Homology, Homotopy and Applications
Volume 20 (2018)
Partial Euler characteristic, normal generations and the stable $D(2)$ problem
Pages: 105 – 114
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
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