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

Andrew Senger; Adela YiYu Zhang

doi:10.1515/crelle-2025-0050

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

Andrew Senger and Adela YiYu Zhang

Published/Copyright: July 31, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 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 In the cases n = 4 , 8 , our results recover unpublished work of Crowley–Nagy and Crowley–Olbermann.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.

References

[1] J. F. Adams, On the groups J ⁢ ( X ) . IV, Topology 5 (1966), 21–71. 10.1016/0040-9383(66)90004-8Search in Google Scholar

[2] D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. 10.2307/1970690Search in Google Scholar

[3] M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins and C. Rezk, Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7 (2014), no. 4, 1077–1117. 10.1112/jtopol/jtu009Search in Google Scholar

[4] M. Ando, M. J. Hopkins and C. Rezk, Multiplicative orientations of k ⁢ o -theory and of the spectrum of topological modular forms, unpublished preprint (2010). Search in Google Scholar

[5] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), 3–38. 10.1016/0040-9383(64)90003-5Search in Google Scholar

[6] J. C. Baez, The octonions, Bull. Amer. Math. Soc. (N. S.) 39 (2002), no. 2, 145–205. 10.1090/S0273-0979-01-00934-XSearch in Google Scholar

[7] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. 10.2307/1970702Search in Google Scholar

[8] M. G. Barratt, J. D. S. Jones and M. E. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension 62, J. Lond. Math. Soc. (2) 30 (1984), no. 3, 533–550. 10.1112/jlms/s2-30.3.533Search in Google Scholar

[9] J. Beardsley, Relative Thom spectra via operadic Kan extensions, Algebr. Geom. Topol. 17 (2017), no. 2, 1151–1162. 10.2140/agt.2017.17.1151Search in Google Scholar

[10] M. Behrens, M. Hill, M. J. Hopkins and M. Mahowald, On the existence of a v 2 32 -self map on M ⁢ ( 1 , 4 ) at the prime 2, Homology Homotopy Appl. 10 (2008), no. 3, 45–84. 10.4310/HHA.2008.v10.n3.a4Search in Google Scholar

[11] T. Bier and N. Ray, Detecting framed manifolds in the 8 and 16 stems, Geometric applications of homotopy theory I (Evanston 1977), Lecture Notes in Math. 657, Springer, Berlin (1978), 32–39. 10.1007/BFb0069226Search in Google Scholar

[12] J. Bowden, D. Crowley, J. Davis, S. Friedl, C. Rovi and S. Tillmann, Open problems in the topology of manifolds, 2019–20 MATRIX annals, MATRIX Book Ser. 4, Springer, Cham (2021), 647–659. 10.1007/978-3-030-62497-2_39Search in Google Scholar

[13] W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. 10.2307/1970686Search in Google Scholar

[14] E. H. Brown, Jr. and F. P. Peterson, The Kervaire invariant of ( 8 ⁢ k + 2 ) -manifolds, Amer. J. Math. 88 (1966), 815–826. 10.2307/2373080Search in Google Scholar

[15] R. Bruner, ext program, available at http://www.rrb.wayne.edu/papers/#code. Search in Google Scholar

[16] R. R. Bruner and J. Rognes, The Adams spectral sequence for topological modular forms, Math. Surveys Monogr. 253, American Mathematical Society, Providence 2021. 10.1090/surv/253Search in Google Scholar

[17] R. Burklund, J. Hahn and A. Senger, Inertia groups in the metastable range, preprint (2020), https://arxiv.org/abs/2010.09869; to appear in Amer. J. Math. Search in Google Scholar

[18] R. Burklund, J. Hahn and A. Senger, On the boundaries of highly connected, almost closed manifolds, Acta Math. 231 (2023), no. 2, 205–344. 10.4310/ACTA.2023.v231.n2.a1Search in Google Scholar

[19] R. Burklund and A. Senger, On the high-dimensional geography problem, Geom. Topol. 28 (2024), no. 9, 4257–4293. 10.2140/gt.2024.28.4257Search in Google Scholar

[20] K. Chang, A v 1 -banded vanishing line for the m ⁢ o ⁢ d ⁢ 2 Moore spectrum, Homology Homotopy Appl. 27 (2025), no. 1, 29–49. 10.4310/HHA.2025.v27.n1.a3Search in Google Scholar

[21] D. Crowley, The smooth structure set of S p × S q , Geom. Dedicata 148 (2010), 15–33. 10.1007/s10711-010-9513-8Search in Google Scholar

