×

A Fourier-Mukai approach to the \(K\)-theory of compact Lie groups. (English) Zbl 1346.19010

The aim of this paper is to provide a simple conceptual proof of the following celebrated result of L. Hodgkin [Topology 6, 1–36 (1967; Zbl 0186.57103)]: the topological \(K\)-theory of a compact Lie group whose fundamental group is torsion free is isomorphic to the exterior algebra over \(\mathbb{Z}\) on \(n\) odd generators, where \(n\) is the rank of \(G\). The original proof of this result boils down in two different parts: the first one is the proof of the torsion-freeness of \(K^*(G)\), and the second part is a detailed analysis of the multiplicative structure of \(K^*(G)\). Before the present paper, alternative proofs of Hodgkin’s theorem have already been provided combining the papers [M. F. Atiyah, Topology 4, 95–99 (1965; Zbl 0136.21001)] for the torsion freeness and [S. Araki, Ann. Math. (2) 85, 508–525 (1967; Zbl 0171.43903)] for the multiplicative structure. Here, the authors give a simpler proof, using the Fourier-Mukai transform as well as duality for torus bundles.

MSC:

19L50 Twisted \(K\)-theory; differential \(K\)-theory
19L64 Geometric applications of topological \(K\)-theory
57T10 Homology and cohomology of Lie groups
22C05 Compact groups
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Araki, S., Hopf structures attached to \(K\)-theory; Hodgkin’s theorem, Ann. of Math. (2), 85, 508-525 (1967) · Zbl 0171.43903
[2] Atiyah, M. F., On the \(K\)-theory of compact Lie groups, Topology, 4, 95-99 (1965) · Zbl 0136.21001
[3] Atiyah, M.; Segal, G., Twisted \(K\)-theory, Ukr. Math. Bull., 1, 3, 291-334 (2004)
[4] Baraglia, D., Topological T-duality for general circle bundles, Pure Appl. Math. Q. (2014), in press · Zbl 1318.19009
[5] Baraglia, D., Topological T-duality for torus bundles with monodromy (2012)
[6] Bouwknegt, P.; Carey, A.; Mathai, V.; Murray, M.; Stevenson, D., Twisted \(K\)-theory and \(K\)-theory of bundle gerbes, Comm. Math. Phys., 228, 1, 17-45 (2002) · Zbl 1036.19005
[7] Bouwknegt, P.; Evslin, J.; Mathai, V., T-duality: topology change from \(H\)-flux, Comm. Math. Phys., 249, 2, 383-415 (2004) · Zbl 1062.81119
[8] Bunke, U.; Rumpf, P.; Schick, T., The topology of T-duality for \(T^n\)-bundles, Rev. Math. Phys., 18, 10, 1103-1154 (2006) · Zbl 1116.55007
[9] Carey, A. L.; Wang, B.-L., Thom isomorphism and push-forward map in twisted \(K\)-theory, J. K-Theory, 1, 2, 357-393 (2008) · Zbl 1204.55006
[10] Freed, D. S.; Hopkins, M. J.; Teleman, C., Loop groups and twisted \(K\)-theory I, J. Topol., 4, 4, 737-798 (2011) · Zbl 1241.19002
[11] Hodgkin, L., On the \(K\)-theory of Lie groups, Topology, 6, 1-36 (1967) · Zbl 0186.57103
[12] Hodgkin, L., The equivariant Künneth theorem in \(K\)-theory, (Topics in \(K\)-theory. Topics in \(K\)-theory, Lecture Notes in Math., vol. 496 (1975), Springer: Springer Berlin), 1-101 · Zbl 0323.55009
[13] Huybrechts, D., Fourier-Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs (2006), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford · Zbl 1095.14002
[14] Kostant, B.; Kumar, S., T-equivariant \(K\)-theory of generalized flag varieties, J. Differential Geom., 32, 2, 549-603 (1990) · Zbl 0731.55005
[15] Mac Lane, S., Homology (1963), Springer-Verlag: Springer-Verlag Berlin, Göttingen, Heidelberg · Zbl 0818.18001
[16] Pittie, H. V., Homogeneous vector bundles on homogeneous spaces, Topology, 11, 199-203 (1972) · Zbl 0229.57017
[17] Rosenberg, J., \(C^\ast \)-algebras, and String Duality, CBMS Regional Conference Series in Mathematics, vol. 111 (2009), American Mathematical Society: American Mathematical Society Providence, RI
[18] Segal, G., Classifying spaces and spectral sequences, Publ. Math. Inst. Hautes Études Sci., 34, 105-112 (1968) · Zbl 0199.26404
[19] Steinberg, R., On a theorem of Pittie, Topology, 14, 173-177 (1975) · Zbl 0318.22010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.