The ring structure for equivariant twisted K-theory

被引:7
|
作者
Tu, Jean-Louis [1 ]
Xu, Ping [2 ]
机构
[1] Univ Paul Verlaine Metz, ISGMP, LMAM CNRS UMR 7122, F-57000 Metz, France
[2] Penn State Univ, Dept Math, University Pk, PA 16802 USA
来源
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2009年 / 635卷
关键词
BAUM-CONNES CONJECTURE; LIE-GROUPS; GROUP COHOMOLOGY; MODULI SPACES; FOLIATIONS; GROUPOIDS; BUNDLES; STACKS; GERBES;
D O I
10.1515/CRELLE.2009.077
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map T(1) : H* (Gamma(center dot); A) -> H*(-1) ((N x Gamma)(center dot);A) for any crossed module N -> Gamma and prove that any element in the image is infinity-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N -> Gamma and any e is an element of Z(3) (Gamma(center dot);G(1)), that the equivariant twisted K-theory group K(e,Gamma)*(N) admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group K([e],G)*(G), defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure K([e],G)(i+d)(G) circle times K([e],G)(j+d)(G) -> K([e],G)(i+j+d)(G), where d = dim G and [c] is an element of H(2)((G x G)(center dot); S(1)). The relation with Freed-Hopkins-Teleman theorem [25] still needs to be explored.
引用
收藏
页码:97 / 148
页数:52
相关论文
共 50 条
  • [31] Equivariant representable K-theory
    Emerson, Heath
    Meyer, Ralf
    JOURNAL OF TOPOLOGY, 2009, 2 (01) : 123 - 156
  • [32] THE SPECTRUM OF EQUIVARIANT K-THEORY
    BOJANOWSKA, A
    MATHEMATISCHE ZEITSCHRIFT, 1983, 183 (01) : 1 - 19
  • [33] EQUIVARIANT K-THEORY FOR CURVES
    ELLINGSRUD, G
    LONSTED, K
    DUKE MATHEMATICAL JOURNAL, 1984, 51 (01) : 37 - 46
  • [34] EQUIVARIANT K-THEORY OF RINGS
    DAVYDOV, AA
    RUSSIAN MATHEMATICAL SURVEYS, 1991, 46 (04) : 167 - 168
  • [35] EQUIVARIANT CONNECTIVE K-THEORY
    Karpenko, Nikita A.
    Merkurjev, Alexander S.
    JOURNAL OF ALGEBRAIC GEOMETRY, 2022, 31 (01) : 181 - 204
  • [36] EQUIVARIANT COMMUNICATION K-THEORY
    DAVYDOV, AA
    VESTNIK MOSKOVSKOGO UNIVERSITETA SERIYA 1 MATEMATIKA MEKHANIKA, 1991, (06): : 90 - 93
  • [37] Twisted K-theory and K-theory of bundle gerbes
    Bouwknegt, P
    Carey, AL
    Mathai, V
    Murray, MK
    Stevenson, D
    COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2002, 228 (01) : 17 - 45
  • [38] Twisted K-Theory and K-Theory of Bundle Gerbes
    Peter Bouwknegt
    Alan L. Carey
    Varghese Mathai
    Michael K. Murray
    Danny Stevenson
    Communications in Mathematical Physics, 2002, 228 : 17 - 49
  • [39] EQUIVARIANT ALGEBRAIC K-THEORY
    FIEDOROWICZ, Z
    HAUSCHILD, H
    MAY, JP
    LECTURE NOTES IN MATHEMATICS, 1982, 967 : 23 - 80
  • [40] K-theory of equivariant quantization
    Tang, Xiang
    Yao, Yi-Jun
    JOURNAL OF FUNCTIONAL ANALYSIS, 2014, 266 (02) : 478 - 486