Abstract
Let M be a smooth, orientable, closed, connected 4-manifold and suppose that H1(M; ℤ) is finitely generated and has no 2-torsion. We give a homotopy decomposition of the suspension of M in terms of spheres, Moore spaces and ΣℂP2. This is used to calculate any reduced generalized cohomology theory of M as a group and to determine the homotopy types of certain current groups and gauge groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 308 (1983), 523–615.
H.-J. Baues and Y. A. Drozd, Classification of sstable homotopy types with torsion-free homology, Topology 40 (2001), 789–821.
H.-J. Baues and M. Hennes, The homotopy classification of (n − 1)-connected (n + 3)-dimensional polyhedral, n ≥ 4, Topology 30 (1991), 373–408.
S. C. Chang, Homotopy invariants and continuous mappings, Proceedings of the Royal Society of London. Series A 202 (1950), 253–263.
I. Etingof and I. B. Frenkel, Central extensions of current groups in two dimensions, Communications in Mathematical Physics 165 (1994), 429–444.
S. Friedl, An introduction to 3-manifolds, https://friedl.app.uni-regensburg.de/papers/muenster.pdf.
D. Gottlieb, Applications of bundle map theory, Transactions of the American Mathematical Society 171 (1972), 23–50.
H. Hamanaka and A. Kono, Unstable K-group and homotopy type of certain gauge groups, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 136 (2006), 149–155.
S. Hasui, D. Kishimoto, T. So and S. Theriault, Odd primary homotopy types of the gauge groups of exceptional Lie groups, Proceedings of the American Mathematical Society 147 (2019), 1751–1762.
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
R. Huang and J. Wu, Cancellation and homotopy rigidity of classic functors, Journal of the London Mathematical Society 99 (2019), 225–248.
R. Kirby, The Topology of 4-Manifolds, Lecture Notes in Mathematics, Vol. 1374, Springer, Berlin, 1989.
D. Kishimoto and A. Kono, On the homotopy types of Sp(n) gauge groups, Algebraic & Geomewtric Topology 19 (2019), 491–502.
D. Kishimoto, S. Theriault and M. Tsutaya, The homotopy types of G2-gauge groups, Topology and its Applications 228 (2017), 92–107.
A. Kono, A note on the homotopy type of certain gauge groups, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 117 (1991), 295–297.
A. Kono and S. Tsukuda, A remark on the homotopy type of certain gauge groups, Journal of Mathematics of Kyoto University 36 (1996), 115–121.
P. Maier and K.-H. Neeb, Central extensions of current groups, Mathematische Annalen 326 (2003), 367–415.
Y. Matsumoto, An Introduction to Morse Theory, Translations of Mathematical Monographs, Vol. 208, American Mathematical Society, Providence, RI, 2002.
J. A. Neisendorfer, Primary homotopy theory, Memoirs of the American Mathematical Society 232 (1980).
J. A. Neisendorfer, Homotopy groups with coefficients, Journal of Fixed Point Theory and Applications 8 (2010), 247–338.
J. Z. Pan and Z. J. Zhu, The homotopy classification of A 4n -polyhedra with 2-torsion free homology, Science China. Mathematics 59 (2016), 1141–1162.
A. Pressley and G. Segal, Loop Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986.
P. Selick, Introduction to Homotopy Theory, Fields Institute Monographs, Vol. 9, American Mathematical Society, Providence, RI, 1997.
T. So, Homotopy types of gauge groups over non-simply-connected closed 4-manifolds, Glasgow Mathematical Journal 61 (2019), 349–371.
T. So and S. Theriault, The homotopy types of Sp(2)-gauge groups over closed, simply-connected four-manifolds, Proceedings of the Steklov Institute of Mathematics 305 (2019), 287–304.
W. A. Sutherland, Function spaces related to gauge groups, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 121 (1992), 185–190.
S. Theriault, Odd primary homotopy decompositions of gauge groups, Algebraic & Geometric Topology 10 (2010), 535–564.
S. Theriault, The homotopy types of Sp(2)-gauge groups, Journal of Mathematics of Kyoto University 50 (2010), 591–605.
S. Theriault, Homotopy types of SU(3)-gauge groups over simply connected 4-manifolds, Publications of the Research Institute for Mathematical Sciences 48 (2012), 543–563.
S. Theriault, The homotopy types of SU(5)-gauge groups, Osaka Journal of Mathematics 52 (2015), 15–29.
S. Theriault, Odd primary homotopy types of SU(n)-gauge groups, Algebraic & Geometric Topology 17 (2017), 1131–1150.
J. H. C. Whitehead, The homotopy type of a special kind of polyhedron, Annales de la Société Polonaise de Mathématique 24 (1948), 176–186.
J. H. C. Whitehead, On simply connected 4-dimensional polyhedra, Commentarii Mathematici Helvetici 22 (1949), 48–92.
Acknowledgments
The first author is funded by PIMS Post-doctoral Fellowship at the University of Regina and was supported by the Oberwolfach Leibniz Fellowship. He would particularly like to thank Mathematisches Forschunginstitut Oberwolfach for the opportunity to work in such a wonderful environment.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
So, T., Theriault, S. The suspension of a 4-manifold and its applications. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2659-0
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-024-2659-0