Showing that two spaces are homotopy equivalent

Question

Let $x_0 \in S^1 \times S^1$. I want to show that $(S^1 \times S^1) - \{x_0\}$ and $S^1 \vee S^1$ are homotopy equivalent.

We have to show that $\exists$ maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f \simeq id_x$ and $f \circ g \simeq id_y$ (with $X = (S^1 \times S^1) - \{x_0\}$ and $Y = S^1 \vee S^1 = S^1 \amalg S^1$).

I want to find continuous functions $f$ and $g$ but cannot think of any. I was just wondering if somebody could give me a hint...

Answer 1

Think of $S^1 \times S^1$ as a square with opposite sides identified (this is precisely what you do when considering a torus), then remove a point inside it. It is now easy to construct a deformation retraction onto its boundary, which is $S^1 \vee S^1$.