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Comparison of relative group (co)homologies

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Abstract

Let G be a discrete group. It is well known that the (co)homology groups of G have both topological and algebraic definitions. Now, consider a subgroup H of G. In the literature, there are two versions of relative (co)homology groups for the pair (G, H), one generalises in a natural way the topological definition, while the other one generalises in a natural way the algebraic definition. In this article, we give a topological definition for the latter one, we give simple examples that show that these theories do not coincide in general, and we give a sufficient condition on the subgroup H in order that both relative group (co)homologies of the pair (G, H) coincide.

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Notes

  1. Or maybe call it AMT relative group (co)homology after Auslander-Massey-Takasu.

  2. According to MathSciNet and Web of Science, it has zero citations.

  3. After Blowers, he used his topological construction in [12] to give a simpler proof of results in [11].

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Acknowledgments

We thank Professor Francisco González Acuña for fruitful discussions. J. L. Cisneros-Molina thanks Galatasaray University and Nesin Mathematics Village, both in Turkey, for their hospitality during part of the writing of this article. We thank the referee for his/her careful reading of the manuscript and for many suggestions to improve this article.

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Authors and Affiliations

  1. Subdirección de Ingeniería y Posgrado, Universidad de Aeronáutica en Querétaro, Carretera Estatal 200 Querétaro-Tequisquiapan No. 22154, Colón, Querétaro, Mexico

    José Antonio Arciniega-Nevárez

  2. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Colonia Lomas de Chamilpa, Cuernavaca, Morelos, Mexico

    José Luis Cisneros-Molina

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  1. José Antonio Arciniega-Nevárez
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  2. José Luis Cisneros-Molina
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Correspondence to José Luis Cisneros-Molina.

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On memory of Sam Gitler.

Research supported by projects UNAM-DGAPA-PAPIIT IN106614 and CONACYT 253506. Part of J. A. Arciniega-Nevárez’s Ph.D. thesis [3]. J. L. Cisneros-Molina is Regular Associate of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

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Arciniega-Nevárez, J.A., Cisneros-Molina, J.L. Comparison of relative group (co)homologies. Bol. Soc. Mat. Mex. 23, 41–74 (2017). https://doi.org/10.1007/s40590-016-0149-z

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  • DOI: https://doi.org/10.1007/s40590-016-0149-z

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