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

Ruberman constructed families $\{g_n\vert n \in \mathbb{N}\} \subset \mathcal{R}^+ (M)$ 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 $\dim (N) \geq 3$. 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 $\pi_1(\mathrm{MTSO(4)})$ 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 $(\nu_N)_*$ on rational homotopy.