[22] D. Crowley and C. M. Escher, A classification of S 3 -bundles over S 4 , Differential Geom. Appl. 18 (2003), no. 3, 363–380. 10.1016/S0926-2245(03)00012-3Search in Google Scholar

[23] D. Crowley and C. Nagy, Inertia groups of 3-connected 8-manifolds, (2020), http://www.dcrowley.net/Crowley-Nagy-3c8m-b.pdf. Search in Google Scholar

[24] D. Crowley and J. Nordström, The classification of 2-connected 7-manifolds, Proc. Lond. Math. Soc. (3) 119 (2019), no. 1, 1–54. 10.1112/plms.12222Search in Google Scholar

[25] D. J. Crowley, The classification of highly connected manifolds in dimensions 7 and 15, Ph.D. Thesis, Indiana University, 2002, Search in Google Scholar

[26] D. M. Davis and M. Mahowald, The image of the stable 𝐽-homomorphism, Topology 28 (1989), no. 1, 39–58. 10.1016/0040-9383(89)90031-1Search in Google Scholar

[27] D. Frank, On Wall’s classification of highly-connected manifolds, Topology 13 (1974), 1–8. 10.1016/0040-9383(74)90033-0Search in Google Scholar

[28] D. L. Frank, An invariant for almost-closed manifolds, Bull. Amer. Math. Soc. 74 (1968), 562–567. 10.1090/S0002-9904-1968-12010-5Search in Google Scholar

[29] V. Giambalvo, On ⟨ 8 ⟩ -cobordism, Illinois J. Math. 15 (1971), 533–541. 10.1215/ijm/1256052508Search in Google Scholar

[30] V. Giambalvo, Correction to my paper: “On < 8 > -cobordism”, Illinois J. Math. 16 (1972), 704–704. 10.1215/ijm/1256065555Search in Google Scholar

[31] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016), no. 1, 1–262. 10.4007/annals.2016.184.1.1Search in Google Scholar

[32] P. J. Hilton, A note on the 𝑃-homomorphism in homotopy groups of spheres, Proc. Cambridge Philos. Soc. 51 (1955), 230–233. 10.1017/S0305004100030085Search in Google Scholar

[33] D. Isaksen, G. Wang and Z. Xu, Classical and ℂ-motivic Adams charts, (2020), https://cpb-us-e1.wpmucdn.com/s.wayne.edu/dist/0/60/files/2020/04/Adamscharts.pdf. Search in Google Scholar

[34] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. 10.2307/1970128Search in Google Scholar

[35] A. Kosiński, On the inertia group of 𝜋-manifolds, Amer. J. Math. 89 (1967), 227–248. 10.2307/2373121Search in Google Scholar

[36] L. Kramer and S. Stolz, A diffeomorphism classification of manifolds which are like projective planes, J. Differential Geom. 77 (2007), no. 2, 177–188. 10.4310/jdg/1191860392Search in Google Scholar

[37] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no. 3, 707–754. 10.2307/121071Search in Google Scholar

[38] N. J. Kuhn, Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces, Adv. Math. 201 (2006), no. 2, 318–378. 10.1016/j.aim.2005.02.005Search in Google Scholar

[39] R. Lampe, Diffeomorphismen auf Sphären und die Milnor-Paarung, Thesis, Johannes Gutenberg University Mainz, 1981. Search in Google Scholar

[40] J. P. Levine, Lectures on groups of homotopy spheres, Algebraic and geometric topology (New Brunswick 1983), Lecture Notes in Math. 1126, Springer, Berlin (1985), 62–95. 10.1007/BFb0074439Search in Google Scholar

[41] M. Mahowald, Description homotopy of the elements in the image of the 𝐽-homomorphism, Proceedings of the International Conference on Manifolds and Related Topics in Topology. Mathematical Society of Japan, Tokyo (1975), 255–263. Search in Google Scholar

[42] M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349–369. 10.1016/0040-9383(67)90023-7Search in Google Scholar

[43] A. Mathew and V. Stojanoska, The Picard group of topological modular forms via descent theory, Geom. Topol. 20 (2016), no. 6, 3133–3217. 10.2140/gt.2016.20.3133Search in Google Scholar

[44] J. P. May, A general algebraic approach to Steenrod operations, The Steenrod algebra and its applications (Columbus 1970), Lecture Notes in Math. 168, Springer, Berlin (1970), 153–231. 10.1007/BFb0058524Search in Google Scholar

