I would like to know if there is a closed form formula for the homotopy type of $\widehat{(\mathbb{C^{\ast}})^n}$? For example, it is not difficult to see that $\widehat{\mathbb{C^{\ast}}}$ has the homotopy type of $S^1\vee S^2$.

My guess is that the formula, for general $n$, should look something like this $$ \bigvee_{i=0}^n \left(\bigvee_{\binom{n}{i}} S^{n+i}\right). $$ The hunch is based on the fact that the compactly supported cohomology of the $n$-torus has the homology of the above wedge of spheres. However, I am not sure how I can prove this.

Note: I did post the same question on Math StackExchange; here is the link.