Cancellation properties for exotic 4-dimensional positive scalar curvature metrics

2025/05/22 by Johannes Ebert, Ebert, Johannes
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic Topology (math.AT) #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2505.16542

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Ruberman constructed families \g_n\vert n ∈ ℕ\ ⊂ R⁺ (M) of metrics of positive scalar curvature on certain 4-manifolds which are concordant but lie in different path components of R⁺ (M). We prove a cancellation result along the following lines. For each closed manifold N, there is a map ν_N: R⁺ (M) → R⁺ (M × 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) ≥ 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 R⁺(L) for high-dimensional N and a calculation of π₁(MTSO(4)) that we also carry out. Recently, Auckly and Ruberman exhibited examples of elements in higher homotopy groups of R⁺(M⁴) 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.

Citations

• Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
• Pseudo-isotopies of simply connected 4-manifolds
• Concordances in Positive Scalar Curvature and Index Theory
• The positive scalar curvature cobordism category
• Parametrized Morse Theory and Positive Scalar Curvature
• Cobordism Categories and Parametrized Morse Theory
• Index theory in spaces of manifolds
• Abelian quotients of mapping class groups of highly connected manifolds
• A vanishing theorem for characteristic classes of odd-dimensional\n manifold bundles
• The stable moduli space of Riemann surfaces: Mumford's conjecture

Related

• The positive scalar curvature cobordism category
• Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
• Concordances in Positive Scalar Curvature and Index Theory
• Moduli Spaces of Positive Curvature Metrics in Dimension Four and Beyond
• Scalar Curvature and Product Manifolds