Algebraic criteria for stable diffeomorphism of spin 4-manifolds

[2006.06127] Algebraic criteria for stable diffeomorphism of spin 4-manifolds Skip to main content Search Submit Donate Log in Mathematics > Geometric Topology arXiv:2006.06127 (math) [Submitted on 11 Jun 2020 ( v1 ), last revised 6 Jun 2024 (this version, v4)] Title: Algebraic criteria for stable diffeomorphism of spin 4-manifolds Authors: Daniel Kasprowski , Mark Powell , Peter Teichner View PDF HTML (experimental) Abstract: We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects $\mathbb{CP}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors. In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $\tau$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z} \times \mathbb{Z}/2$. Comments: 102 pages. Version 2: Some results on the Kervaire-Milnor invariant have been extracted to create arXiv:2105.12153 . A new Chapter 7 gives an application of our theory. Version 3: Changes following a referee report. Accepted for publication in Memoirs of the American Mathematical Society Subjects: Geometric Topology (math.GT) ; Algebraic Topology (math.AT) MSC classes: 57K40 Report number: MPIM-Bonn-2021 Cite as: arXiv:2006.06127 [math.GT] (or arXiv:2006.06127v4 [math.GT] for this version) https://doi.org/10.48550/arXiv.2006.06127 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Mem. Amer. Math. Soc. 312 (2025), no. 1579 Related DOI : https://doi.org/10.1090/memo/1579 Focus to learn more DOI(s) linking to related resources Submission history From: Daniel Kasprowski [ view email ] [v1] Thu, 11 Jun 2020 00:31:35 UTC (393 KB) [v2] Sat, 29 May 2021 13:25:10 UTC (86 KB) [v3] Tue, 19 Oct 2021 22:47:40 UTC (86 KB) [v4] Thu, 6 Jun 2024 11:28:09 UTC (89 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GT < prev | next > new | recent | 2020-06 Change to browse by: math math.AT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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