This question already has an answer here:
If $\mathrm{GL}_n(\mathbb{R})$ is homeomorphic to $\mathrm{GL}_m(\mathbb{R})$, then (using that these are $n^2$-dimensional resp. $m^2$-dimensional manifolds) it follows that there is a homeomorphism between $\mathbb{R}^{n^2}$ and $\mathbb{R}^{m^2}$. It is well-known that this implies $n=m$ and there a lot of proofs for this, also without homology. See SE/24873 and arXiv:1310.8090.