**Hatcher Exercise 2.2.29**

The surface of genus (i.e. a connect sum of tori), when embedded in , bounds a compact solid region . Let be two copies of this space with their boundaries (the actual surface) identified via the identity map, giving a three-manifold. We calculate the homology. Let be one region with an open neighborhood into the other, and likewise for . The region bounded by deformation retracts onto its core – a wedge of circles. So, and deformation retract onto a wedge of circles. deformation retracts onto itself. So , and (and others zero). Meyer-Vietoris gives:

First note .
Now the details of the map .
is generated by the and in the familiar representation as a polygon, i.e. .
has the core circles as generators in both summands, so write it as .
For each ‘sub-torus’ of the genus surface, the deformation retraction takes one of or *to* the core circle (in both regions), and contracts the other onto a single point.
Say the ’s are mapped onto the core circle.
With these assumptions, we have and .
Firstly, we have .
Secondly, we compute .
We have: