Proof of the Brouwer fixed point theorem:

Assume that there does exist a map from with no fixed point. Then let be the following map: Start at , draw the ray going through and then let be the first intersection of that line with the sphere. This map is continuous and well defined only because fixes no point. Also, it is not hard to see that it must be the identity on the boundary sphere. Thus we have a map , which is the identity on , that is, a retraction. Now, if is the inclusion map, . Applying the reduced homology functor, we find that , where indicates the induced map on homology.

But, it is a well-known fact that (since is contractible), and that . Thus we have an isomorphism of a non-zero group onto itself factoring through a trivial group, which is clearly impossible. Thus we have a contradiction, and no such map exists.