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The mod-p homology of the classifying spaces of certain gauge groups

Published online by Cambridge University Press:  19 September 2022

Daisuke Kishimoto
Affiliation:
Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan (kishimoto@math.kyushu-u.ac.jp)
Stephen Theriault
Affiliation:
Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom (s.d.theriault@soton.ac.uk)

Abstract

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.

Type
Research Article
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Atiyah, M. F. and Bott, R.. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523615.Google Scholar
Atiyah, M. F. and Jones, J. D. S.. Topological aspects of Yang-Mills theory. Commun. Math. Phys. 61 (1978), 97118.CrossRefGoogle Scholar
Choi, Y.. Homology of the classifying space of $Sp(n)$ gauge groups. Israel J. Math. 151 (2006), 167177.CrossRefGoogle Scholar
Cohen, F. R., Lada, T. J. and May, J. P.. The homology of iterated loop spaces, Lecture Notes in Math. Vol. 533 (Springer-Verlag, 1976).CrossRefGoogle Scholar
Donaldson, S. K.. Connections, cohomology and the intersection forms on $4$-manifolds. J. Differ. Geom. 24 (1986), 275341.Google Scholar
Floer, A.. An instanton invariant for $3$-manifolds. Commun. Math. Phys. 118 (1988), 215240.CrossRefGoogle Scholar
Gottlieb, D. H.. Applications of bundle map theory. Trans. Amer. Math. Soc. 171 (1972), 2350.CrossRefGoogle Scholar
Gray, B. and Theriault, S.. An elementary construction of Anick's fibration. Geom. Topol. 14 (2010), 243276.CrossRefGoogle Scholar
Hasui, S., Kishimoto, D. and Ohsita, A.. Samelson products in $p$-regular exceptional Lie groups. Topology Appl. 178 (2014), 1729.CrossRefGoogle Scholar
Kaji, S. and Kishimoto, D.. Homotopy nilpotency in $p$-regular loop spaces. Math. Z. 264 (2010), 209224.CrossRefGoogle Scholar
Kishimoto, D. and Kono, A.. Splitting of gauge groups. Trans. Amer. Math. Soc. 362 (2010), 67156731.CrossRefGoogle Scholar
Kishimoto, D. and Tsutaya, M.. Infiniteness of $A_\infty$-types of gauge groups. J. Topol. 9 (2016), 181191.CrossRefGoogle Scholar
Kishimoto, D. and Tsutaya, M.. Samelson products in $p$-regular $\mathrm {SO}(2n)$ and its homotopy normality. Glasg. Math. J. 60 (2018), 165174.CrossRefGoogle Scholar
Kono, A.. A note on the homotopy type of certain gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295297.CrossRefGoogle Scholar
Lang, G. E.. The evaluation map and $EHP$ sequences. Pacific J. Math. 44 (1973), 201210.CrossRefGoogle Scholar
Masbaum, G.. On the cohomology of the classifying space of the gauge group over some $4$-complexes. Bull. Soc. Math. France 119 (1991), 131.CrossRefGoogle Scholar
McGibbon, C. A.. Homotopy commutativity in localized groups. Amer. J. Math. 106 (1984), 665687.CrossRefGoogle Scholar
Selick, P. S.. A reformulation of the Arf invariant one mod-$p$ problem and applications to atomic spaces. Pacific J. Math. 108 (1983), 431450.CrossRefGoogle Scholar
Theriault, S. D.. Atomicity for Anick's spaces. J. Pure Appl. Alg. 219 (2015), 23462358.CrossRefGoogle Scholar
Toda, H.. Composition methods in homotopy groups of spheres, Annals of Math. Studies No. Vol. 49 (Princeton University Press, Princeton NJ, 1962).Google Scholar