Algebraic & Geometric Topology

One-point reductions of finite spaces, $h$–regular CW–complexes and collapsibility

Jonathan Barmak and Elias Minian

Full-text: Open access

Abstract

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h–regular CW–complex, generalizing the concept of regular CW–complex, and prove that the h–regular CW–complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 3 (2008), 1763-1780.

Dates
Received: 12 March 2008
Revised: 5 September 2008
Accepted: 5 September 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796905

Digital Object Identifier
doi:10.2140/agt.2008.8.1763

Mathematical Reviews number (MathSciNet)
MR2448871

Zentralblatt MATH identifier
1227.55005

Subjects
Primary: 55U05: Abstract complexes 55P15: Classification of homotopy type 57Q05: General topology of complexes 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]
Secondary: 06A06: Partial order, general 52B70: Polyhedral manifolds

Keywords
finite topological spaces simplicial complexes regular CW-complexes collapses weak homotopy types posets

Citation

Barmak, Jonathan; Minian, Elias. One-point reductions of finite spaces, $h$–regular CW–complexes and collapsibility. Algebr. Geom. Topol. 8 (2008), no. 3, 1763--1780. doi:10.2140/agt.2008.8.1763. https://projecteuclid.org/euclid.agt/1513796905


Export citation

References

  • J J Andrews, M L Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965) 192–195
  • J A Barmak, E G Minian, Minimal finite models, J. Homotopy Relat. Struct. 2 (2007) 127–140
  • J A Barmak, E G Minian, Simple homotopy types and finite spaces, Adv. Math. 218 (2008) 87–104
  • A Bj örner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984) 7–16
  • M M Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics 10, Springer, New York (1973)
  • D N Kozlov, Collapsing along monotone poset maps, Int. J. Math. Math. Sci. (2006) Art. ID 79858, 8
  • J May, Finite spaces and simplicial complexes, Notes for REU (2003) Available at \setbox0\makeatletter\@url http://www.math.uchicago.edu/~may/MISCMaster.html {\unhbox0
  • J May, Finite topological spaces, Notes for REU (2003) Available at \setbox0\makeatletter\@url http://www.math.uchicago.edu/~may/MISCMaster.html {\unhbox0
  • M C McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966) 465–474
  • J Milnor, Construction of universal bundles. II, Ann. of Math. $(2)$ 63 (1956) 430–436
  • T Osaki, Reduction of finite topological spaces, Interdiscip. Inform. Sci. 5 (1999) 149–155
  • D Quillen, Homotopy properties of the poset of nontrivial $p$–subgroups of a group, Adv. in Math. 28 (1978) 101–128
  • R E Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966) 325–340
  • R E Stong, Group actions on finite spaces, Discrete Math. 49 (1984) 95–100
  • J Walker, Topology and combinatorics of ordered sets, PhD thesis, MIT Cambridge, MA (1981)
  • V Welker, Constructions preserving evasiveness and collapsibility, Discrete Math. 207 (1999) 243–255