Algebraic & Geometric Topology Volume 26, issue 2 (2026)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

$\mathrm{KSp}$-characteristic classes determine $\mathrm{Spin}^h$ cobordism

Jonathan Buchanan and Stephen McKean

Algebraic & Geometric Topology 26 (2026) 485–551

DOI: 10.2140/agt.2026.26.485

Bibliography

1	J F Adams, Stable homotopy and generalised homology, University of Chicago Press (1974) MR402720
2	M Albanese, A Milivojević, Spin^h and further generalisations of spin, J. Geom. Phys. 164 (2021) 104174 MR4226400
3	D W Anderson, E H Brown Jr., F P Peterson, SU-cobordism, KO-characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966) 54 MR189043
4	D W Anderson, E H Brown Jr., F P Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967) 271 MR219077
5	D W Anderson, E H Brown Jr., F P Peterson, Pin cobordism and related topics, Comment. Math. Helv. 44 (1969) 462 MR261613
6	M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3 MR167985
7	A Back, P G Freund, M Forger, New gravitational instantons and universal spin structures, Phys. Lett. B 77 (1978) 181
8	A Bahri, P Gilkey, The eta invariant, Pin^c bordism, and equivariant Spin^c bordism for cyclic 2-groups, Pacific J. Math. 128 (1987) 1 MR883375
9	C Bär, Elliptic symbols, Math. Nachr. 201 (1999) 7 MR1680857
10	A Beaudry, J A Campbell, A guide for computing stable homotopy groups, from: "Topology and quantum theory in interaction" (editors D Ayala, D S Freed, R E Grady), Contemp. Math. 718, Amer. Math. Soc. (2018) 89 MR3869642
11	A Borel, F Hirzebruch, Characteristic classes and homogeneous spaces, II, Amer. J. Math. 81 (1959) 315 MR110105
12	E H Brown Jr., The cohomology of BSO_n and BO_n with integer coefficients, Proc. Amer. Math. Soc. 85 (1982) 283 MR652459
13	J Buchanan, mspinh, GitHub repository (2023)
14	U Bunke, N Naumann, Secondary invariants for string bordism and topological modular forms, Bull. Sci. Math. 138 (2014) 912 MR3278648
15	H Cartan, Détermination des algèbres H_∗(π,n;Z), from: "Séminaire Henri Cartan, 1954/1955", Secrét. Math. (1955)
16	X Chen, Bundles of irreducible Clifford modules and the existence of spin structures, PhD thesis, State University of New York at Stony Brook (2017) MR3705968
17	A Debray, How to calculate Spin-U₂ cobordism, lecture notes (2021)
18	A Debray, C Krulewski, Smith homomorphisms and spin^h structures, Proc. Amer. Math. Soc. 153 (2025) 897 MR4852808
19	A Dold, Lectures on algebraic topology, Band 200, Springer (1972) MR415602
20	H Duan, The characteristic classes and Weyl invariants of Spinor groups, preprint (2018) arXiv:1810.03799
21	D S Freed, M J Hopkins, Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165 MR4268163
22	H Gupta, An asymptotic formula in partitions, J. Indian Math. Soc. (N.S.) 10 (1946) 73 MR22867
23	G H Hardy, S Ramanujan, Asymptotic formulæ in combinatory analysis, from: "Collected papers of Srinivasa Ramanujan" (editors G H Hardy, P V Seshu Aiyar, B M Wilson), AMS Chelsea (2000) 276 MR2280879
24	G H Hardy, S Ramanujan, Une formule asymptotique pour le nombre des partitions de n, from: "Collected papers of Srinivasa Ramanujan" (editors G H Hardy, P V Seshu Aiyar, B M Wilson), AMS Chelsea (2000) 239 MR2280874
25	A Hatcher, Spectral sequences, book project (2004)
26	S W Hawking, C N Pope, Generalized spin structures in quantum gravity, Phys. Lett. B 73 (1978) 42 MR496137
27	M J Hopkins, M A Hovey, Spin cobordism determines real K-theory, Math. Z. 210 (1992) 181 MR1166518
28	J Hu, Quaternionic Clifford modules, spin^h manifolds, and symplectic K-theory, notes (2022)
29	J Hu, Invariants of real vector bundles, preprint (2023) arXiv:2310.05061
30	M Joachim, Higher coherences for equivariant K-theory, from: "Structured ring spectra" (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 87 MR2122155
31	R C Kirby, L R Taylor, A calculation of Pin⁺ bordism groups, Comment. Math. Helv. 65 (1990) 434 MR1069818
32	G Laures, An E_∞ splitting of spin bordism, Amer. J. Math. 125 (2003) 977 MR2004426
33	H B Lawson Jr., Spin^h manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 19 (2023) 012 MR4563424
34	H B Lawson Jr., M L Michelsohn, Spin geometry, 38, Princeton Univ. Press (1989) MR1031992
35	C I Lazaroiu, C S Shahbazi, Real pinor bundles and real Lipschitz structures, Asian J. Math. 23 (2019) 749 MR4095184
36	H R Margolis, Spectra and the Steenrod algebra : modules over the Steenrod algebra and the stable homotopy category, 29, North-Holland (1983) MR738973
37	K Mills, The structure of the Spin^h bordism spectrum, preprint (2023) arXiv:2306.17709
38	J Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958) 150 MR99653
39	J Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963) 198 MR157388
40	J W Milnor, J D Stasheff, Characteristic classes, 76, Princeton Univ. Press (1974) MR440554
41	M Nagase, Spin^q structures, J. Math. Soc. Japan 47 (1995) 93 MR1304190
42	C Okonek, A Teleman, Quaternionic monopoles, Comm. Math. Phys. 180 (1996) 363 MR1405956
43	H Shapourian, K Shiozaki, S Ryu, Many-body topological invariants for fermionic symmetry-protected topological phases, Phys. Rev. Lett. 118 (2017) 216402 MR3693816
44	R E Stong, Relations among characteristic numbers, II, Topology 5 (1966) 133 MR192516
45	R E Stong, Notes on cobordism theory, Princeton Univ. Press (1968) MR248858
46	N Strickland, Bott periodicity and Hopf rings, PhD thesis, University of Manchester (1992)
47	R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17 MR61823
48	C T C Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960) 292 MR120654