Two remarks on Wall's D2 problem

@article{Hambleton2018TwoRO,
  title={Two remarks on Wall's D2 problem},
  author={Ian Hambleton},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2018},
  volume={167},
  pages={361 - 368}
}
  • I. Hambleton
  • Published 28 August 2017
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract If a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then X ∨ S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X). 
The homotopy type of a finite 2-complex with non-minimal Euler characteristic
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 aExpand
Homotopy classification of 4-manifolds whose fundamental group is dihedral.
We show that the homotopy type of an oriented Poincare 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. This applies inExpand
Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group
We consider the Euler characteristics χ(M) of closed orientable topological 2n–manifolds with (n−1)–connected universal cover and a given fundamental group G of type Fn. We define q2n(G), aExpand
Projective modules and the homotopy classification of $(G,n)$-complexes
If $G$ has periodic cohomology then, for appropriate $n$, the tree of algebraic $n$-complexes is isomorphic as a graded tree to the stable class $[P]$ of a projective module representing the SwanExpand
On CW-complexes over groups with periodic cohomology
  • John Nicholson
  • Mathematics
  • Transactions of the American Mathematical Society
  • 2021
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 quaternionicExpand
2D problems in groups
We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version ofExpand
An exotic presentation of Q28
We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.
A T ] 1 2 O ct 2 02 1 An exotic presentation of Q 28
  • 2021

References

SHOWING 1-10 OF 45 REFERENCES
Partial Euler Characteristic, Normal Generations and the stable D(2) problem
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 ofExpand
The D.2/ property for D8
Wall’s D2 problem asks if a cohomologically 2–dimensional geometric 3–complex is necessarily homotopy equivalent to a geometric 2–complex. We solve part of the problem when the fundamental group isExpand
Quillen's plus construction and the D(2) problem
Given a finite connected 3–complex with cohomological dimension 2, we show it may be constructed up to homotopy by applying the Quillen plus construction to the Cayley complex of a finite groupExpand
Minimal algebraic complexes over D4n
In 1965 Wall showed that for n > 2, if a finite cell complex is cohomologically n– dimensional (in the sense of having no non-trivial cohomology in dimensions above n with respect to any coefficientExpand
On the homotopy type of CW-complexes with aspherical fundamental group
Abstract This paper is concerned with the homotopy type distinction of finite CW-complexes. A ( G , n ) -complex is a finite n -dimensional CW-complex with fundamental-group G and vanishing higherExpand
Stable Modules and the D(2)-Problem
1. Orders in semisimple algebras 2. Representation of finite groups 3. Stable modules and cancellation theorems 4. Relative homological algebra 5. The derived category of a finite group 6.Expand
Equivariant CW-complexes and the orbit category
We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S5 admits a nite G-CW complex X homotopy equivalent to a sphere,Expand
Cancellation of lattices and finite two-complexes.
This is the first in a series of three papers (referred to below s [I], [II] and [III]) on certain cancellation problems which arise in algebra and topology. For example, if M, M\ N are modules withExpand
Realizing algebraic 2-complexes by cell complexes
  • W. Mannan
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2009
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. WeExpand
Minimal resolutions for finite groups
(1) ~~*+Fz-+F,+F,+Z--+O where G acts trivially on Z and each F, is ZG-free. For computation of the cohomology of G it is convenient to choose a resolution (1) in which the Fi do not have too manyExpand
...
1
2
3
4
5
...