I'm going through some applications of Poincaré duality for de Rham cohomology in Greub's Connections, Curvature and Cohomology. I was wondering about the homotopy invariance of the signature of a compact oriented $4k$-dimensional manifold and found this old question. In this case the signature $\mathrm{sig}(M)$ of $M$ is defined as $\mathrm{sig}(M):=\mathrm{sig}(P_{M}^{2k})$ where the non-degenerate symmetric bilinear form $$P_{M}^{2k}:H^{2k}(M)\times H^{2k}(M)\to \mathbb{R}$$ is given by $$([\alpha],[\beta])\mapsto \int_{M}\alpha\wedge\beta$$
I cannot seem to understand why the signature is a homotopy invariant, as stated in the first answer of the aforementioned question.
Any help would be appreciated.
The definition you've given for the signature is phrased in terms of de Rham cohomology and is indeed not obviously homotopy invariant. However, the pairing you've given has a "lift" to singular cohomology: $([\alpha],[\beta]) \rightarrow \langle [\alpha] \cup [\beta], [M] \rangle$. Up to a sign, homotopy equivalences of compact manifolds always preserve the fundamental class, and so this definition of the bilinear form is a homotopy invariant, up to a sign.