# 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.

$\square$