Hatcher-Algebraic-Topology · Problem 2.3.2

Homology

Mathematics — Graph Theory · homology-theory-axioms

Problem

Define a candidate for a reduced homology theory on CW complexes by h~n(X)=∏iH~i(X)/⨁iH~i(X)\tilde{h}_n(X) = \prod_i \widetilde{H}_i(X) \big/ \bigoplus_i \widetilde{H}_i(X). Thus h~n(X)\tilde{h}_n(X) is independent of nn and is zero if XX is finite-dimensional, but is not identically zero, for example for X=⋁iSiX = \bigvee_i S^i. Show that the axioms for a homology theory are satisfied except that the wedge axiom fails.

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