Algebraic & Geometric Topology

Poincaré duality complexes in dimension four

Hans Joachim Baues and Beatrice Bleile

Full-text: Open access


Generalising Hendriks’ fundamental triples of PD3–complexes, we introduce fundamental triples for PDn–complexes and show that two PDn–complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n–dimensional manifolds. Another main result describes chain complexes with additional algebraic structure which classify homotopy types of PD4–complexes. Up to 2–torsion, homotopy types of PD4–complexes are classified by homotopy types of chain complexes with a homotopy commutative diagonal.

Article information

Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2355-2389.

Received: 26 February 2008
Revised: 20 October 2008
Accepted: 26 October 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57P10: Poincaré duality spaces
Secondary: 55S35: Obstruction theory 55S45: Postnikov systems, $k$-invariants

homotopy types of manifolds PD complex degree 1 map chain complex 4-dimensional manifold


Baues, Hans Joachim; Bleile, Beatrice. Poincaré duality complexes in dimension four. Algebr. Geom. Topol. 8 (2008), no. 4, 2355--2389. doi:10.2140/agt.2008.8.2355.

Export citation


  • H J Baues, Combinatorial homotopy and $4$–dimensional complexes, de Gruyter Expositions in Math. 2, de Gruyter, Berlin (1991) With a preface by R Brown
  • H J Baues, D Conduché, The central series for Peiffer commutators in groups with operators, J. Algebra 133 (1990) 1–34
  • W Browder, Poincaré spaces, their normal fibrations and surgery, Invent. Math. 17 (1972) 191–202
  • A Cavicchioli, F Hegenbarth, On $4$–manifolds with free fundamental group, Forum Math. 6 (1994) 415–429
  • A Cavicchioli, F Spaggiari, On the homotopy type of Poincaré spaces, Ann. Mat. Pura Appl. $(4)$ 180 (2001) 331–358
  • I Hambleton, M Kreck, On the classification of topological $4$–manifolds with finite fundamental group, Math. Ann. 280 (1988) 85–104
  • I Hambleton, M Kreck, P Teichner, Topological $4$–manifolds with geometrically $2$–dimensional fundamental groups
  • F Hegenbarth, S Piccarreta, On Poincaré four-complexes with free fundamental groups, Hiroshima Math. J. 32 (2002) 145–154
  • H Hendriks, Obstruction theory in $3$–dimensional topology: an extension theorem, J. London Math. Soc. $(2)$ 16 (1977) 160–164
  • J A Hillman, ${\rm PD}\sb 4$–complexes with free fundamental group, Hiroshima Math. J. 34 (2004) 295–306
  • J A Hillman, ${\rm PD}\sb 4$–complexes with fundamental group of ${\rm PD}\sb 2$–group, Topology Appl. 142 (2004) 49–60
  • J A Hillman, Strongly minimal ${\rm PD}^4$–complexes, Preprint (2008)
  • A Ranicki, The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. $(3)$ 40 (1980) 87–192
  • A Ranicki, Algebraic Poincaré cobordism, from: “Topology, geometry, and algebra: interactions and new directions”, Contemp. Math. 279, Amer. Math. Soc. (2001) 213–255
  • G A Swarup, On a theorem of C B Thomas, J. London Math. Soc. $(2)$ 8 (1974) 13–21
  • P Teichner, Topological four-manifolds with finite fundamental group, PhD thesis, Johannes Gutenberg Uniersität Mainz (1992)
  • C B Thomas, The oriented homotopy type of compact $3$–manifolds, Proc. London Math. Soc. $(3)$ 19 (1969) 31–44
  • V G Turaev, Three-dimensional Poincaré complexes: homotopy classification and splitting, Mat. Sb. 180 (1989) 809–830
  • C T C Wall, Finiteness conditions for ${\rm CW}$–complexes, Ann. of Math. $(2)$ 81 (1965) 56–69
  • C T C Wall, Poincaré complexes. I, Ann. of Math. $(2)$ 86 (1967) 213–245
  • C T C Wall, Poincaré duality in dimension 3, from: “Proceedings of the Casson Fest”, Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 1–26
  • J H C Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949) 453–496
  • J H C Whitehead, A certain exact sequence, Ann. of Math. $(2)$ 52 (1950) 51–110