Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$.
I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces are equal. How can I show it? maybe using fixed point theorem?
The space $S^1\vee S^1\vee S^2$ is the wedge sum of two circles and one sphere. In particular this space is also known as mouse space.
It is quite easy to see that the fundamental group of $S^1\vee S^1\vee S^2$ is $\mathbb{Z}*\mathbb{Z}\cong\langle a,b| \emptyset \rangle$ ( you can use the Seifert-VanKampen theorem to see it ); whereas the fundamental group of the torus is $\mathbb{Z}\times\mathbb{Z}=\langle a,b | [a,b]=1\rangle$, which is not isomorphic to the previous.
If there was a homotopic equivalence between them; their fundamental groups needs to be isomorphic; but this is not true. So there is not any homotopic equivalence.