Classifying Spaces and Homology Decompositions

Dwyer, W. G.

doi:10.1007/978-3-0348-8356-6_1

Cited by 12 publications

(6 citation statements)

References 28 publications

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“…Note that since the subgroup families that we take always include the trivial subgroup, they are ample collections, and hence the cohomology of the homotopy colimit of classifying spaces of subgroups in the family is isomorphic to the cohomology of the group. More details on this can be found in [3].…”

Section: Relative Group Cohomology and The Orbit Category 3239mentioning

confidence: 98%

Relative Group Cohomology and The Orbit Category

Pamuk

Yalcin

2014

Communications in Algebra

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Let G be a finite group and ℱ be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ℱ-projective resolution for ℤ when ℱ is the family of all subgroups H ≤ G with rk H ≤ rkG - 1. We answer this question negatively by calculating the relative group cohomology ℱH*(G, F{double-struck}2) where G = ℤ/2 × ℤ/2 and ℱ is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ℱH*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ℱ. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E 2 page is isomorphic to the relative group cohomology of G. © 2014 Copyright Taylor & Francis Group, LLC

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Section: Relative Group Cohomology and The Orbit Category 3239mentioning

confidence: 98%

Relative Group Cohomology and The Orbit Category

Pamuk

Yalcin

2014

Communications in Algebra

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“…To begin, we present definitions of abstract simplicial complexes and ordered simplicial complex. A more complete reference is [Dwy01].…”

Section: (∞ 1)-categoriesmentioning

confidence: 99%

Euler characteristic for weak n-categories and $\left(\infty,1\right)$-categories

Gonzalez,

Necoechea,

Stratmann

2015

Preprint

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The Euler characteristic was defined for finite strict n-categories by Leinster using the theory of enriched categories. This was an extension of some of his earlier work, which defined Euler characteristic for finite categories. Building on Leinster's work, we extend the notion of Euler characteristic to certain finite weak 2-categories and present a sketch of a similar extension to weak n-categories. We also discuss the extension of the Euler characteristic to certain finite (∞, 1)-categories.

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“…Hence . Therefore the Bousfield-Kan spectral sequence for H * (BG) is a spectral sequence of finitely generated H * (|L|)−modules, the E 2 term with E s,t 2 = lim s C H t (F (−); F p ) is concentrated in the first two columns and E 2 = E ∞ for placement reasons, see [8] for a reference for the Bousfield-Kan spectral sequence of a homotopy colimit. Therefore H * (BG) is a finitely generated module over H * (|L|) and in particular noetherian.…”

Section: Homology Decomposition For Robinson's Modelsmentioning

confidence: 99%

“…X be the cover of BG with fundamental group K. Then, using [2, VII.3.2], we have that X is p−good and X ∧ p is simply connected. Hence for placement reasons, see [8] for a reference for the Bousfield-Kan spectral sequence of a homotopy colimit. Therefore H * (BG) is a finitely generated module over H * (|L|) and in particular noetherian.…”

Section: Homology Decomposition For Robinson's Modelsmentioning

confidence: 99%