Section 4. (0N63) · ScalingStacks

4. Nonabelian Poincaré duality

Applying Theorem 3.24, we offer a slightly different perspective, and proof, of the nonabelian Poincaré duality of Salvatore [Sa], Segal [Se3], and Lurie [Lu2], which calculate factorization homology with coefficients in iterated loop spaces as a compactly supported mapping space.

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Definition 4.1. For a space $B$ , the $\infty$ -category $\Space_{B}$ is that of retractive spaces over $B$ . The $\oo$ -category $\Space^{\geq n}_{B}$ is the full $\oo$ -subcategory of $\Space_{B}$ consisting of those $X\rightleftarrows B$ for which the retraction is $n$ -connective, that is, $\pi_{*}X\rightarrow\pi_{*}B$ is an isomorphism for $*<n$ with any choice of base-point of $B$ .

Equivalently, an object $X\in\Space_{B}^{\geq n}$ may be thought of as a fibration $X\rightarrow B$ with a distinguished section for which, for each $b\in B$ , the fiber $X_{b}$ is $n$ -connective.

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Definition 4.2. For $X\in\Space_{B}$ and $M\rightarrow B$ a space over $B$ , the space $\Gammac(M,X)$ of compactly supported sections of $X$ over $M$ is the subspace of $\Map_{/B}(M,X)$ consisting of those maps $f\colon M\to X$ over $B$ for which there is a compact subspace $K\subset M$ with the property that the restriction factors through the section: $f_{|M\smallsetminus K}:M\to B\to X$ .

For $B\rightarrow\BTop(n)$ as before, note that $\Gammac(-,X)$ defines a covariant functor $\mfld_{n}^{B}\rightarrow\Space$ . By inspection, this functor carries finite disjoint unions to finite products of spaces, which is to say that $\Gammac(-,X)$ is symmetric monoidal with respect to the Cartesian monoidal structure on the $\infty$ -category of spaces.

To easily state the next result, Theorem 4.4, we introduce some terminology. Each point $g\in B$ determines a symmetric monoidal functor $\disk_{n}^{\fr}\to\disk_{n}^{B}$ . Thereafter, each symmetric monoidal functor $A\colon\disk_{n}^{B}\to\spaces$ determines the associative monoid

\pi_{0}(A,g)\colon\disk_{1}^{\fr}\xrightarrow{\mathbb{R}^{n-1}\times-}\disk_{n}^{\fr}\longrightarrow\disk_{n}^{B}\xrightarrow{~A~}\spaces\xrightarrow{~\pi_{0}~}{\sf Set}~.

We say $A$ is group-like if, for each $g\in B$ , this monoid $\pi_{0}(A,g)$ is a group.

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Example 4.3. In the case $B\simeq\ast$ so that a $B$ -framing is a framing in the standard sense, a symmetric monoidal functor $A\colon\disk_{n}^{\sf fr}\to\spaces$ is the data of an $\mathcal{E}_{n}$ -algebra, and this $\mathcal{E}_{n}$ -algebra is group-like in the standard sense if and only if $A$ is group-like in the the sense just above.

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Theorem 4.4. The functor $\Gammac:\Space_{B}\rightarrow\Fun^{\otimes}(\mfld_{n}^{B},\Space)$ restricts as a fully faithful functor

\Space_{B}^{\geq n}~\hookrightarrow~\mathbf{H}(\mfld_{n}^{B},\Space)

from $n$ -connective retractive spaces over $B$ to homology theories. The essential image consists of those $\mathcal{F}$ for which the restriction $\mathcal{F}_{|\disk_{n}^{B}}$ is group-like.

The following is the core technical detail in the proof of Theorem 4.4, that the assignment of compactly supported sections $\Gammac(-,X)$ is $\otimes$ -excisive in our sense, provided the retraction $X\rightarrow B$ is sufficiently connected. The fullness of the functor above is a parametrized form of May’s theorem from [Ma], identifying $n$ -connective objects as $\disk^{\fr}_{n}$ -algebras, which is Theorem 5.1.3.6 of [Lu2].

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Lemma 4.5. Let $M$ be $B$ -framed manifold, equipped with a collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ . Let $X\in\Space^{\geq n}_{B}$ be an $n$ -connective space over $B$ . There is a natural weak homotopy equivalence

\Gammac(M^{\prime},X)\underset{\Gammac(M_{0}\times\mathbb{R},X)}{\times}\Gammac(M^{\prime\prime},X)~\simeq~\Gammac(M,X)

between the quotient of the product $\Gammac(M^{\prime},X)\times\Gammac(M^{\prime\prime},X)$ by the diagonal action of $\Gammac(M_{0}\times\mathbb{R},X)$ and the space of compactly supported sections of $X$ over $M$ .

