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Homotopy of gauge groups over high-dimensional manifolds

Part of: Fiber spaces and bundles Homotopy theory Maps and general types of spaces defined by maps Differential topology

Published online by Cambridge University Press:  05 February 2021

Ruizhi Huang
Ruizhi Huang*
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China ([email protected])
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Abstract

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

Keywords

Gauge groupsHomotopy suspensionsHomotopy decompositionsHomotopy exponents

MSC classification

Primary: 55P15: Classification of homotopy type 55P40: Suspensions 54C35: Function spaces
Secondary: 55R25: Sphere bundles and vector bundles 57R19: Algebraic topology on manifolds 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 182 - 208
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Adams, J. F.. On the groups J(X)-IV. Topology 5 (1966), 21–71.CrossRefGoogle Scholar
Atiyah, M. and Bott, R.. The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci. 308 (1983), 523–615.Google Scholar
Bott, R. and Samelson, H.. Application of the theory of Morse to symmetric spaces. Am. J. Math. 80 (1958), 964–1029.CrossRefGoogle Scholar
Cohen, R. L. and Milgram, R. J.. The homotopy type of gauge theoretic moduli spaces, from: ‘Algebraic topology and its applications’. In Math. Sci. Res. Inst. Publ. (eds. Carlsson, G. E., Cohen, R. L., Hsiang, W. C. and Jones, J. D. S.), vol. 27, pp. 15–55 (New York: Springer, 1994).Google Scholar
Cohen, F. R., Moore, J. C. and Neisendorfer, J. A.. The double suspension and exponents of the homotopy group of spheres. Ann. Math. 109 (1979), 549–565.CrossRefGoogle Scholar
Crabb, M. C. and Sutherland, W. A.. Counting homotopy types of gauge groups. Proc. London Math. Soc. 81 (2000), 747–768.CrossRefGoogle Scholar
Davis, D. M. and Theriault, S. D.. Odd-primary homotopy exponents of simple compact Lie groups. Geom. Topol. Monographs 13 (2008), 195–201.CrossRefGoogle Scholar
Donaldson, S. K.. Connections, cohomology and the intersection forms of 4-manifolds. J. Differ. Geom. 24 (1986), 275–341.Google Scholar
Duan, H. and Wang, S.. The degrees of maps between manifolds. Math. Z. 244 (2003), 67–89.CrossRefGoogle Scholar
Farrell, T. and Basu, S.. Introductory lectures on surgery theory, from: ‘Introduction to Modern Mathematics’. In ALM (eds. Cheng, S.-Y., Ji, L., Poon, Y.-S., Xiao, J., Yang, L. and Yau, S.-T.), vol. 33, pp. 3–108 (Beijing: Higher Education Press, 2015).Google Scholar
Freedman, M. H.. The topology of 4-manifolds. J. Differ. Geom. 17 (1982), 337–453.Google Scholar
Gottlieb, D. H.. Applications of bundle map theory. Trans. Am. Math. Soc. 171 (1972), 23–50.CrossRefGoogle Scholar
Gray, B. I.. On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219–232.CrossRefGoogle Scholar
Haefliger, A.. Differential imbeddings. Bull. Am. Math. Soc. 67 (1961), 109–112.CrossRefGoogle Scholar
Hamanaka, H. and Kono, A.. Homotopy type of gauge groups of SU(3)-bundles over S 6. Topol. Appl. 154 (2007), 1377–1380.CrossRefGoogle Scholar
Hamanaka, H., Kaji, S. and Kono, A.. Samelson products in Sp(2). Topol. Appl. 155 (2008), 1207–1212.CrossRefGoogle Scholar
Harris, B.. On the homotopy groups of the classical groups. Ann. Math. 74 (1961), 407–413.CrossRefGoogle Scholar
Hasui, S., Kishimoto, D., Kono, A. and Sato, T.. The homotopy types of PU(3)- and PSp(2)-gauge groups. Algebr. Geom. Topol. 16 (2016), 1813–1825.CrossRefGoogle Scholar
