Poincaré duality IV (Ex)
(Created page with "<wikitex>; {{beginthm|Exercise}} \label{chain_map} Let $ C_* $ and $ D_* $ be $ \Zz\pi $-chain complexes and $$ s: \Zz^{\omega} \otimes_{\Zz\pi}(C_* \otimes_{\Zz} D_*) \righta...") |
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Show that $s$ is a $ \Zz $-chain map. | Show that $s$ is a $ \Zz $-chain map. | ||
{{endthm}} | {{endthm}} | ||
− | |||
− | The exercises on this page were sent by Alex Koenen and Arkadi Schelling. | + | '''Hint''': Note that $ \Zz^{\omega} \otimes_{\Zz \pi}(C_* \otimes_{\Zz} D_*)_n=\prod_{k \in \Zz} \Zz^{\omega} \otimes_{\Zz\pi}(C_{n-k} \otimes_{\Zz} D_k) $ with boundary map $ d=id \otimes ((-1)^kd_C \otimes id + id \otimes d_D) $ and that $Hom_{\Zz\pi}(C^{-*},D_*)_n=\prod_{k \in \Zz}Hom_{\Zz\pi}((C^{-*})_{k-n},D_k)$ with boundary map $d(f)=d_D \circ f-(-1)^nf \circ d_{C^{-*}}$. |
− | == References == | + | |
− | {{#RefList:}} | + | |
+ | {{beginthm|Exercise}} \label{homol_hom} | ||
+ | Let $ C_* $ and $ D_* $ be $ \Zz\pi $-chain complexes. Show that for the $ n $-th homology of the complex $ hom_{\Zz\pi}(C_*,D_*) $ we have | ||
+ | $$ H_n(hom_{\Zz\pi}(C_*,D_*))= [ \Sigma^n C_*,D_*]_{\Zz\pi} $$ | ||
+ | where $ (\sum^nC_*)_k:=C_{k-n} $ denotes the shifted complex with boundary map $ d_{\sum C}:=(-1)^nd_C $. | ||
+ | {{endthm}} | ||
+ | |||
+ | '''Hint''': use the boundary map of Exercise \ref{chain_map}. | ||
+ | |||
+ | {{beginthm|Exercise}} | ||
+ | Deduce from the Poincare homotopy equivalence $ ? \cap [X]:C^{n-*}(\widetilde X) \rightarrow C_*(\widetilde X) $ that | ||
+ | $$ H_k(X,\Zz^{\omega}) \cong H^{n-k}(X,\Zz):=H_k(C^{n-*}(X)) $$ | ||
+ | as $ \Zz $-Modules. | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Exercise}} | ||
+ | Let $(X,A)$ be a Poincare pair with $X$ an $n$-dimensional, connected, finite CW-complex and $A\subset X$ an $(n-1)$-dimensional subcomplex, an orientation homomorphism $\omega_X : \pi \longrightarrow \{\pm 1\}$ and a fundamental class $[X,A]\in H_n(X,A;\Zz^{\omega_X}):=H_n(\Zz^{\omega_X}\otimes_{\Zz\pi}C_*(\tilde{X},\tilde{A}))$. That means, for a universal covering $p:\tilde{X}\longrightarrow X$ and $\tilde{A}:=p^{-1}(A)$ the $\Zz\pi$-chain maps $?\cap [X,A]:C^{n-*}(\tilde{X},\tilde{A})\longrightarrow C_*(\tilde{X})$ and $?\cap [X,A]:C^{n-*}(\tilde{X})\longrightarrow C_*(\tilde{X},\tilde{A})$ are $\Zz\pi$-chain homotopy equivalences. | ||
+ | |||
+ | Show that the components $C\in\pi_0(A)$ of $A$ inherit the structure of a finite $(n-1)$-dimensional Poincare complex, i.e. that there is an induced orientation homomorphism $\omega_C : \pi_1(C) \longrightarrow \{\pm 1\}$ and an induced fundamental class $[C]\in H_{n-1}(C;\Zz^{\omega_C})$, such that | ||
+ | $$ ?\cap [C]:C^{n-1-*}(\tilde{C})\longrightarrow C_*(\tilde{C})$$ | ||
+ | is a $\Zz\pi$-chain homotopy equivalence. | ||
+ | {{endthm}} | ||
+ | |||
+ | '''Hint''': Tensorize both sides with $ \Zz^{\omega} $ and consider the induced map. | ||
+ | |||
+ | The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling. | ||
+ | </wikitex> | ||
+ | <!-- == References == | ||
+ | {{#RefList:}} --> | ||
[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises without solution]] |
Latest revision as of 14:57, 1 April 2012
Exercise 0.1. Let and be -chain complexes and
be defined by sending to the map
Show that is a -chain map.
Hint: Note that with boundary map and that with boundary map .
Exercise 0.2. Let and be -chain complexes. Show that for the -th homology of the complex we have
where denotes the shifted complex with boundary map .
Hint: use the boundary map of Exercise 0.1.
Exercise 0.3. Deduce from the Poincare homotopy equivalence that
as -Modules.
Exercise 0.4. Let be a Poincare pair with an -dimensional, connected, finite CW-complex and an -dimensional subcomplex, an orientation homomorphism and a fundamental class . That means, for a universal covering and the -chain maps and are -chain homotopy equivalences.
Show that the components of inherit the structure of a finite -dimensional Poincare complex, i.e. that there is an induced orientation homomorphism and an induced fundamental class , such that
is a -chain homotopy equivalence.
Hint: Tensorize both sides with and consider the induced map.
The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.