Simple homotopy types of even dimensional manifolds

[2312.00322] Simple homotopy types of even dimensional manifolds Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Topology arXiv:2312.00322 (math) [Submitted on 1 Dec 2023 ( v1 ), last revised 10 Apr 2026 (this version, v4)] Title: Simple homotopy types of even dimensional manifolds Authors: Csaba Nagy , John Nicholson , Mark Powell View a PDF of the paper titled Simple homotopy types of even dimensional manifolds, by Csaba Nagy and 2 other authors View PDF HTML (experimental) Abstract: Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map, and the homotopy automorphisms of the manifold. We use this to construct the first examples, for all $n \ge 4$ even, of closed $n$-manifolds that are homotopy equivalent but not simple homotopy equivalent. In fact, we construct infinite families of manifolds that are all homotopy equivalent but pairwise not simple homotopy equivalent, and our examples can be taken to be smooth for $n \geq 6$. Our examples are homotopy equivalent to the product of a circle and a lens space. We analyse the simple homotopy manifold sets of these manifolds, determining exactly when they are trivial, finite, or infinite, and investigating their asymptotic behaviour. The proofs involve integral representation theory and class numbers of cyclotomic fields. We also compare with the relation of $h$-cobordism, and produce similar detailed quantitative descriptions of the manifold sets that arise. Comments: 56 pages. v4. Final version, to appear in Annales Scientifiques de l'École Normale Supérieure Subjects: Algebraic Topology (math.AT) ; Geometric Topology (math.GT); K-Theory and Homology (math.KT); Number Theory (math.NT) MSC classes: 57N65, 57Q10, 19A31, 19B28, 19J10 Cite as: arXiv:2312.00322 [math.AT] (or arXiv:2312.00322v4 [math.AT] for this version) https://doi.org/10.48550/arXiv.2312.00322 Focus to learn more arXiv-issued DOI via DataCite Submission history From: John Nicholson [ view email ] [v1] Fri, 1 Dec 2023 03:36:06 UTC (96 KB) [v2] Wed, 31 Jul 2024 18:29:15 UTC (71 KB) [v3] Fri, 24 Oct 2025 11:56:23 UTC (72 KB) [v4] Fri, 10 Apr 2026 11:52:49 UTC (74 KB) Full-text links: Access Paper: View a PDF of the paper titled Simple homotopy types of even dimensional manifolds, by Csaba Nagy and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AT < prev | next > new | recent | 2023-12 Change to browse by: math math.GT math.KT math.NT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

Related documents

• Simple homotopy types#610666
• Autour de la conjecture standard de type Hodge pour les variétés abéliennes#268741
• Table of mean motion of Mercury anomaly in months of non-leap years by Hadrava (Petr); Hadravová (Alena) (2026) | Type-A (ID n° 1156)#440584
• Logarithmic Lefschetz fixed point formulae and resonant boundary indices#1117528
• “Population” Is Not a Natural Kind of Kinds#753792