Algebraic & Geometric Topology

The LS category of the product of lens spaces

Alexander N Dranishnikov

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


We reduce Rudyak’s conjecture that a degree-one map between closed manifolds cannot raise the Lusternik–Schnirelmann category to the computation of the category of the product of two lens spaces Lpn × Lqn with relatively prime p and q. We have computed cat(Lpn × Lqn) for values p, q > n2. It turns out that our computation supports the conjecture.

For spin manifolds M we establish a criterion for the equality catM = dimM 1, which is a K–theoretic refinement of the Katz–Rudyak criterion for catM = dimM. We apply it to obtain the inequality cat(Lpn × Lqn) 2n 2 for all odd n and odd relatively prime p and q.

Article information

Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2983-3008.

Received: 15 October 2014
Revised: 17 February 2015
Accepted: 20 February 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}

Lusternik–Schnirelmann category lens spaces inessential manifolds ko-theory


Dranishnikov, Alexander N. The LS category of the product of lens spaces. Algebr. Geom. Topol. 15 (2015), no. 5, 2983--3008. doi:10.2140/agt.2015.15.2985.

Export citation


  • I,K Babenko, Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 707–751 In Russian; translated in Russian Acad. Sci. Izv. Math. 41 (1993) 1–38
  • I Berstein, On the Lusternik–Schnirelmann category of Grassmannians, Math. Proc. Cambridge Philos. Soc. 79 (1976) 129–134
  • D Bolotov, A Dranishnikov, On Gromov's scalar curvature conjecture, Proc. Amer. Math. Soc. 138 (2010) 1517–1524
  • G,E Bredon, Sheaf theory, 2nd edition, Graduate Texts in Mathematics 170, Springer, New York (1997)
  • K,S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1994)
  • H Cartan, S Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ (1956)
  • O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Mathematical Surveys and Monographs 103, Amer. Math. Soc. (2003)
  • A,N Dranishnikov, M Katz, Y,B Rudyak, Small values of the Lusternik–Schnirelman 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 Ewing, S Moolgavkar, L Smith, R,E Stong, Stable parallelizability of lens spaces, J. Pure Appl. Algebra 10 (1977/78) 177–191
  • A Franc, Spin structures and Killing spinors on lens spaces, J. Geom. Phys. 4 (1987) 277–287
  • M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1–147
  • M Katz, Y,B Rudyak, Lusternik–Schnirelmann category and systolic category of low-dimensional manifolds, Comm. Pure Appl. Math. 59 (2006) 1433–1456
  • A,A Kosinski, Differential manifolds, Pure and Applied Mathematics 138, Academic Press, Boston (1993)
  • J,H Kwak, The stable parallelizability of a smooth homotopy lens space, J. Pure Appl. Algebra 50 (1988) 155–169
  • R Newton, On Lusternik–Schnirelmann category of connected sums, preprint
  • P Olum, Mappings of manifolds and the notion of degree, Ann. of Math. 58 (1953) 458–480
  • Y,B Rudyak, On Thom spectra, orientability, and cobordism, Springer, Berlin (1998)
  • Y,B Rudyak, On category weight and its applications, Topology 38 (1999) 37–55
  • J-P Serre, Homologie singulière des espaces fibrés: applications, Ann. of Math. 54 (1951) 425–505
  • W Singhof, Minimal coverings of manifolds with balls, Manuscripta Math. 29 (1979) 385–415
  • A,S Švarc, The genus of a fibered space, Trudy Moskov. Mat. Obšč. 10, 11 (1961, 1962) 217–272, 99–126 In Russian; translated in Amer. Math. Soc. Transl. 55 (1966) 49–140
  • C,T,C Wall, Surgery on compact manifolds, 2nd edition, Mathematical Surveys and Monographs 69, Amer. Math. Soc. (1999)