Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

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

Pages: 105 – 114

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n2.a6


Feng Ji (Infinitus, Nanyang Technological University, Singapore)

Shengkui Ye (Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, Jiangsu, China)


We study the interplay among Wall’s $D(2)$ problem, the 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 $S^2$.


$D(2)$ problem, cohomological dimensions, Quillen’s plus construction

2010 Mathematics Subject Classification

57M05, 57M20

The second author is supported by Jiangsu Natural Science Foundation (No. BK20140402) and NSFC (Nos. 11501459, 11771345, 11771022).

Received 3 November 2017

Received revised 28 December 2017

Published 9 May 2018