Poincaré duality IV (Ex)
m |
m (changed \sum to \Sigma for the suspension of a chain complex) |
||
Line 13: | Line 13: | ||
{{beginthm|Exercise}} \label{homol_hom} | {{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 | 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 | ||
− | $$ H_n(hom_{\Zz\pi}(C_*,D_*))= [ \ | + | $$ 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 $. | where $ (\sum^nC_*)_k:=C_{k-n} $ denotes the shifted complex with boundary map $ d_{\sum C}:=(-1)^nd_C $. | ||
{{endthm}} | {{endthm}} |
Revision as of 21:51, 26 March 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.