[45] J. P. May and R. J. Milgram, The Bockstein and the Adams spectral sequences, Proc. Amer. Math. Soc. 83 (1981), no. 1, 128–130. 10.1090/S0002-9939-1981-0619997-8Search in Google Scholar

[46] J. P. May and J. Sigurdsson, Parametrized homotopy theory, Math. Surveys Monogr. 132, American Mathematical Society, Providence 2006. 10.1090/surv/132Search in Google Scholar

[47] R. M. F. Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970), 283–310. 10.1007/BF01129978Search in Google Scholar

[48] J. R. Munkres, Concordance inertia groups, Adv. Math. 4 (1970), 224–235. 10.1016/0001-8708(70)90024-1Search in Google Scholar

[49] C. Nagy, The classification of 8-dimensional E-manifolds, Ph.D. Thesis, The University of Melbourne, 2020. Search in Google Scholar

[50] C. Nagy, Extended surgery theory for simply-connected 4 ⁢ k -manifolds, preprint (2024), https://arxiv.org/abs/2402.13394. Search in Google Scholar

[51] S. P. Novikov, Homotopy properties of the group of diffeomorphisms of the sphere, Dokl. Akad. Nauk SSSR 148 (1963), 32–35. Search in Google Scholar

[52] P. Pstragowski, Synthetic spectra and the cellular motivic category, Invent. Math. 232 (2023), no. 2, 553–681. 10.1007/s00222-022-01173-2Search in Google Scholar

[53] K. Ramesh, Inertia groups and smooth structures of ( n − 1 ) -connected 2 ⁢ n -manifolds, Osaka J. Math. 53 (2016), no. 2, 309–319. Search in Google Scholar

[54] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure Appl. Math. 121, Academic Press, Orlando 1986. Search in Google Scholar

[55] R. Schultz, Composition constructions on diffeomorphisms of S p × S q , Pacific J. Math. 42 (1972), 739–754. 10.2140/pjm.1972.42.739Search in Google Scholar

[56] J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–MacLane, Comment. Math. Helv. 27 (1953), 198–232. 10.1007/BF02564562Search in Google Scholar

[57] S. Stolz, Hochzusammenhängende Mannigfaltigkeiten und ihre Ränder, Lecture Notes in Math. 1116, Springer, Berlin 1985. 10.1007/BFb0101604Search in Google Scholar

[58] S. Stolz, A note on the b ⁢ P -component of ( 4 ⁢ n − 1 ) -dimensional homotopy spheres, Proc. Amer. Math. Soc. 99 (1987), no. 3, 581–584. 10.1090/S0002-9939-1987-0875404-7Search in Google Scholar

[59] R. E. Stong, Determination of H ∗ ⁢ ( BO ⁢ ( k , … , ∞ ) , Z 2 ) and H ∗ ⁢ ( BU ⁢ ( k , … , ∞ ) , Z 2 ) , Trans. Amer. Math. Soc. 107 (1963), 526–544. Search in Google Scholar

[60] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Stud. 49, Princeton University, Princeton 1962. 10.1515/9781400882625Search in Google Scholar

[61] C. T. C. Wall, Classification of ( n − 1 ) -connected 2 ⁢ n -manifolds, Ann. of Math. (2) 75 (1962), 163–189. 10.2307/1970425Search in Google Scholar

[62] C. T. C. Wall, The action of Γ 2 ⁢ n on ( n − 1 ) -connected 2 ⁢ n -manifolds, Proc. Amer. Math. Soc. 13 (1962), 943–944. 10.1090/S0002-9939-1962-0143223-8Search in Google Scholar

[63] C. T. C. Wall, Classification problems in differential topology. VI. Classification of ( s − 1 ) -connected ( 2 ⁢ s + 1 ) -manifolds, Topology 6 (1967), 273–296. 10.1016/0040-9383(67)90020-1Search in Google Scholar

[64] D. L. Wilkens, Closed ( s − 1 ) -connected ( 2 ⁢ s + 1 ) -manifolds, Ph.D. Thesis, University of Liverpool, 1971. Search in Google Scholar

[65] D. L. Wilkens, Closed ( s − 1 ) -connected ( 2 ⁢ s + 1 ) -manifolds, s = 3 , 7 , Bull. Lond. Math. Soc. 4 (1972), 27–31. 10.1112/blms/4.1.27Search in Google Scholar

Received: 2023-12-06

Revised: 2025-05-29

Published Online: 2025-07-31

Published in Print: 2025-11-01

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