Representabilidad de Brown y espacios sobre una categoría

Brown Representability and Spaces over a Category

Probamos un teorema de representabilidad de Brown en el contexto de espacios sobre una categoría. Discutimos dos aplicaciones a la representabilidad de teorías de cohomología, con énfasis en cohomología de Bredon con coeficientes locales.

We prove a Brown Representability Theorem in the context of spaces over a category. We discuss two applications to the representability of equivariant cohomology theories, with emphasis on Bredon cohomology with local coefficients.

Brown Representability and Spaces over a Category Representabilidad de Brown y espacios sobre una categoría NOÉ BÁRCENAS¹

¹Centro de Ciencias Matemáticas UNAM, Morelia, Michoacán, México. Email: barcenas@matmor.unam.mx

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Abstract

We prove a Brown Representability Theorem in the context of spaces over a category. We discuss two applications to the representability of equivariant cohomology theories, with emphasis on Bredon cohomology with local coefficients.

Key words: Brown Representability, Spaces over a category, Bredon Cohomology with local coefficients.

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2000 Mathematics Subject Classification: 53N91, 55N25.

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Resumen

Probamos un teorema de representabilidad de Brown en el contexto de espacios sobre una categoría. Discutimos dos aplicaciones a la representabilidad de teorías de cohomología, con énfasis en cohomología de Bredon con coeficientes locales.

Palabras clave: Representabilidad de Brown, espacios sobre una categoría, cohomología de Bredon con coeficientes locales.

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Texto completo disponible en PDF

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References

[1] N. Barcenas, J. Espinoza, B. Uribe, and M. Velasquez, Segal's Spectral Sequence in Twisted Equivariant K Theory for Proper Actions, `preprint, arXiv:1307.1003, math.AT', (2013).

[2] S. Basu and D. Sen, Representing Bredon Cohomology with local Coefficients by Crossed Complexes and Parametrized Spectra, `ArXiv:1206.2781v1', (2012).

[3] G. E. Bredon, Equivariant Cohomology Theories, Vol. 34 of Lecture Notes in Mathematics, Berlin, Germany, 1967.

[4] J. F. Davis and W. Lück, `Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory', K-Theory 15, 3 (1998), 201-252.

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[7] W. Lück, `Equivariant Cohomological Chern Characters', Internat. J. Algebra Comput. 15, 5-6 (2005a), 1025-1052.

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[10] T. Matumoto, `On G-CW Complexes and a Theorem of J. H. C. Whitehead', J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18, (1971), 363-374.

[11] J. P. May, Equivariant Homotopy and Cohomology Theory, Vol. 91 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, D.C., 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner

[12] M. C. McCord, `Classifying Spaces and Infinite Symmetric Products', Trans. Amer. Math. Soc. 146, (1969), 273-298.

[13] I. Moerdijk and J. A. Svensson, `The Equivariant Serre Spectral Sequence', Proc. Amer. Math. Soc. 118, 1 (1993), 263-278.

[14] G. Mukherjee and N. Pandey, `Equivariant Cohomology with Local Coefficients', Proc. Amer. Math. Soc. 130, 1 (2002), 227-232.

[15] A. Neeman, `The Grothendieck Duality Theorem via Bousfield Techniques and Brown Representability', Journal of the American Mathematical Society 9, 1 (1996), 205-236.

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(Recibido en marzo de 2013. Aceptado en noviembre de 2013)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv48n1a04,
    AUTHOR  = {Bárcenas, Noé},
    TITLE   = {{Brown Representability and Spaces over a Category}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2014},
    volume  = {48},
    number  = {1},
    pages   = {55--77}
}

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