Abstract
Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper, we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity. Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.
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Adams J F. On the non-existence of elements of Hopf invariant one. Ann of Math (2), 1960, 72: 20–104
Atiyah M, Bott R. The Yang-Mills equations over Riemann surfaces. Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 1983, 308: 523–615
Barden D. Simply connected five-manifolds. Ann of Math (2), 1965, 82: 365–385
Blakers A L, Massey W S. The homotopy groups of a triad II. Ann of Math (2), 1952, 55: 192–201
Browder W. Poincaré spaces, their normal fibrations and surgery. Invent Math, 1972, 17: 191–202
Cohen F R, Moore J C, Neisendorfer J A. Torsion in homotopy groups. Ann of Math (2), 1979, 109: 121–168
Cohen F R, Moore J C, Neisendorfer J A. The double suspension and exponents of the homotopy group of spheres. Ann of Math (2), 1979, 109: 549–565
Cohen R L, Milgram R J. The homotopy type of gauge theoretic moduli spaces. In: Algebraic Topology and Its Applications. Mathematical Sciences Research Institute Publications, vol. 27. New York: Springer, 1994, 15–55
Crabb M C, Sutherland W A. Counting homotopy types of gauge groups. Proc Lond Math Soc (3), 2000, 81: 747–768
Davis D M, Theriault S D. Odd-primary homotopy exponents of simple compact Lie groups. Geom Topol Monogr, 2008, 13: 195–201
Duan H, Liang C. Circle bundles over 4-manifolds. Arch Math (Basel), 2005, 85: 278–282
Félix Y, Halperin S, Thomas J-C. Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Berlin: Springer, 2000
Félix Y, Oprea J. Rational homotopy of gauge groups. Proc Amer Math Soc, 2009, 137: 1519–1527
Gitler S, Stasheff J D. The first exotic class of BF. Topology, 1965, 4: 257–266
Gottlieb D H. Applications of bundle map theory. Trans Amer Math Soc, 1972, 171: 23–50
Gray B I. On the sphere of origin of infinite families in the homotopy groups of spheres. Topology, 1969, 8: 219–232
Hamanaka H, Kono A. Unstable K 1-group and homotopy type of certain gauge groups. Proc Roy Soc Edinburgh Sect A, 2006, 136: 149–155
Hambleton I, Su Y. On certain 5-manifolds with fundamental group of order 2. Q J Math, 2013, 64: 149–175
Harper J. Secondary Cohomology Operations. Graduate Studies in Mathematics, vol. 49. Providence: Amer Math Soc, 2002
Harris B. On the homotopy groups of the classical groups. Ann of Math (2), 1961, 74: 407–413
Hasui S, Kishimoto D, Kono A, et al. The homotopy types of PU(3)- and PSp(2)-gauge groups. Algebr Geom Topol, 2016, 16: 1813–1825
Hasui S, Kishimoto D, So T, et al. Odd primary homotopy types of the gauge groups of exceptional Lie groups. Proc Amer Math Soc, 2019, 147: 1751–1762
Hatcher A. Algebraic Topology. Cambridge: Cambridge University Press, 2002
Hirsch M W. Immersions of manifolds. Trans Amer Math Soc, 1959, 93: 242–276
Huang R. Homotopy of gauge groups over high dimensional manifolds. ArXiv:1805.04879, 2018
Huang R, Wu J. Cancellation and homotopy rigidity of classical functors. J Lond Math Soc (2), 2019, 99: 225–248
Kasprowski D, Land M, Powell M, et al. Stable classification of 4-manifolds with 3-manifold fundamental groups. J Topol, 2017, 10: 827–881
Kervaire M A, Milnor J W. Groups of homotopy spheres: I. Ann of Math (2), 1963, 77: 504–537
Kirby R C, Siebenmann L C. On the triangulation of manifolds and the Hauptvermutung. Bull Amer Math Soc (NS), 1969, 75: 742–749
Kishimoto D, Kono A. On the homotopy types of Sp(n) gauge groups. Algebr Geom Topol, 2019, 19: 491–502
Kishimoto D, Theriault S, Tsutaya M. The homotopy types of G 2-gauge groups. Topology Appl, 2017, 228: 92–107
Kono A. A note on the homotopy type of certain gauge groups. Proc Roy Soc Edinburgh Sect A, 1991, 117: 295–297
