Poincaré duality IV (Ex)

Exercise 0.1. Let $C_*$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}C_*$ and $D_*$$D_*$ be $\Zz\pi$$\Zz\pi$-chain complexes and

$\displaystyle s: \Zz^{\omega} \otimes_{\Zz\pi}(C_* \otimes_{\Zz} D_*) \rightarrow Hom_{\Zz\pi}(C^{-*},D_*)$ $$\displaystyle s: \Zz^{\omega} \otimes_{\Zz\pi}(C_* \otimes_{\Zz} D_*) \rightarrow Hom_{\Zz\pi}(C^{-*},D_*)$$

be defined by sending $1 \otimes x \otimes y \in \Zz^{\omega} \otimes_{\Zz \pi}(C_{n-k} \otimes_{\Zz} D_k)$$1 \otimes x \otimes y \in \Zz^{\omega} \otimes_{\Zz \pi}(C_{n-k} \otimes_{\Zz} D_k)$ to the map

$\displaystyle s(1 \otimes x \otimes y): (C^{-*})_{k-n} \rightarrow D_k, (\phi:C_{n-k} \rightarrow \Zz \pi) \mapsto \overline{\phi(x)}\cdot y$ $$\displaystyle s(1 \otimes x \otimes y): (C^{-*})_{k-n} \rightarrow D_k, (\phi:C_{n-k} \rightarrow \Zz \pi) \mapsto \overline{\phi(x)}\cdot y$$

Show that $s$$s$ is a $\Zz$$\Zz$-chain map.

Hint: Note that $\Zz^{\omega} \otimes_{\Zz \pi}(C_* \otimes_{\Zz} D_*)_n=\prod_{k \in \Zz} \Zz^{\omega} \otimes_{\Zz\pi}(C_{n-k} \otimes_{\Zz} D_k)$$\Zz^{\omega} \otimes_{\Zz \pi}(C_* \otimes_{\Zz} D_*)_n=\prod_{k \in \Zz} \Zz^{\omega} \otimes_{\Zz\pi}(C_{n-k} \otimes_{\Zz} D_k)$ with boundary map $d=id \otimes ((-1)^kd_C \otimes id + id \otimes d_D)$$d=id \otimes ((-1)^kd_C \otimes id + id \otimes d_D)$ and that $Hom_{\Zz\pi}(C^{-*},D_*)_n=\prod_{k \in \Zz}Hom_{\Zz\pi}((C^{-*})_{k-n},D_k)$$Hom_{\Zz\pi}(C^{-*},D_*)_n=\prod_{k \in \Zz}Hom_{\Zz\pi}((C^{-*})_{k-n},D_k)$ with boundary map $d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$$d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$.

Exercise 0.2. Let $C_*$$C_*$ and $D_*$$D_*$ be $\Zz\pi$$\Zz\pi$-chain complexes. Show that for the $n$$n$-th homology of the complex $hom_{\Zz\pi}(C_*,D_*)$$hom_{\Zz\pi}(C_*,D_*)$ we have

$\displaystyle H_n(hom_{\Zz\pi}(C_*,D_*))= [ \Sigma^n C_*,D_*]_{\Zz\pi}$ $$\displaystyle H_n(hom_{\Zz\pi}(C_*,D_*))= [ \Sigma^n C_*,D_*]_{\Zz\pi}$$

where $(\sum^nC_*)_k:=C_{k-n}$$(\sum^nC_*)_k:=C_{k-n}$ denotes the shifted complex with boundary map $d_{\sum C}:=(-1)^nd_C$$d_{\sum C}:=(-1)^nd_C$.

Hint: use the boundary map of Exercise 0.1.

Exercise 0.3. Deduce from the Poincare homotopy equivalence $? \cap [X]:C^{n-*}(\widetilde X) \rightarrow C_*(\widetilde X)$$? \cap [X]:C^{n-*}(\widetilde X) \rightarrow C_*(\widetilde X)$ that

$\displaystyle H_k(X,\Zz^{\omega}) \cong H^{n-k}(X,\Zz):=H_k(C^{n-*}(X))$ $$\displaystyle H_k(X,\Zz^{\omega}) \cong H^{n-k}(X,\Zz):=H_k(C^{n-*}(X))$$

as $\Zz$$\Zz$-Modules.

Exercise 0.4. Let $(X,A)$$(X,A)$ be a Poincare pair with $X$$X$ an $n$$n$-dimensional, connected, finite CW-complex and $A\subset X$$A\subset X$ an $(n-1)$$(n-1)$-dimensional subcomplex, an orientation homomorphism $\omega_X : \pi \longrightarrow \{\pm 1\}$$\omega_X : \pi \longrightarrow \{\pm 1\}$ and a fundamental class $[X,A]\in H_n(X,A;\Zz^{\omega_X}):=H_n(\Zz^{\omega_X}\otimes_{\Zz\pi}C_*(\tilde{X},\tilde{A}))$$[X,A]\in H_n(X,A;\Zz^{\omega_X}):=H_n(\Zz^{\omega_X}\otimes_{\Zz\pi}C_*(\tilde{X},\tilde{A}))$. That means, for a universal covering $p:\tilde{X}\longrightarrow X$$p:\tilde{X}\longrightarrow X$ and $\tilde{A}:=p^{-1}(A)$$\tilde{A}:=p^{-1}(A)$ the $\Zz\pi$$\Zz\pi$-chain maps $?\cap [X,A]:C^{n-*}(\tilde{X},\tilde{A})\longrightarrow C_*(\tilde{X})$$?\cap [X,A]:C^{n-*}(\tilde{X},\tilde{A})\longrightarrow C_*(\tilde{X})$ and $?\cap [X,A]:C^{n-*}(\tilde{X})\longrightarrow C_*(\tilde{X},\tilde{A})$$?\cap [X,A]:C^{n-*}(\tilde{X})\longrightarrow C_*(\tilde{X},\tilde{A})$ are $\Zz\pi$$\Zz\pi$-chain homotopy equivalences.

Show that the components $C\in\pi_0(A)$$C\in\pi_0(A)$ of $A$$A$ inherit the structure of a finite $(n-1)$$(n-1)$-dimensional Poincare complex, i.e. that there is an induced orientation homomorphism $\omega_C : \pi_1(C) \longrightarrow \{\pm 1\}$$\omega_C : \pi_1(C) \longrightarrow \{\pm 1\}$ and an induced fundamental class $[C]\in H_{n-1}(C;\Zz^{\omega_C})$$[C]\in H_{n-1}(C;\Zz^{\omega_C})$, such that

$\displaystyle ?\cap [C]:C^{n-1-*}(\tilde{C})\longrightarrow C_*(\tilde{C})$ $$\displaystyle ?\cap [C]:C^{n-1-*}(\tilde{C})\longrightarrow C_*(\tilde{C})$$

is a $\Zz\pi$$\Zz\pi$-chain homotopy equivalence.

Hint: Tensorize both sides with $\Zz^{\omega}$$\Zz^{\omega}$ and consider the induced map.

The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.