Induced character in equivariant K-theory: wreath products and pullback of groups

• German Combariza ^[1] ; Juan Rodriguez ^[2] ; Mario Velasquez ^[3]
  1. [1] Fundación Universitaria Konrad Lorenz

     Fundación Universitaria Konrad Lorenz

     Colombia

     
  2. [2] École Normale Supérieure de Lyon

     École Normale Supérieure de Lyon

     Arrondissement de Lyon, Francia

     
  3. [3] Universidad Nacional de Colombia

     Universidad Nacional de Colombia

     Colombia

     

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• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 56, Nº. 1, 2022, págs. 35-61
• Idioma: inglés
• DOI: 10.15446/recolma.v56n1.105613
• Títulos paralelos:
  • Carácter inducido en K-teoría equivariante: productos wreath y pullbacks de grupos
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• Resumen
  • español

    Sea G un grupo finito y X un G-espacio compacto. En esta nota estudiamos el álgebra (Z+ × Z/2Z)-graduada FqG (X) = ⊕n ≤ 0 qn · KG∫Gn(Xn) ⊗ C, definida en términos de K-teoría equivariante con respecto a productos guirnalda, como un álgebra simétrica, revisamos algunas de las propiedades de FqG (X) probadas por Segal y Wang. Probamos una formula tipo Kunneth para estas álgebras graduadas, más específcamente, sea H otro grupo finito y Y un H-espacio compacto, nosotros damos una descomposición de FqG × H (X × Y) en términos de FqG (X) y FqH (Y), para esto, debemos estudiar la teoría de representaciones de pullbacks de grupos. Discutimos también algunas aplicaciones de los resultados anteriores a K-homología equivariante conectiva.

  • English

    Let G be a finite group and let X be a compact G-space. In this note we study the (Z+ × Z/2Z)-graded algebra FqG (X) = ⊕n ≤ 0 qn · KG∫Gn(Xn) ⊗ C, defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra, we review some properties of FqG (X) proved by Segal and Wang. We prove a Kunneth type formula for this graded algebras, more specifically, let H be another finite group and let Y be a compact H-space, we give a decomposition of FqG × H (X × Y) in terms of FqG (X) and FqH (Y). For this, we need to study the representation theory of pullbacks of groups. We discuss also some applications of the above result to equivariant connective K-homology.

• Referencias bibliográficas
  • M. F. Atiyah, K-theory, second ed., Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989,...
  • M. F. Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671-677. MR MR1076708 (92c:19005) DOI:...
  • P. Baum, N. Higson, and T. Schick, On the equivalence of geometric and analytic K-homology, Pure Appl. Math. Q. 3 (2007), no. 1, part 3, 1-24....
  • G. Combariza, Pullbacks with kernel s3, https://sites.google.com/site/combariza/research/pullbacks-with-kernel-s3, 2019.
  • T. T. Dieck, Transformation groups, de Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR MR889050 (89c:57048)
  • The GAP Group, GAP { Groups, Algorithms, and Programming, Version 4.8.10, 2018.
  • M. J. Hopkins, N. J.Kuhn, and D. C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13...
  • N. J. Kuhn, Character rings in algebraic topology, Advances in homotopy theory (Cortona, 1988), London Math. Soc. Lecture Note Ser., vol....
  • W. Lück, Chern characters for proper equivariant homology theories and applications to K- and L-theory, J. Reine Angew. Math. 543 (2002),...
  • I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press,...
  • J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR MR0174052 (30 #4259) DOI: https://doi.org/10.2307/1970615
  • H. Minami, A Künneth formula for equivariant K-theory, Osaka J. Math. 6 (1969), 143-146. MR 0259897
  • G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 129-151. MR 0234452 (38 #2769) DOI: https://doi.org/10.1007/BF02684593
  • G. Segal, Equivariant K-theory and symmetric products, Preprint, 1996.
  • J.-P. Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977, Translated from the second French edition...
  • M. Velásquez, A configuration space for equivariant connective Khomology, J. Noncommut. Geom. 9 (2015), no. 4, 1343-1382. MR 3448338 DOI:...
  • M. Velásquez, A description of the assembly map for the Baum-connes conjecture with coefficients, New York Journal of Mathematics 29 (2019),...
  • W. Wang, Equivariant K-theory, wreath products, and Heisenberg algebra, Duke Math. J. 103 (2000), no. 1, 1-23. MR MR1758236 (2001b:19005)...