq ((E \wedge S[G^{\ell r}])_{hG_\ell})^{hG} \mathrel{\mathop{\rightarrow}^{\eta^{hG}}_{\sim}} ((E \wedge S^\tau[G^{\ell r}])^{hG_\ell})^{hG}\\ & \quad\cong (E \wedge S^\tau[G^{\ell r}])^{h(G \times G)} , \end{align*}

where $\eta$ is the norm equivalence applied to $E \wedge S^\tau [G^{\ell r}]$.

Proof. Note that the diagram is natural in $E$ and the functor $E\mapsto E^{h(G \times G)}$ is represented by $(EG_+)^{[2]} = EG_+ \wedge EG_+$, where to avoid clutter, we have identified $EG_+$ with $S\wedge EG_+$.

Consequently, by the Yoneda lemma, it suffices to show that the two composites

are homotopic. Using that $EG_+$ is the unit we can simplify the diagram to

where the upper composite is adjoint to $\Delta _!$. Note that the unit $S \to (EG_+ \wedge EG_+)^{h(G \times G)}$ is mapped to the unit in $(EG_+)^{hG}$ under the diagonal map. Thus the lower composite is the unit $S \to (EG_+)^{hG}$ composed with the equivalence

\begin{align*} & (EG_+)^{hG} \simeq ((EG_+{\wedge} S[G^{\ell r}])_{hG_\ell})^{hG} \mathrel{\mathop{\rightarrow}^{\eta^{hG}}_{\sim}} ((EG_+{\wedge} S^\tau[G^{\ell r}])^{hG_\ell})^{hG} \cong \\ & \cong (EG_+{\wedge} S^\tau[G^{\ell r}])^{h(G \times G)} \cong (S^\tau[G^{\ell r}])^{h(G \times G)}. \end{align*}

Observe that the last equivalence is obtained from the norm equivalence

\[ \eta \colon\! EG_+ \mathop{\to}\limits^{{\simeq}} (S^\tau[G^{\ell r}])^{hG_\ell} \]

by passing to $G$-homotopy fixed points. Thus the lower composite is adjoint to the $(G \times G)$-equivariant map

\[ EG_+{\wedge} EG_+ \overset{\operatorname{id}\wedge \eta}\longrightarrow EG_+{\wedge} (S^\tau[G^{\ell r}])^{hG_\ell} \to S^\tau[G^{\ell r}], \]

which was the definition of $\Delta _!$ (here, the second displayed map is given by evaluation).

5.1 The diagonal embedding obstruction $\mu _M$

In [Reference Klein17] the first author defines an obstruction to finding a Poincaré embedding of the diagonal as the reduced Hopf invariant of the $(\pi \times \pi )$-equivariant stable map given by inducing $\Delta _!$ along the homomorphism $G\times G \to \pi \times \pi$. After some minor rewriting, takes the form

where $\bar M \to M$ is the universal cover, $\hat M = \bar M \times _\pi (\pi \times \pi )$ and $\hat M^\tau = S^\tau \wedge _{G} ((\pi \times \pi )_+)$. It is straightforward to check that the Hopf invariants in each case coincide (cf. remark 5.2). Hence,

\[ H(\Delta_!) = H(\Delta_! \wedge_{G\times G}( (\pi \times \pi)_+)). \]

It immediately follows that

\[ \mu_M = H(\Delta_!), \]

where $\mu _M$ is the complete obstruction to finding a Poincaré embedding of the diagonal.

Remark 5.4 In the above, we have implicitly used lemma 3.2, remark 3.3 and the proof of proposition 3.4 to define a preferred isomorphism identify $\mathbb {Z}\langle \bar \pi \rangle$ with

\[ \mathbb{Z}\langle \bar \pi \rangle \cong \{EG_+{\wedge} EG_+, S^\tau[G^{\ell r}] \wedge S^\tau[G^{\ell r}] \}_{G} , \]

as well as a preferred isomorphism

\[ Q_d(\pi) \cong \{EG_+{\wedge} EG_+, D_2(S^\tau[G^{\ell r}])\}_{G} . \]

We refer the reader to [Reference Klein17, §6] for more details (note that the discussion in [Reference Klein17, §6] carries over mutatis mutandis from $\pi$-spectra to $G$-spectra).