Algebraic & Geometric Topology

Even triangulations of $n$–dimensional pseudo-manifolds

J Hyam Rubinstein and Stephan Tillmann

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

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

Abstract

This paper introduces even triangulations of n–dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n–manifolds, an interesting connection between very short hierarchies and even triangulations is observed.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2947-2982.

Dates
Received: 13 October 2014
Revised: 3 February 2015
Accepted: 5 February 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841038

Digital Object Identifier
doi:10.2140/agt.2015.15.2949

Mathematical Reviews number (MathSciNet)
MR3426699

Zentralblatt MATH identifier
1346.57014

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M60: Group actions in low dimensions 57Q15: Triangulating manifolds

Keywords
3-manifold n-manifold triangulation even triangulation normal surface normal hypersurface representations of the fundamental group

Citation

Rubinstein, J Hyam; Tillmann, Stephan. Even triangulations of $n$–dimensional pseudo-manifolds. Algebr. Geom. Topol. 15 (2015), no. 5, 2947--2982. doi:10.2140/agt.2015.15.2949. https://projecteuclid.org/euclid.agt/1510841038


Export citation

References

  • I Agol, M Culler, P,B Shalen, Singular surfaces, mod $2$ homology, and hyperbolic volume, I, Trans. Amer. Math. Soc. 362 (2010) 3463–3498
  • I,R Aitchison, S Matsumoto, J,H Rubinstein, Immersed surfaces in cubed manifolds, Asian J. Math. 1 (1997) 85–95
  • L Bartolini, J,H Rubinstein, One-sided Heegaard splittings of $\Bbb R{\rm P}\sp 3$, Algebr. Geom. Topol. 6 (2006) 1319–1330
  • B,A Burton, R Budney, W Pettersson, et al., Regina: Software for $3$–manifold topology and normal surface theory, open source software (1999–2014) Available at \setbox0\makeatletter\@url http://regina.sourceforge.net {\unhbox0
  • B,A Burton, B Foozwell, J,H Rubinstein, Normal $3$–manifolds in triangulated $4$–manifolds, in preparation
  • D Cooper, S Tillmann, Transversely oriented normal surfaces, in preparation
  • D Cooper, S Tillmann, The Thurston norm via normal surfaces, Pacific J. Math. 239 (2009) 1–15
  • M Culler, P,B Shalen, Singular surfaces, mod $2$ homology, and hyperbolic volume, II, Topology Appl. 158 (2011) 118–131
  • G David, R Kirby, Trisecting $4$–manifolds (2013)
  • B Foozwell, H Rubinstein, Introduction to the theory of Haken $n$–manifolds, from: “Topology and geometry in dimension three”, (W Li, L Bartolini, J Johnson, F Luo, R Myers, J,H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc. (2011) 71–84
  • F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239–1284
  • I Izmestiev, M Joswig, Branched coverings, triangulations, and $3$–manifolds, Adv. Geom. 3 (2003) 191–225
  • W Jaco, H Rubinstein, S Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009) 157–180
  • W Jaco, J,H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
  • W Jaco, J,H Rubinstein, S Tillmann, Coverings and minimal triangulations of $3$–manifolds, Algebr. Geom. Topol. 11 (2011) 1257–1265
  • W Jaco, J,H Rubinstein, S Tillmann, $\Z\sb 2$–Thurston norm and complexity of $3$–manifolds, Math. Ann. 356 (2013) 1–22
  • M Joswig, Projectivities in simplicial complexes and colorings of simple polytopes, Math. Z. 240 (2002) 243–259
  • E Kang, J,H Rubinstein, Ideal triangulations of $3$–manifolds, I: spun normal surface theory, from: “Proceedings of the Casson Fest”, (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7 (2004) 235–265
  • M Lackenby, Finite covering spaces of $3$–manifolds, from: “Proceedings of the International Congress of Mathematicians, Vol II”, (R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, editors), Hindustan Book Agency, New Delhi (2010) 1042–1070
  • M Lackenby, Finding disjoint surfaces in $3$–manifolds, Geom. Dedicata 170 (2014) 385–401
  • A Lubotzky, On finite index subgroups of linear groups, Bull. London Math. Soc. 19 (1987) 325–328
  • P Orlik, Seifert manifolds, Lecture Notes in Mathematics 291, Springer, Berlin (1972)
  • J,H Rubinstein, One-sided Heegaard splittings of $3$–manifolds, Pacific J. Math. 76 (1978) 185–200
  • J,H Rubinstein, On $3$–manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979) 129–137
  • J,H Rubinstein, J,S Birman, One-sided Heegaard splittings and homeotopy groups of some $3$–manifolds, Proc. London Math. Soc. 49 (1984) 517–536
  • J,H Rubinstein, S Tillmann, Multisections of piecewise linear manifolds, in preparation
  • H Seifert, W Threlfall, Lehrbuch der Topologie, B,G Teubner, Leipzig (1934)
  • P,B Shalen, P Wagreich, Growth rates, $\mathbb{Z}_p$–homology, and volumes of hyperbolic $3$–manifolds, Trans. Amer. Math. Soc. 331 (1992) 895–917
  • S Tillmann, Normal surfaces in topologically finite $3$–manifolds, Enseign. Math. 54 (2008) 329–380