Algebraic & Geometric Topology

Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

John Oprea and Jeff Strom

Full-text: Open access

Abstract

In this paper, we study the growth with respect to dimension of quite general homotopy invariants Q applied to the CW skeleta of spaces. This leads to upper estimates analogous to the classical “dimension divided by connectivity” bound for Lusternik–Schnirelmann category. Our estimates apply, in particular, to the Clapp–Puppe theory of A–category. We use cat1(X) (which is A–category with A the collection of 1–dimensional CW complexes), to reinterpret in homotopy-theoretical terms some recent work of Dranishnikov on the Lusternik–Schnirelmann category of spaces with fundamental groups of finite cohomological dimension. Our main result is the inequality cat(X) dim(Bπ1(X))+ cat1(X), which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1165-1186.

Dates
Received: 15 December 2009
Revised: 22 April 2010
Accepted: 24 April 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715129

Digital Object Identifier
doi:10.2140/agt.2010.10.1165

Mathematical Reviews number (MathSciNet)
MR2653059

Zentralblatt MATH identifier
1204.55004

Subjects
Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 55P99: None of the above, but in this section

Keywords
Lusternik–Schnirelmann category skeleta fundamental group symplectic manifold

Citation

Oprea, John; Strom, Jeff. Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov. Algebr. Geom. Topol. 10 (2010), no. 2, 1165--1186. doi:10.2140/agt.2010.10.1165. https://projecteuclid.org/euclid.agt/1513715129


Export citation

References

  • P S Alexandrov, Combinatorial topology. Vol. 1, 2 and 3, Dover Publications, Mineola, NY (1998)
  • C Allday, J Oprea, A c-symplectic free $S\sp 1$–manifold with contractible orbits and ${\rm cat}=\frac12$ DIM, Proc. Amer. Math. Soc. 134 (2006) 599–604
  • I Berstein, On the Lusternik–Schnirelmann category of Grassmannians, Math. Proc. Cambridge Philos. Soc. 79 (1976) 129–134
  • M Clapp, D Puppe, Invariants of the Lusternik–Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986) 603–620
  • O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Math. Surveys and Monogr. 103, Amer. Math. Soc. (2003)
  • A N Dranishnikov, On the Lusternik–Schnirelmann category of spaces with $2$–dimensional fundamental group, Proc. Amer. Math. Soc. 137 (2009) 1489–1497
  • A N Dranishnikov, The Lusternik–Schnirelmann category and the fundamental group, Algebr. Geom. Topol. 10 (2010) 917–924
  • A N Dranishnikov, M G Katz, Y B Rudyak, Small values of the Lusternik–Schnirelmann category for manifolds, Geom. Topol. 12 (2008) 1711–1727
  • A N Dranishnikov, Y B Rudyak, On the Berstein–Svarc theorem in dimension 2, Math. Proc. Cambridge Philos. Soc. 146 (2009) 407–413
  • J C Gómez-Larrañaga, F González-Acuña, Lusternik–Schnirelmann category of $3$–manifolds, Topology 31 (1992) 791–800
  • R E Gompf, A new construction of symplectic manifolds, Ann. of Math. $(2)$ 142 (1995) 527–595
  • D P Grossman, An estimation of the category of Lusternik–Shnirelman, C. R. $($Doklady$)$ Acad. Sci. URSS $($N.S.$)$ 54 (1946) 109–112
  • J A Hillman, ${\rm PD}\sb 4$–complexes with free fundamental group, Hiroshima Math. J. 34 (2004) 295–306
  • R Ibáñez, J Kędra, Y Rudyak, A Tralle, On fundamental groups of symplectically aspherical manifolds, Math. Z. 248 (2004) 805–826
  • A Lundell, S Weingram, The topology of CW complexes, Univ. Ser. Higher Math. VIII, Van Nostrand Reinhold, New York (1969)
  • T Matumoto, A Katanaga, On $4$–dimensional closed manifolds with free fundamental groups, Hiroshima Math. J. 25 (1995) 367–370
  • R Nendorf, N Scoville, J Strom, Categorical sequences, Algebr. Geom. Topol. 6 (2006) 809–838
  • J Oprea, Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group, Proc. Amer. Math. Soc. 130 (2002) 833–839
  • J Oprea, Y Rudyak, Detecting elements and Lusternik–Schnirelmann category of $3$–manifolds, from: “Lusternik–Schnirelmann category and related topics (South Hadley, MA, 2001)”, (O Cornea, G Lupton, J Oprea, D Tanré, editors), Contemp. Math. 316, Amer. Math. Soc. (2002) 181–191
  • P A Ostrand, Dimension of metric spaces and Hilbert's problem $13$, Bull. Amer. Math. Soc. 71 (1965) 619–622
  • F Roth, On the category of Euclidean configuration spaces and associated fibrations, from: “Groups, homotopy and configuration spaces”, (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13, Geom. Topol. Publ., Coventry (2008) 447–461
  • Y B Rudyak, J Oprea, On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z. 230 (1999) 673–678
  • H Seifert, W Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Math. 89, Academic Press, New York (1980) With a preface by Joan S Birman, With “Topology of $3$–dimensional fibered spaces” by Seifert
  • B Strom, Personal communication (2008)
  • J Strom, Lusternik–Schnirelmann category of spaces with free fundamental group, Algebr. Geom. Topol. 7 (2007) 1805–1808
  • A S Švarc, The genus of a fibered space, Trudy Moskov. Mat. Obšč. 10 (1961) 217–272