Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

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

Pages: 105 – 114

DOI: http://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

Full Text (PDF format)

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