Algebraic & Geometric Topology

The Lusternik–Schnirelmann category and the fundamental group

Alexander N Dranishnikov

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Abstract

We prove that

cat LS X cd ( π 1 ( X ) ) + dim X 1 2

for every CW–complex X where cd(π1(X)) denotes the cohomological dimension of the fundamental group of X.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 917-924.

Dates
Received: 23 September 2009
Revised: 16 February 2010
Accepted: 2 March 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715120

Digital Object Identifier
doi:10.2140/agt.2010.10.917

Mathematical Reviews number (MathSciNet)
MR2629770

Zentralblatt MATH identifier
1194.55008

Subjects
Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space

Keywords
Lusternik–Schnirelmann category cohomological dimension fundamental group

Citation

Dranishnikov, Alexander N. The Lusternik–Schnirelmann category and the fundamental group. Algebr. Geom. Topol. 10 (2010), no. 2, 917--924. doi:10.2140/agt.2010.10.917. https://projecteuclid.org/euclid.agt/1513715120


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References

  • G E Bredon, Introduction to compact transformation groups, 46, Academic Press, New York (1972)
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
  • O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Math. Surveys and Monogr. 103, Amer. Math. Soc. (2003)
  • A Dold, Partitions of unity in the theory of fibrations, Ann. of Math. $(2)$ 78 (1963) 223–255
  • 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, M G Katz, Y B Rudyak, Small values of the Lusternik–Schnirelmann category for manifolds, Geom. Topol. 12 (2008) 1711–1727
  • S Eilenberg, T Ganea, On the Lusternik–Schnirelmann category of abstract groups, Ann. of Math. $(2)$ 65 (1957) 517–518
  • D P Grossman, An estimation of the category of Lusternik–Schnirelmann, C. R. $($Doklady$)$ Acad. Sci. URSS $($N.S.$)$ 54 (1946) 109–112
  • W Hurewicz, On the concept of fiber space, Proc. Nat. Acad. Sci. USA 41 (1955) 956–961
  • P A Ostrand, Dimension of metric spaces and Hilbert's problem $13$, Bull. Amer. Math. Soc. 71 (1965) 619–622
  • E H Spanier, Algebraic topology, McGraw-Hill, New York (1966)
  • J Stallings, Groups of dimension 1 are locally free, Bull. Amer. Math. Soc. 74 (1968) 361–364
  • 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
  • R G Swan, Groups of cohomological dimension one, J. Algebra 12 (1969) 585–610
  • G W Whitehead, The homology suspension, from: “Colloque de topologie algébrique, Louvain, 1956”, Georges Thone, Liège (1957) 89–95