Cancellation properties for exotic $4$-dimensional positive scalar curvature metrics

Ruberman constructed families $\{g_n\mathrm{\mid }n\in \mathbb{N}\}\subset {\mathcal{R}}^{+}(M)$ of metrics of positive scalar curvature on certain $4$-manifolds which are concordant but lie in different path components of ${\mathcal{R}}^{+}(M)$. We prove a cancellation result along the following lines. For each closed manifold $N$, there is a map ${\nu }_N:{\mathcal{R}}^{+}(M)\to {\mathcal{R}}^{+}(M\times N)$, well-defined up to homotopy, that takes the product with $N$. We prove that when $N$ has positive dimension ${\nu }_N$ takes all metrics of Ruberman’s family to the same path component. This is trivial when $N$ has a psc metric and follows from pseudoisotopy theory when $\dim (N)\ge 3$. Our proof is cobordism theoretic in nature and also applies to $\dim (N)=1,2$. The proof relies on rigidity properties for the action of the diffeomorphism group on ${\mathcal{R}}^{+}(L)$ for high-dimensional $N$ and a calculation of ${\pi }_1(\mathrm{M}\mathrm{T}\mathrm{S}\mathrm{O}(4))$ that we also carry out. Recently, Auckly and Ruberman exhibited examples of elements in higher homotopy groups of ${\mathcal{R}}^{+}(M^4)$ for certain $M$. Using the same method, we also prove that these elements lie in the kernel of the induced map $({\nu }_N{)}_{\ast }$ on rational homotopy.