Hatcher Exercise 2.2.28


(a) Let be the space obtained from the torus with a Mobius band attached by gluing the its single boundary component to . We compute the homology using Meyer-Vietoris. As usual, let be with a small neighborhood inside , and the Mobius band with a small neighborhood inside . deformation retracts to . We have – let and be the generators, with the edge attached to the boundary of . deformation retracts onto . We have , let represent the single generator (the core circle of the strip). Finally, deformation retracts onto a circle, so and is the generator. The second homology is for and zero for the others, so we have .

since is clearly zero for everything as well. In terms of generators, then:

First, note that , since the boundary of traverses the core circle of twice. So, we have that is injective, and by exactness so we must have . Again by exactness we must have

(b) Now is the space obtained by attaching a Mobius strip to by attaching its boundary to the copy of . Let be defined as above, so that deformation retracts onto and deformation retracts onto . We have – let be the generator (which is attached to the boundary of ). Again and let be its generator (the core circle of ). Finally deformation retracts onto a circle and so with generator . This time the second homology is zero for all parts. So, we have:

First note that . First by exactness we must have . is injective . Secondly, we must have . The image of in the free module is . So we have