Kreck M, Su Y. On 5-manifolds with free fundamental group and simple boundary links in S5. Geom Topol, 2017, 21: 2989–3008
Kumpel P G. On p-equivalences of mod pH-spaces. Q J Math, 1972, 23: 173–178
Lang G E Jr. The evaluation map and EHP sequences. Pacific J Math, 1973, 44: 201–210
Madsen I, Milgram R J. The Classifying Spaces for Surgery and Cobordism of Manifolds. Annals of Mathematics Studies, vol. 92. Princeton: Princeton University Press, 1979
Milnor J, Stasheff J. Characteristic Classes. Annals of Mathematics Studies, vol. 76. Princeton: Princeton University Press, 1975
Neisendorfer J. 3-primary exponents. Math Proc Cambridge Philos Soc, 1981, 90: 63–83
Neisendorfer J. Properties of certain H-spaces. Q J Math, 1983, 34: 201–209
Neisendorfer J. Algebraic Methods in Unstable Homotopy Theory. Cambridge: Cambridge University Press, 2010
Neisendorfer J. Homotopy groups with coefficients. J Fixed Point Theory Appl, 2010, 8: 247–338
Selick P S. Odd primary torsion in πk(S 3). Topology, 1978, 17: 407–412
Serre J P. Groupes d’homotopie et classes de groups abéliens. Ann of Math (2), 1953, 58: 258–294
So T L. Homotopy types of gauge groups over non-simply-connected closed 4-manifolds. Glasg Math J, 2019, 61: 349–371
Spivak M. Spaces satisfying Poincaré duality. Topology, 1967, 6: 77–101
Terzić S. The rational topology of gauge groups and of spaces of connections. Compos Math, 2005, 141: 262–270
Theriault S D. Homotopy exponents of Harper’s spaces. J Math Kyoto Univ, 2004, 44: 33–42
Theriault S D. The odd primary H-structure of low rank Lie groups and its application to exponents. Trans Amer Math Soc, 2007, 359: 4511–4535
Theriault S D. Odd primary homotopy decompositions of gauge groups. Algebr Geom Topol, 2010, 10: 535–564
Theriault S D. The homotopy types of Sp(2)-gauge groups. Kyoto J Math, 2010, 50: 591–605
Theriault S D. The homotopy types of SU(5)-gauge groups. Osaka J Math, 2015, 52: 15–31
Theriault S D. Odd primary homotopy types of SU(n)-gauge groups. Algebr Geom Topol, 2017, 17: 1131–1150
Toda H. Composition Methods in Homotopy Groups of Spheres. Annals of Mathematics Studies, vol. 49. Princeton: Princeton University Press, 1962
Whitehead G. Elements of Homotopy Theory. Graduate Texts in Mathematics, vol. 62. Berlin: Springer-Verlag, 1978
Wu J. Homotopy theory and the suspensions of the projective plane. Mem Amer Math Soc, 2003, 162: 1–130
Wu W-T. On Pontrjagin classes, II. Scientia Sinica, 1955, 4: 455–490
Acknowledgments
This work was supported by Postdoctoral International Exchange Program for Incoming Postdoctoral Students under Chinese Postdoctoral Council and Chinese Postdoctoral Science Foundation, Chinese Postdoctoral Science Foundation (Grant No. 2018M631605) and National Natural Science Foundation of China (Grant No. 11801544). The author thanks Haibao Duan for bringing the non-simply connected 5-manifolds to his attention. He is indebted to Daisuke Kishimoto for suggestions about adding more applications of the decompositions on an early version of this paper. He also thanks Fred Cohen, and the anonymous referees most warmly for their careful reading of the manuscript and many helpful suggestions which have improved the paper.
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Huang, R. Homotopy of gauge groups over non-simply-connected five-dimensional manifolds. Sci. China Math. 64, 1061–1092 (2021). https://doi.org/10.1007/s11425-019-9540-3
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DOI: https://doi.org/10.1007/s11425-019-9540-3
Keywords
- gauge group
- 5-manifolds
- homotopy decompositions
- homotopy exponents
- homotopy groups
- rational homotopy theory
- Bott periodicity