CW complex with no cells in dimension $n$

Question

Hi need some help with the following problem:

if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group.

thanx.

Answer 1

The cellular chain complex is composed of sets of the form $H_n(X_n,X_{n-1})$. One can prove using excision and homotopy invariance that $H_n(X_n,X_{n-1}) = H_n(X_n/X_{n-1})$, where this last set is the wedge product of all the $n$ cells: $$ H_n(\bigvee_{\omega:\,\, n-\text{cell}} D_{\omega}^{n}/S_{\omega}^{n-1}) = \bigoplus_{\omega:\,\, n-\text{cell}}H_n(S_{\omega}^{n}) = \bigoplus_{\omega:\,\, n-\text{cell}}G $$ Now, if there are no $n$-cells, then the direct sum is empty and thus the last term equals zero. Therefore $H_n(X;G) = 0$.