# 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

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.