Realizing algebraic 2-complexes by cell complexes

Mannan, W. H.

doi:10.1017/s0305004108002107

Cited by 18 publications

(13 citation statements)

References 5 publications

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“…if G has the D2 property. In particular, this shows the following which was proven by Johnson in the case of finite groups [19] and was later extended by Mannan [24].…”

Section: Polarised Homotopy Types and Algebraic 2-complexessupporting

confidence: 62%

On CW-complexes over groups with periodic cohomology

Nicholson

2019

Preprint

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If G has 4-periodic cohomology, then D2 complexes over G are determined up to polarised homotopy by their Euler characteristic if and only if G has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincaré 3-complex X with π 1 (X) = G admits a cell structure with a single 3-cell. The proof involves cancellation theorems for ZG modules where G has periodic cohomology.

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“…if G has the D2 property. In particular, this shows the following which was proven by Johnson in the case of finite groups [19] and was later extended by Mannan [24].…”

Section: Polarised Homotopy Types and Algebraic 2-complexessupporting

confidence: 62%

On CW-complexes over groups with periodic cohomology

Nicholson

2019

Preprint

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“…Although the original version is stated for complexes with finite groups, it does hold for complexes with finitely presentable groups (cf. [8], appendix B and Mannan [14]).…”

Section: It Follows Thatmentioning

confidence: 99%

Partial Euler characteristic, normal generations and the stable $D(2)$ problem

2018

Homology, Homotopy and Applications

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We study the interplay among Wall's D(2) problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan's problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex X of cohomological dimension 2 with finite fundamental group, assuming the Wiegold conjecture holds, we prove that X is homotopy equivalent to a finite 2-complex after wedging a copy of sphere S 2 .

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“…Recall that a finitely presented group G has the D2 property if every finite CWcomplex X such that π 1 (X) ∼ = G, H i ( X; Z) = 0 for i > 2 and H n+1 (X; M ) = 0 for all finitely generated ZG-modules M is homotopy equivalent to a finite 2-complex. The following is a mild improvement of Wall's results on finiteness conditions for CW-complexes due to Johnson [29] and Mannan [37]. This precise version follows from [31,Corollary 8.27] in the case n ≥ 3 and [41, Theorem 2.1] in the case n = 2.…”

Section: Algebraic Classification Of Finite (G N)-complexesmentioning

confidence: 71%

The homotopy type of a finite 2-complex with non-minimal Euler characteristic

Nicholson¹

2021

Preprint

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We resolve two long-standing and closely related problems concerning stably free ZG-modules and the homotopy type of finite 2-complexes. In particular, for all k ≥ 1, we show that there exists a group G and a non-free stably free ZG-module of rank k. We use this to show that, for all k ≥ 0, there exists homotopically distinct finite 2-complexes with fundamental group G and with Euler characteristic k greater than the minimal value over G. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.

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Realizing algebraic 2-complexes by cell complexes

Abstract: The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact equivalent for all finitely presented groups.

Cited by 18 publications

References 5 publications

On CW-complexes over groups with periodic cohomology

On CW-complexes over groups with periodic cohomology

Partial Euler characteristic, normal generations and the stable $D(2)$ problem

The homotopy type of a finite 2-complex with non-minimal Euler characteristic

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