I am interested in the homotopy types of maps $m:T^n\setminus \left\lbrace x_0,x_1, ... x_k \right\rbrace \rightarrow S^2$.
My first try was to figure out how the n-dimensional torus looks like, if n-points are removed.
I know that for the two dimensional torus I get the wedge of $n+1$ circles.
I suppose now, that $T^3 \setminus \left\lbrace x_0 \right\rbrace$ retracts to $\left(S^1 \times S^1 \right)\vee \left( S^1 \times S^1 \right)$. Is this right? And how does it look like for k points?
I would be wery happy about some advice.
Best, w
