A specific cell complex

Question

We construct a cell complex by attaching the boundary of a two dimensional disk $D$ to $S^1$ by $z\to z^n$ ($n>2$). This cell complex seems to be closed, compact and connected. But it's fundamental group $\mathbb{Z}_n$ is different from the fundamental group of any surface in the classification of closed, compact surfaces (a 2-sphere, a connected sum of tori, or a connected some of projective planes). Unfortunately I cannot tell where I'm making a mistake. Any help would be appreciated.

Answer 1

The answer is simple: if $n > 2$ then this cell complex $X$ is not a surface.

In fact, if $x \in X$ is any point of the 1-skeleton of $X$ then the local homology group $H_1(X,X-x)$ is free of rank $n-1 \ge 2$.

On the other hand if $X$ is a surface and if $x \in S$ then $H_1(X,X-x)$ is free of rank $1$.

Of course, these local homology groups are invariants of the pair $(X,x)$ up to homeomorphism.