Cancellation properties for exotic $4$-dimensional positive scalar curvature metrics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908582868418560 |
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| author | Ebert, Johannes |
| author_facet | Ebert, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cancellation properties for exotic $4$-dimensional positive scalar curvature metrics Ebert, Johannes Algebraic Topology Differential Geometry 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. |
| title | Cancellation properties for exotic $4$-dimensional positive scalar curvature metrics |
| topic | Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2505.16542 |