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 \(\nu _N: \mathcal {R}^+ (M) \rightarrow \mathcal {R}^+ (M \times N)\) , 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 \(\dim (N) \ge 3\) . More generally, we give easily checkable conditions on diffeomorphisms \(f:M \rightarrow M\) of 1-connected 4-manifolds which guarantee that for each closed N and each \(g \in \mathcal {R}^+ (M)\) , the metrics \(\nu _N (g)\) and \(\nu _N (f^* g)\) lie in the same path component of \(\mathcal {R}^+ (M \times N)\) . The conditions are (A) the cobordism class of the mapping torus \([T(f)] \in \Omega _5^\textrm{SO}\) vanishes and (B) the induced map \(f^*: H^2 (M;\mathbb {R}) \rightarrow H^2(M;\mathbb {R})\) belongs to the unit component of \(\textrm{Aut}(I_M)\) , the orthogonal group of the intersection form of M. Depending on whether \(\dim (N)=1\) or \(\dim (N) \ge 2\) and on whether or not M is spin, various combinations of (A) and (B) are needed. The proof uses rigidity properties for the diffeomorphism action on \(\mathcal {R}^+ (M)\) for high–dimensional M, which in the necessary generality were established by Frenck and Bantje. These rigidity properties reduce the problem to a calculation of the homotopy group \(\pi _1 (\textrm{MTSO}(4))\) of the Madsen–Tillmann spectrum which 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.
