Poincaré duality IV (Ex)
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
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.