$S^2$ and a point are not homotopy equivalent Is there a way to show that $S^2$ does not have the same homotopy type of a point without using homology groups?

If $S^2$ were homotopy equivalent to a point, then every vector bundle over $S^2$ would be trivial.  But, by the hairy ball theorem, the tangent bundle of $S^2$ is nontrivial.
The hairy ball theorem can be proven via Brouwer's fixed point theorem for maps $f:D^2\rightarrow D^2$, which doesn't require the knowledge of homology groups of $S^2$.  
