Simplicial Complex and CW's

Question

I read somewhere that every finite CW complex (together with another family of topological spaces) is a retract of a finite simplicial complex. Before reading that I thought that every CW complex could be given the structure of a simplicial complex. Are there finite CW complexes that cannot be given the structure of a simplicial complex?

Answer 1

Let $X$ be the CW complex with two $0$-cells "$-1$" and "$1$", one $1$-cell $D^1$ connecting both $0$-cells, and one $\text{$2$-cell}$ whose attaching map $\phi:S^1\to D^1$ is $$ (x,y) \mapsto \tfrac12(x+1) \sin\left(\tfrac\pi{x+1}\right) $$ Then $X$ has for every $n\in\mathbb N$ a point $x\in X$ such that the local homology group satisfies $$ H_2(X,X-\{x\})\approx \Bbb Z^{2n+1} $$ If we wish this to hold in a simplicial complex, then we would need for every $n\in\Bbb N$ an edge which is in the boundary of $2n+2$ triangles, which is not possible in a compact and thus finite simplicial complex.