If you are clever you can sometimes use homotopy invariants to give you homeomorphic information. The answer of @anomaly alludes to one example, but here is an even simpler example: a proof that if $\mathbb R^m$ is homeomorphic to $\mathbb R^n$ then $m=n$.

Of course $\mathbb R^m$ and $\mathbb R^n$ are homotopy equivalent, in fact both are contractible, so that does not help.

Instead we make special use of relative homology, which is a homotopy invariant in the category of topological pairs.

Suppose that there did exist a homeomorphism $f : \mathbb R^m \to \mathbb R^n$. Choosing any $p \in \mathbb R^m$, and letting $q=f(p)$, we obtain a homeomorphism $f : (\mathbb R^m,\mathbb R^m - \{p\}) \to (\mathbb R^n,\mathbb R^n - \{q\})$ in the category of pairs. Therefore, $f$ is also a homotopy equivalence in the category of pairs, and so we obtain an induced isomorphism of relative homology groups (i.e. an isomorphism of "local homology groups"): $$f_* : H_i(\mathbb R^m,\mathbb R^m - \{p\}) \to H_i(\mathbb R^n,\mathbb R^n - \{q\}) $$ It follows that $m=n$, because of the following calculation: for any point $a \in \mathbb R^j$ we have $$H_i(\mathbb R^j,\mathbb R^j - \{a\}) = \begin{cases} \mathbb Z \quad\text{if $i=j$} \\ 0 \quad\text{if $i \ne j$} \end{cases} $$

If you are clever you can sometimes use homotopy theory to give you complete homeomorphic information. The answer of @anomaly alludes to one example, but here is an even simpler example: a proof that if $\mathbb R^m$ is homeomorphic to $\mathbb R^n$ then $m=n$.

Of course $\mathbb R^m$ and $\mathbb R^n$ are homotopy equivalent, in fact both are contractible, so that does not help.

Instead we make special use of relative homology, which is a homotopy invariant in the category of topological pairs.

Suppose that there did exist a homeomorphism $f : \mathbb R^m \to \mathbb R^n$. Choosing any $p \in \mathbb R^m$, and letting $q=f(p)$, we obtain a homeomorphism $f : (\mathbb R^m,\mathbb R^m - \{p\}) \to (\mathbb R^n,\mathbb R^n - \{q\})$ in the category of pairs. Therefore, $f$ is also a homotopy equivalence in the category of pairs. Relative homology is a homotopy functor of the category of pairs, and so we obtain an induced isomorphism of relative homology groups (i.e. an isomorphism of "local homology groups"): $$f_* : H_i(\mathbb R^m,\mathbb R^m - \{p\}) \to H_i(\mathbb R^n,\mathbb R^n - \{q\}) $$ It follows that $m=n$, because of the following calculation: for any point $a \in \mathbb R^j$ we have $$H_i(\mathbb R^j,\mathbb R^j - \{a\}) = \begin{cases} \mathbb Z \quad\text{if $i=j$} \\ 0 \quad\text{if $i \ne j$} \end{cases} $$