Partial Euler Characteristic, Normal Generations and the stable D(2) problem

TL;DR

This paper explores the relationships between the D(2) problem, normal generation of perfect groups, and partial Euler characteristics, proving a homotopy equivalence result for certain 3-complexes under the Wiegold conjecture.

Contribution

It establishes that, assuming the Wiegold conjecture, certain 3-dimensional complexes with finite fundamental groups are homotopy equivalent to finite 2-complexes after wedging with a sphere.

Findings

Proves homotopy equivalence to finite 2-complex after wedging with S^2

Links the D(2) problem with normal generation and Euler characteristic conjectures

Provides conditions under which 3-complexes simplify topologically

Abstract

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 a 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 $S^2$.