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Proof. Since $M_{0}\hookrightarrow M$ is a proper embedding, a compactly supported section over $M$ can be restricted to obtain a compactly supported section over $M_{0}$ , as well as over $M\smallsetminus M^{\prime}$ and over $M\smallsetminus M^{\prime\prime}$ . Namely, there is a diagram among spaces of compactly supported sections

By inspection, the bottom horizontal sequence is a fiber sequence, as is the right vertical sequence, as is the diagonal sequence. Also, the inner square is pullback because $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ is a pushout. Because $M_{0}\subset M$ is equipped with a regular neighborhood, these fiber sequences are in fact Serre fibration sequences, and so the inner square is a weak homotopy pullback square. In particular, there is a right homotopy coherent action of $\Omega\Gammac(M_{0},X)$ on $\Gammac(M^{\prime},X)$ , a left homotopy coherent action of $\Omega\Gammac(M_{0},X)$ on $\Gammac(M^{\prime\prime},X)$ , and a continuous map of topological spaces

(7)

\Gammac(M^{\prime},X)\underset{\Omega\Gammac(M_{0},X)}{\times}\Gammac(M^{\prime\prime},X)\longrightarrow\Gammac(M,X)

from the balanced homotopy coinvariants. Because $X\to B$ is $n$ -connective and $M_{0}$ is $(n-1)$ -dimensional, the base $\Gammac(M_{0},X)$ is connected. It follows that the map (7) is in fact a weak homotopy equivalence. The assertion follows after the canonical identification $\Omega\Gammac(M_{0},X)\cong\Gammac(M_{0}\times\mathbb{R},X)$ as group-like $\mathcal{E}_{1}$ -spaces.

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As a consequence, we recover the following theorem of Salvatore [Sa], Segal [Se3], and Lurie [Lu2].

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Corollary 4.6 (Nonabelian Poincaré duality). For any $X$ in $\Space_{B}^{\geq n}$ , with associated $\disk^{B}_{n}$ -algebra $\Omega_{B}^{n}X\in\Alg_{\disk^{B}_{n}}(\Space)$ , there is a natural equivalence

\int_{M}\Omega_{B}^{n}X~\simeq~\Gammac(M,X)

between the factorization homology of a $B$ -framed $n$ -manifold $M$ with coefficients in $\Omega^{n}_{B}X$ and the space of compactly supported sections of $X$ over $M$ .

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Proof. We apply Theorem 3.24: Since $\Gammac(-,X)$ is a homology theory, it is equivalent to factorization homology with coefficients in $\Gammac(\mathbb{R}^{n},X)$ , which is identified as the $n$ -fold loop space of the fiber of the map $X\rightarrow B$ , $\Gammac(\mathbb{R}^{n},X)\simeq\Omega^{n}_{B}X$ . ∎

This result specializes to Poincaré duality between twisted homology and compactly supported cohomology, as we will now explain. Given $X\simeq\BTop(n)\times K(A,i)$ a product with an Eilenberg-MacLane space, then $\Omega^{n}_{\BTop(n)}X\simeq\Mapc(\mathbb{R}^{n},K(A,i))\simeq K(A,i-n)$ is an $n$ -disk algebra in spaces, where the multiplication is the usual group structure on $K(A,i-n)$ but is equipped with a nontrivial action of $\Top(n)$ . We then have an equivalence of spaces

\int_{M}\Mapc(\mathbb{R}^{n},K(A,i))\simeq\Mapc(M,K(A,i))

which is the space level version of the equivalence $\mathsf{H}_{*}^{\tau}(M,A[i-n])\simeq\mathsf{H}^{*}_{\sf c}(M,A[i])$ , obtained by applying $\Omega^{\infty}$ to the spectrum level equivalence given by Atiyah duality. Given an $A$ -orientation of $M$ , then one can additionally untwist the lefthand side, as usual.

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Remark 4.7. The factorization homology $\int_{M}\Omega^{n}_{B}X$ is built from configuration spaces of disks in $M$ with labels defined by $X\to B$ , and the preceding result thereby has roots in the configuration space models of mapping spaces dating to the work of Segal, May, McDuff and others in the 1970s, see [Se1], [Ma], [Mc], and [Bö]. Factorization homology is not a generalization of the classical configuration spaces with labels, as described in [Bö], because the configuration space with labels in $X$ models a mapping space with target the $n$ -fold suspension of $X$ , rather than into $X$ itself. Instead, factorization homology generalizes the configuration spaces with summable or amalgamated labels of Salvatore [Sa] and Segal [Se3].

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Proof of Theorem 4.4. Corollary 4.6 identifies the functor $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\to\mathbf{H}(\mfld_{n}^{B},\Space)$ as the composition $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\xrightarrow{\Omega^{n}_{B}}\disk_{n}^{B}\xrightarrow{\int}\mathbf{H}(\mfld_{n}^{B},\Space)$ . Theorem 3.24 gives that $\int$ is fully faithful, so it remains to argue that $\Omega_{B}^{n}$ is fully faithful with essential image the group-like $\disk_{n}^{B}$ -algebras in spaces. This is immediate because, for instance, $\spaces_{/B}$ is an $\infty$ -topos (Theorem 5.1.3.6 of [Lu2]).

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David Ayala, John Francis

Original source: arXiv:1206.5522v6