Inertia groups of 
                  
                     
                        
                           (
                           
                              n
                              −
                              1
                           
                           )
                        
                     
                     
                     
                  
               -connected 
                  
                     
                        
                           2
                           ⁢
                           n
                        
                     
                     
                     
                  
               -manifolds

Andrew Senger; Adela YiYu Zhang

doi:10.1515/crelle-2025-0050

Artikel

Inertia groups of ( n − 1 ) -connected 2 ⁢ n -manifolds

Andrew Senger und Adela YiYu Zhang

Veröffentlicht/Copyright: 31. Juli 2025

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2025 Heft 828

Abstract

In this paper, we compute the inertia groups of ( n − 1 ) -connected, smooth, closed, oriented 2 ⁢ n -manifolds, where n ≥ 3 . As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the 126-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for n ≠ 4 , 8 , 9 ; for n ≫ 0 , this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When n = 4 , 8 , 9 , we apply Kreck’s modified surgery and a special case of Crowley’s 𝑄-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases n = 4 , 8 , our results recover unpublished work of Crowley–Nagy and Crowley–Olbermann. In contrast, we show that the homotopy and concordance inertia groups of ( n − 1 ) -connected, smooth, closed, oriented 2 ⁢ n -manifolds with n ≥ 3 always vanish.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2103236

Award Identifier / Grant number: DMS-1906072

Funding source: Danmarks Grundforskningsfond

Award Identifier / Grant number: DRNF151

Funding source: H2020 Marie Skłodowska-Curie Actions

Award Identifier / Grant number: 101150469

Funding statement: During the course of this work, the first named author was partially supported by NSF grant DMS-2103236, and the second named author was partially supported by NSF grant DMS-1906072, the DNRF through the Copenhagen Centre for Geometry and Topology (DRNF151), and the European Union via the Marie Skłodowska-Curie postdoctoral fellowship (project 101150469).

Acknowledgements

We thank Robert Burklund and Sanath Devalapurkar for helpful discussions regarding the contents of this paper. We are particularly grateful to Diarmuid Crowley for assistance with the application of modified surgery and the 𝑄-form conjecture in Section 7. We would also like to thank the referee for their careful reading of the paper and their many helpful comments.

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Received: 2023-12-06

Revised: 2025-05-29

Published Online: 2025-07-31

Published in Print: 2025-11-01

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https://doi.org/10.1515/crelle-2025-0050