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

TL;DR

This work establishes a cobordism-based cancellation principle for exotic 4-dimensional positive scalar curvature metrics under stabilization by products and connected sums. It proves that for any closed manifold $N$, the product map $\nu_N: \mathcal{R}^+(M) \to \mathcal{R}^+(M\times N)$ sends Ruberman’s concordant but nonisotopic families to a single path component when $\dim N>0$, with triviality guaranteed if $N$ admits a psc metric; for higher dimensions, pseudoisotopy and cobordism tools are used, including calculations in $\pi_1(\mathrm{MT\,SO}(4))$ and related spectra. The paper develops and employs rigidity results for the diffeomorphism action on the PSC metric space, derives a bordism-based criterion for triviality of the action, and computes related invariants from Madsen–Tillmann theory and family index theorems to control mapping tori. It also extends these ideas to show that certain higher-homotopy elements identified by Auckly–Ruberman lie in the kernel of the rationalized product-map-induced homotopy, reinforcing the stability of these exotic phenomena under stabilization. Overall, the results illuminate how stabilization regularizes exotic 4-manifold psc phenomena and connect geometric questions to cobordism and index-theoretic frameworks with explicit computations in MT-spectra.

Abstract

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 $ν_N: \mathcal{R}^+ (M) \to \mathcal{R}^+ (M \times N)$, well-defined up to homotopy, that takes the product with $N$. We prove that when $N$ has positive dimension $ν_N$ 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 $π_1(\mathrm{MTSO(4)})$ 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 $(ν_N)_*$ on rational homotopy.

Theorems & Definitions (32)

• Theorem A
• Example 1.1
• Theorem 1.2: Ruberman
• Remark 1.5
• Theorem 1.6
• Theorem B
• Theorem 1.9: Auckly--Ruberman
• Theorem C
• proof : Proof of Theorem \ref{['thm:existence-questionstably']}
• Definition 3.2