Algebraic Topology · Problem 2.2.28

Homology — Allen Hatcher

Mathematics — Topology · mayer-vietoris-sequence

Problem

(a) Use the Mayer--Vietoris sequence to compute the homology groups of the space obtained from a torus

S^1 \times S^1

by attaching a M\"obius band via a homeomorphism from the boundary circle of the M\"obius band to the circle

S^1 \times \{x_0\}

in the torus. (b) Do the same for the space obtained by attaching a M\"obius band to

\mathbb{RP}^2

via a homeomorphism of its boundary circle to the standard

\mathbb{RP}^1 \subset \mathbb{RP}^2