Poincaré Duality Spaces

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2 Definition

A Poincaré pair of dimension $d$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}d$ consists of a finitely dominated CW pair $(X,\partial X)$$(X,\partial X)$ for which there exists

$\displaystyle (\mathcal{L},[X])$ $$\displaystyle (\mathcal{L},[X])$$

in which

• $\mathcal{L}$$\mathcal{L}$ is a bundle of local coefficients on $X$$X$ which is free abelian of rank one, and

• $[X] \in H_d(X,\partial X;\mathcal {L})$$[X] \in H_d(X,\partial X;\mathcal {L})$ is a class such that

$\displaystyle \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$ $$\displaystyle \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$

and

$\displaystyle \cap [X] : H^*(X,\partial X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$ $$\displaystyle \cap [X] : H^*(X,\partial X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$

are isomorphisms.

Here, $\mathcal B$$\mathcal B$ is allowed to range over all local coefficient bundles on $X$$X$, but in fact it is sufficient to check the condition when $\mathcal{B}$$\mathcal{B}$ is the local coefficient bundle over $X$$X$ associated with $\Bbb Z[\pi]$$\Bbb Z[\pi]$, where $\pi$$\pi$ is the fundamental groupoid of $X$$X$.

3 Notes

• If $\partial X = \emptyset$$\partial X = \emptyset$, one says that $X$$X$ is a Poincaré duality space. (In view of this, perhaps better terminology would be to call $(X,\partial X)$$(X,\partial X)$ a Poincaré duality space with boundary.)

• $\mathcal L$$\mathcal L$ is called an orientation sheaf and $[X]$$[X]$ is called a fundamental class. The pair $(\mathcal L,[X])$$(\mathcal L,[X])$ is unique up to unique isomorphism.

• If $(X,\partial X)$$(X,\partial X)$ with respect to $(\mathcal{L},[X])$$(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$$d$, then $\partial X$$\partial X$ is a Poincaré space of dimension $d-1$$d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$$(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$$\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism.

• A finite CW complex $X$$X$ admits the structure of a Poincaré duality space of dimension $n$$n$ if and only if there exists a framed compact smooth manifold $M$$M$ of dimension $m \ge n+3$$m \ge n+3$ such $M$$M$ is homotopy equivalent to $X$$X$ and the inclusion $\partial M \subset M$$\partial M \subset M$ has homotopy fiber homotopy equivalent to $S^{m-n-1}$$S^{m-n-1}$.

4 Example

A compact (smooth, PL, TOP or homology) manifold $(X,\partial X)$$(X,\partial X)$ of dimension $d$$d$ is a Poincaré duality pair of dimension $d$$d$, where $\mathcal L$$\mathcal L$ is the orientation sheaf of $X$$X$ and $[X]$$[X]$ is the manifold fundamental class.