## Algebraic & Geometric Topology

### Quillen's plus construction and the D(2) problem

W H Mannan

#### Abstract

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 group presentation. This reduces the D(2) problem to a question about perfect normal subgroups.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1399-1411.

Dates
Revised: 21 February 2009
Accepted: 9 June 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513797033

Digital Object Identifier
doi:10.2140/agt.2009.9.1399

Mathematical Reviews number (MathSciNet)
MR2520404

Zentralblatt MATH identifier
1176.57004

#### Citation

Mannan, W H. Quillen's plus construction and the D(2) problem. Algebr. Geom. Topol. 9 (2009), no. 3, 1399--1411. doi:10.2140/agt.2009.9.1399. https://projecteuclid.org/euclid.agt/1513797033

#### References

• F R Beyl, N Waller, Examples of exotic free $2$–complexes and stably free nonfree modules for quaternion groups, Algebr. Geom. Topol. 8 (2008) 1–17
• M R Bridson, M Tweedale, Deficiency and abelianized deficiency of some virtually free groups, Math. Proc. Cambridge Philos. Soc. 143 (2007) 257–264
• D E Cohen, Combinatorial group theory: a topological approach, London Math. Soc. Student Texts 14, Cambridge Univ. Press (1989)
• T Edwards, Generalised Swan modules and the D(2) problem, Algebr. Geom. Topol. 6 (2006) 71–89
• R H Fox, Free differential calculus. V. The Alexander matrices re-examined, Ann. of Math. $(2)$ 71 (1960) 408–422
• J Harlander, Some aspects of efficiency, from: “Groups–-Korea '98 (Pusan)”, (Y G Baik, D L Johnson, A C Kim, editors), de Gruyter, Berlin (2000) 165–180
• F E A Johnson, Explicit homotopy equivalences in dimension two, Math. Proc. Cambridge Philos. Soc. 133 (2002) 411–430
• F E A Johnson, Stable modules and the $D(2)$–problem, London Math. Soc. Lecture Note Ser. 301, Cambridge Univ. Press (2003)
• W H Mannan, The $D(2)$ property for $D\sb 8$, Algebr. Geom. Topol. 7 (2007) 517–528
• J Rosenberg, Algebraic $K$–theory and its applications, Graduate Texts in Math. 147, Springer, New York (1994)
• J R Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. $(2)$ 88 (1968) 312–334
• R G Swan, Groups of cohomological dimension one, J. Algebra 12 (1969) 585–610
• C T C Wall, Finiteness conditions for ${\rm CW}$–complexes, Ann. of Math. $(2)$ 81 (1965) 56–69