Servicios Personalizados
Revista
Articulo
Indicadores
- Citado por SciELO
- Accesos
Links relacionados
- Citado por Google
- Similares en SciELO
- Similares en Google
Compartir
Revista Colombiana de Matemáticas
versión impresa ISSN 0034-7426
Rev.colomb.mat. vol.56 no.1 Bogotá ene./jun. 2022 Epub 03-Ene-2023
https://doi.org/10.15446/recolma.v56n1.105613
Original articles
Induced character in equivariant K-theory, wreath products and pullback of groups
Carácter inducido en K-teoría equivariante, productos wreath y pullbacks de grupos
1 Fundación Universitaria Konrad Lorenz, Bogotá, Colombia
2 École normale supérieure de Lyon, Lyon, France
3 Universidad Nacional de Colombia, Bogotá, Colombia
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
defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra, we review some properties of F q G (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 F q G(H (X ( Y) in terms of F q G (X) and F q H (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.
Keywords: equivariant K-theory; wreath products; Fock space
Sea G un grupo finito y X un G-espacio compacto. En esta nota estudiamos el álgebra (Z + ( Z /2Z)-graduada
Definida en términos de K-teoría equivariante con respecto a productos guirnalda, como un álgebra simétrica, revisamos algunas de las propiedades de F q G (X) probadas por Segal y Wang. Probamos una formula tipo Kunneth para estas álgebras graduadas, más específicamente, sea H otro grupo finito y Y un H-espacio compacto, nosotros damos una descomposición de F q G(H (X(Y) en términos de F q G (X) y F q H (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.
Palabras clave: K-teoría equivariante; productos wreath; espacio de Fock
REFERENCES
1. M. F. Atiyah, K-theory, second ed., Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989, Notes by D. W. Anderson. MR 1043170 (90m:18011) [ Links ]
2. M. F. Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671-677. MR MR1076708 (92c:19005) [ Links ]
3. 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. MR MR2330153 (2008d:58015) [ Links ]
4. G. Combariza, Pullbacks with kernel s3, https://sites.google.com/site/combariza/research/pullbacks-with-kernel-s3, 2019. [ Links ]
5. T. T. Dieck, Transformation groups, de Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR MR889050 (89c:57048) [ Links ]
6. The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.10, 2018. [ Links ]
7. M. J. Hopkins, N. J.Kuhn, and D. C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553-594. MR 1758754 [ Links ]
8. N. J. Kuhn, Character rings in algebraic topology, Advances in homotopy theory (Cortona, 1988), London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 111-126. MR 1055872 [ Links ]
9. W. Lück, Chern characters for proper equivariant homology theories and applications to K- and L-theory, J. Reine Angew. Math. 543 (2002), 193-234. MR 1887884 (2003a:19003) [ Links ]
10. I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144]. MR 3443860 [ Links ]
11. 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) [ Links ]
12. H. Minami, A Künneth formula for equivariant K-theory, Osaka J. Math. 6 (1969), 143-146. MR 0259897 [ Links ]
13. G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 129-151. MR 0234452 (38 #2769) [ Links ]
14. G. Segal, Equivariant K-theory and symmetric products, Preprint, 1996. [ Links ]
15. J.-P. Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42. MR 0450380 [ Links ]
16. M. Velásquez, A configuration space for equivariant connective Khomology, J. Noncommut. Geom. 9 (2015), no. 4, 1343-1382. MR 3448338 [ Links ]
17. M. Velásquez, A description of the assembly map for the Baum-connes conjecture with coefficients, New York Journal of Mathematics 29 (2019), 668-686. [ Links ]
18. W. Wang, Equivariant K-theory, wreath products, and Heisenberg algebra, Duke Math. J. 103 (2000), no. 1, 1-23. MR MR1758236 (2001b:19005) [ Links ]
Received: November 01, 2021; Accepted: May 27, 2022