Algebraic & Geometric Topology

Indecomposable $\mathrm{PD}_3$–complexes

Jonathan A Hillman

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Abstract

We show that if X is an indecomposable PD3–complex and π1(X) is the fundamental group of a reduced finite graph of finite groups but is neither nor 2 then X is orientable, the underlying graph is a tree, the vertex groups have cohomological period dividing 4 and all but at most one of the edge groups is 2. If there are no exceptions then all but at most one of the vertex groups is dihedral of order 2m with m odd. Every such group is realized by some PD3–complex. Otherwise, one edge group may be 6. We do not know of any such examples.

We also ask whether every PD3–complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold, and we propose a strategy for tackling this question.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 1 (2012), 131-153.

Dates
Received: 27 January 2009
Revised: 23 October 2011
Accepted: 28 October 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715335

Digital Object Identifier
doi:10.2140/agt.2012.12.131

Mathematical Reviews number (MathSciNet)
MR2916272

Zentralblatt MATH identifier
1252.57009

Subjects
Primary: 57M05: Fundamental group, presentations, free differential calculus 57M99: None of the above, but in this section
Secondary: 57P10: Poincaré duality spaces

Keywords
degree–$1$ map Dehn surgery graph of groups indecomposable $3$–manifold $\mathrm{PD}_3$–complex $\mathrm{PD}_3$–group periodic cohomology virtually free

Citation

Hillman, Jonathan A. Indecomposable $\mathrm{PD}_3$–complexes. Algebr. Geom. Topol. 12 (2012), no. 1, 131--153. doi:10.2140/agt.2012.12.131. https://projecteuclid.org/euclid.agt/1513715335


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References

  • B H Bowditch, Planar groups and the Seifert conjecture, J. Reine Angew. Math. 576 (2004) 11–62
  • W Browder, Poincaré spaces, their normal fibrations and surgery, Invent. Math. 17 (1972) 191–202
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
  • M Carlette, The automorphism group of accessible groups
  • J Crisp, The decomposition of $3$–dimensional Poincaré complexes, Comment. Math. Helv. 75 (2000) 232–246
  • C W Curtis, I Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Math. XI, Wiley Interscience, New York-London (1962)
  • J F Davis, R J Milgram, A survey of the spherical space form problem, Math. Reports 2, Harwood Academic, Chur, Switzerland (1985)
  • M W Davis, Poincaré duality groups, from: “Surveys on surgery theory, Vol. 1”, (S Cappell, A Ranicki, J Rosenberg, editors), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 167–193
  • W Dicks, M J Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Math. 17, Cambridge Univ. Press (1989)
  • B Eckmann, P Linnell, Poincaré duality groups of dimension two, II, Comment. Math. Helv. 58 (1983) 111–114
  • B Eckmann, H Müller, Poincaré duality groups of dimension two, Comment. Math. Helv. 55 (1980) 510–520
  • D Gabai, The simple loop conjecture, J. Differential Geom. 21 (1985) 143–149
  • S Gadgil, Degree-one maps, surgery and four-manifolds, Bull. Lond. Math. Soc. 39 (2007) 419–424
  • D Groves, personal communication (2011)
  • I Hambleton, I Madsen, Actions of finite groups on ${\bf R}\sp {n+k}$ with fixed set ${\bf R}\sp k$, Canad. J. Math. 38 (1986) 781–860
  • J-C Hausmann, P Vogel, Geometry on Poincaré spaces, Math. Notes 41, Princeton Univ. Press (1993)
  • H Hendriks, Applications de la théorie d'obstruction en dimension $3$, Bull. Soc. Math. France Mém. (1977) 81–196
  • H Hendriks, Obstruction theory in $3$–dimensional topology: an extension theorem, J. London Math. Soc. 16 (1977) 160–164
  • J A Hillman, On $3$–dimensional Poincaré duality complexes and $2$–knot groups, Math. Proc. Cambridge Philos. Soc. 114 (1993) 215–218
  • J A Hillman, Four-manifolds, geometries and knots, Geom. Topol. Monogr. 5, Geom. Topol. Publ., Coventry (2002)
  • J A Hillman, An indecomposable $\rm PD\sb 3$-complex: II, Algebr. Geom. Topol. 4 (2004) 1103–1109
  • J A Hillman, An indecomposable $\rm PD\sb 3$–complex whose fundamental group has infinitely many ends, Math. Proc. Cambridge Philos. Soc. 138 (2005) 55–57
  • J A Hillman, D H Kochloukova, Finiteness conditions and ${\rm PD}\sb r$-group covers of ${\rm PD}\sb n$–complexes, Math. Z. 256 (2007) 45–56
  • B Jahren, S Kwasik, Three-dimensional surgery theory, UNil-groups and the Borel conjecture, Topology 42 (2003) 1353–1369
  • R C Kirby, L R Taylor, A survey of $4$–manifolds through the eyes of surgery, from: “Surveys on surgery theory, Vol. 2”, (A A Ranicki, editor), Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 387–421
  • J Milnor, Groups which act on $S\sp n$ without fixed points, Amer. J. Math. 79 (1957) 623–630
  • D J S Robinson, A course in the theory of groups, Graduate Texts in Math. 80, Springer, New York (1982)
  • C B Thomas, On Poincaré $3$–complexes with binary polyhedral fundamental group, Math. Ann. 226 (1977) 207–221
  • V G Turaev, Three-dimensional Poincaré complexes: homotopy classification and splitting, Mat. Sb. 180 (1989) 809–830 In Russian; translated in Math. USSR-Sbornik 67 (1990) 261-282
  • C T C Wall, Poincaré complexes: I, Ann. of Math. 86 (1967) 213–245
  • B Zimmermann, Das Nielsensche Realisierungsproblem für hinreichend groß e $3$–Mannigfaltigkeiten, Math. Z. 180 (1982) 349–359