Hatcher 2.1.22

Hatcher, Algebraic Topology, Chapter 2, Section 1

22. Prove by induction on dimension the following facts about the homology of a finite-dimensional CW complex X, using the observation that X^n/X^{n-1} is a wedge sum of n-spheres:

(a) If X has dimension n then H_i(X)=0 for i>n and H_n(X) is free.

(b) H_n(X) is free with basis in bijective correspondence with the n-cells if there are no cells of dimension n-1 or n+1.

(c) If X has k n-cells, then H_n(X) is generated by at most k elements.
Proof.

\square

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