Algebraic & Geometric Topology

Poincaré duality complexes with highly connected universal cover

Beatrice Bleile, Imre Bokor, and Jonathan A Hillman

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension 3 ( PD 3 –complexes) generalize to PD n –complexes with ( n 2 ) –connected universal cover for n 3 . Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev’s conjecture on classification.

We prove Turaev’s conjectures on realization and splitting. We show that a triple ( G , ω , μ ) , comprising a group G , a cohomology class ω H 1 ( G ; 2 ) and a homology class μ H n ( G ; ω ) , can be realized by a PD n –complex with ( n 2 ) –connected universal cover if and only if the Turaev map applied to μ yields an equivalence. We show that a PD n –complex with ( n 2 ) –connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups.

We then consider the indecomposable PD n –complexes of this type. When n is odd the results are similar to those for the case n = 3 . The indecomposables are either aspherical or have virtually free fundamental group. When n is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order > 2 then it has two ends.

Article information

Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3749-3788.

Received: 31 October 2016
Revised: 7 July 2018
Accepted: 27 July 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N65: Algebraic topology of manifolds 57P10: Poincaré duality spaces

Poincaré duality complex (or PD-complex) $(n{-}2)$–connected fundamental triple realization theorem splitting theorem indecomposable graph of groups periodic cohomology virtually free


Bleile, Beatrice; Bokor, Imre; Hillman, Jonathan A. Poincaré duality complexes with highly connected universal cover. Algebr. Geom. Topol. 18 (2018), no. 7, 3749--3788. doi:10.2140/agt.2018.18.3749.

Export citation


  • H J Baues, Combinatorial homotopy and $4$–dimensional complexes, De Gruyter Expositions in Mathematics 2, de Gruyter, Berlin (1991)
  • H J Baues, B Bleile, Poincaré duality complexes in dimension four, Algebr. Geom. Topol. 8 (2008) 2355–2389
  • B Bleile, Poincaré duality pairs of dimension three, PhD thesis, The University of Sydney (2004)
  • B Bleile, Poincaré duality pairs of dimension three, Forum Math. 22 (2010) 277–301
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic, New York (1972)
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982)
  • I M Chiswell, Exact sequences associated with a graph of groups, J. Pure Appl. Algebra 8 (1976) 63–74
  • J Crisp, The decomposition of $3$–dimensional Poincaré complexes, Comment. Math. Helv. 75 (2000) 232–246
  • W Dicks, M J Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge Univ. Press (1989)
  • M Golasiński, D L Gonçalves, Free and properly discontinuous actions of groups $G\rtimes\mathbb{Z}^m$ and $G_1\ast_{G_0}G_2$, J. Homotopy Relat. Struct. 11 (2016) 803–824
  • F González-Acuña, C M Gordon, J Simon, Unsolvable problems about higher-dimensional knots and related groups, Enseign. Math. 56 (2010) 143–171
  • I Hambleton, I Madsen, Actions of finite groups on ${\mathbb R}^{n+k}$ with fixed set ${\mathbb R}^k$, Canad. J. Math. 38 (1986) 781–860
  • I Hambleton, E K Pedersen, More examples of discrete co-compact group actions, from “Algebraic topology: applications and new directions” (U Tillmann, S Galatius, D Sinha, editors), Contemp. Math. 620, Amer. Math. Soc., Providence, RI (2014) 133–143
  • J A Hillman, Four-manifolds, geometries and knots, Geometry & Topology Monographs 5, Geometry & Topology Publications, Coventry (2002)
  • J A Hillman, Indecomposable $\mathrm{PD}_3$–complexes, Algebr. Geom. Topol. 12 (2012) 131–153
  • S Plotnick, Homotopy equivalences and free modules, Topology 21 (1982) 91–99
  • D J S Robinson, A course in the theory of groups, Graduate Texts in Mathematics 80, Springer (1982)
  • J-P Serre, Trees, Springer (2003)
  • R G Swan, Periodic resolutions for finite groups, Ann. of Math. 72 (1960) 267–291
  • L R Taylor, Unoriented geometric functors, Forum Math. 20 (2008) 457–467
  • V G Turaev, Three-dimensional Poincaré complexes: homotopy classification and splitting, Mat. Sb. 180 (1989) 809–830 In Russian; translated in Math. USSR-Sb. 67 (1990) 261–282
  • C T C Wall, Finiteness conditions for $\mathrm{CW}$–complexes, Ann. of Math. 81 (1965) 56–69
  • C T C Wall, Finiteness conditions for $\mathrm{CW}$ complexes, II, Proc. Roy. Soc. Ser. A 295 (1966) 129–139
  • C T C Wall, Poincaré complexes, I, Ann. of Math. 86 (1967) 213–245
  • C T C Wall, On the structure of finite groups with periodic cohomology, from “Lie groups: structure, actions, and representations” (A Huckleberry, I Penkov, G Zuckerman, editors), Progr. Math. 306, Springer (2013) 381–413