Paper 3, Section II, H

Algebraic Topology
Part II, 2010

Suppose XX is a finite simplicial complex and that H(X)H_{*}(X) is a free abelian group for each value of *. Using the Mayer-Vietoris sequence or otherwise, compute H(S1×X)H_{*}\left(S^{1} \times X\right) in terms of H(X)H_{*}(X). Use your result to compute H(Tn)H_{*}\left(T^{n}\right).

[Note that Tn=S1××S1T^{n}=S^{1} \times \ldots \times S^{1}, where there are nn factors in the product.]