Huang, R.. Homotopy of gauge groups over non-simply-connected five-dimensional manifolds. Sci. China Math. (2019), https://doi.org/10.1007/s11425-019-9540-3.Google Scholar
James, I. M. and Whitehead, J. H. C.. The homotopy theory of sphere bundles over spheres (I). Proc. London Math. Soc. 3 (1954), 196–218.CrossRefGoogle Scholar
Kishimoto, D., Kono, A. and Tsutaya, M.. Mod p decompositions of gauge groups. Algebr. Geom. Topol. 13 (2013), 1757–1778.CrossRefGoogle Scholar
Kishimoto, D., Kono, A. and Tsutaya, M.. On p-local homotopy types of gauge groups. Proc. R. Soc. Edinburgh A 144 (2014), 149–160.CrossRefGoogle Scholar
Kono, A.. A note on the homotopy type of certain gauge groups. Proc. R. Soc. Edinburgh A 117 (1991), 295–297.CrossRefGoogle Scholar
Kumpel, P. G.. On p-equivalences of mod p H-spaces. Quart. J. Math. 23 (1972), 173–178.CrossRefGoogle Scholar
Lang, Jr G. E.. The evaluation map and EHP sequences. Pacific J. Math. 44 (1973), 201–210.CrossRefGoogle Scholar
Membrillo-Solis, I.. Homotopy types of gauge groups related to S 3-bundles over S 4. Topol Appl. 255 (2019), 56–85.CrossRefGoogle Scholar
Milnor, J. and Stasheff, J.. Characteristic classes. In Ann. math. studies, vol. 76, pp. vii+331 (Princeton, NJ: Princeton University Press and Univ. Tokyo Press, 1975).Google Scholar
Mimura, M., Nishida, G. and Toda, H.. Mod p decomposition of compact Lie groups. Publ. RIMS, Kyoto Univ. 13 (1977), 627–680.CrossRefGoogle Scholar
Mukai, J.. The S 1-transfer map and homotopy groups of suspended complex projective spaces. Math. J. Okayama Univ. 24 (1982), 179–200.Google Scholar
Neisendorfer, J. A.. 3-primary exponents. Math. Proc. Camb. Phil. Soc. 90 (1981), 63–83.CrossRefGoogle Scholar
Quillen, D.. The Adams conjecture. Topology 10 (1971), 67–80.CrossRefGoogle Scholar
Serre, J. P.. Groupes d'homotopie et classes des groups abéliens. Ann. Math. 58 (1953), 258–294.CrossRefGoogle Scholar
So, T. L.. Homotopy types of gauge groups over non-simply-connected closed 4-manifolds. Glasg. Math. J. 61 (2019), 349–371.CrossRefGoogle Scholar
So, T. L. and Theriault, S.. The suspension of a four-manifold and its applications, preprint (2019), arXiv:1909.11129.Google Scholar
Theriault, S. D.. Homotopy exponents of Harper's spaces. J. Math. Kyoto Univ. 44 (2004), 33–42.Google Scholar
Theriault, S. D.. The odd primary H-structure of low rank Lie groups and its application to exponents. Trans. Am. Math. Soc. 359 (2007), 4511–4535.CrossRefGoogle Scholar
Theriault, S.. Odd primary homotopy decompositions of gauge groups. Algebr. Geom. Topol. 10 (2010), 535–564.CrossRefGoogle Scholar
Theriault, S.. Odd primary homotopy types of SU(n)-gauge groups. Algebr. Geom. Topol. 17 (2017), 1131–1150.CrossRefGoogle Scholar
Toda, H.. Composition methods in homotopy groups of spheres. In Ann. Math. Studies, vol. 49, pp. 193 (Princeton, NJ: Princeton Univ. Press, 1962).Google Scholar
Wall, C. T. C.. Classification of (n − 1)-connected 2n-manifolds. Ann. Math. 75 (1962), 163–198.CrossRefGoogle Scholar
West, M.. Homotopy decompositions of gauge groups over real surfaces. Algebr. Geom. Topol. 17 (2017), 2429–2480.CrossRefGoogle Scholar
Whitehead, G. W.. Generalization of the Hopf invariant. Ann. Math. 51 (1950), 192–237.CrossRefGoogle Scholar
Whitehead, G.. Elements of homotopy theory GTM 62 (Berlin-Heidelberg, New York: Springer-Verlag, 1978).CrossRefGoogle Scholar
Wu, W. T.. On Pontrjagin classes, II. Sci. Sin. 4 (1955), 455–490.Google Scholar
Wu, J.. Homotopy theory and the suspensions of the projective plane. In Memoirs AMS, vol. 162, pp. 130, No. 769, 2003).Google Scholar