Algebraic Topology · Problem 2.2.30

Homology — Allen Hatcher

Mathematics — Topology · mayer-vietoris-sequence

Problem

For the mapping torus

T_f

of a map

f: X \to X

, we constructed in Example 2.48 a long exact sequence

\cdots \to H_n(X) \xrightarrow{\mathbb{1} - f_*} H_n(X) \to H_n(T_f) \to H_{n-1}(X) \to \cdots

. Use this to compute the homology of the mapping tori of the following maps: (a) A reflection

S^2 \to S^2

. (b) A map

S^2 \to S^2

of degree 2. (c) The map

S^1 \times S^1 \to S^1 \times S^1

that is the identity on one factor and a reflection on the other. (d) The map

S^1 \times S^1 \to S^1 \times S^1

that is a reflection on each factor. (e) The map

S^1 \times S^1 \to S^1 \times S^1

that interchanges the two factors and then reflects one of the factors.