Topological complexity of 𝐻-spaces

Lupton, Gregory; Scherer, Jérôme

doi:10.1090/S0002-9939-2012-11454-6

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

Topological complexity of -spaces

Authors: Gregory Lupton and Jérôme Scherer
Journal: Proc. Amer. Math. Soc. 141 (2013), 1827-1838
MSC (2010): Primary 55M30, 55S40, 57T99, 70Q05
Posted: October 23, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a (not-necessarily homotopy-associative) -space. We show that $\mathrm {TC}_{n+1}(X) = \mathrm {cat}(X^n)$ , for $n \geq 1$ , where $\mathrm {TC}_{n+1}(-)$ denotes the so-called higher topological complexity introduced by Rudyak, and $\mathrm {cat}(-)$ denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for $\mathrm {TC}_{n+1}(X)$ , in the setting of a space acting on .

1. I. Basabe, J. González, Y. Rudyak, and D. Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres, preprint, arXiv:1009.1851v5[math.AT], 2010.
2. Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003. MR 1990857 (2004e:55001)
3. M. Cristina Costoya-Ramos, Catégorie de Lusternik-Schnirelmann et genre des 𝐻₀-espaces, Proc. Amer. Math. Soc. 131 (2003), no. 2, 637–645 (electronic) (French, with French summary). MR 1933357 (2003f:55005), http://dx.doi.org/10.1090/S0002-9939-02-06533-4
4. Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221. MR 1957228 (2004c:68132), http://dx.doi.org/10.1007/s00454-002-0760-9
5. Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245–266. MR 2074919 (2005g:68166), http://dx.doi.org/10.1016/j.topol.2003.07.011
6. Michael Farber, Topology of robot motion planning, Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht, 2006, pp. 185–230. MR 2276952 (2008d:68141), http://dx.doi.org/10.1007/1-4020-4266-3_05
7. Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014)
8. Lucía Fernández-Suárez, Antonio Gómez-Tato, Jeffrey Strom, and Daniel Tanré, The Lusternik-Schnirelmann category of 𝑆𝑝(3), Proc. Amer. Math. Soc. 132 (2004), no. 2, 587–595. MR 2022385 (2004m:55005), http://dx.doi.org/10.1090/S0002-9939-03-07019-9
9. Pierre Ghienne, The Lusternik-Schnirelmann category of spaces in the Mislin genus of 𝑆𝑝(3), Lusternik-Schnirelmann category and related topics (South Hadley, MA, 2001), Contemp. Math., vol. 316, Amer. Math. Soc., Providence, RI, 2002, pp. 121–126. MR 1962158 (2004b:55006), http://dx.doi.org/10.1090/conm/316/05500
10. Norio Iwase and Akira Kono, Lusternik-Schnirelmann category of 𝑆𝑝𝑖𝑛(9), Trans. Amer. Math. Soc. 359 (2007), no. 4, 1517–1526 (electronic). MR 2272137 (2008b:55007), http://dx.doi.org/10.1090/S0002-9947-06-04120-1
11. I. M. James, Multiplication on spheres. II, Trans. Amer. Math. Soc. 84 (1957), 545–558. MR 0090812 (19,875e), http://dx.doi.org/10.1090/S0002-9947-1957-0090812-8
12. I. M. James, On 𝐻-spaces and their homotopy groups, Quart. J. Math. Oxford Ser. (2) 11 (1960), 161–179. MR 0133132 (24 #A2966)
13. Yuli B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916–920. MR 2593704 (2011c:55009), http://dx.doi.org/10.1016/j.topol.2009.12.007
14. Yuli B. Rudyak, Erratum to “On higher analogs of topological complexity” [Topology Appl. 157 (5) (2010) 916–920] [MR2593704], Topology Appl. 157 (2010), no. 6, 1118. MR 2593724 (2011c:55010), http://dx.doi.org/10.1016/j.topol.2010.02.004
15. Wilhelm Singhof, On the Lusternik-Schnirelmann category of Lie groups, Math. Z. 145 (1975), no. 2, 111–116. MR 0391075 (52 #11897)
16. Masahiro Sugawara, On a condition that a space is an 𝐻-space, Math. J. Okayama Univ. 6 (1957), 109–129. MR 0086303 (19,160c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55M30, 55S40, 57T99, 70Q05

Retrieve articles in all journals with MSC (2010): 55M30, 55S40, 57T99, 70Q05

Additional Information

Gregory Lupton
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
Email: G.Lupton@csuohio.edu

Jérôme Scherer
Affiliation: SB Mathgeom, Ma B3 455, Station 8, EPFL, CH-1015 Lausanne, Switzerland
Email: jerome.scherer@epfl.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11454-6
PII: S 0002-9939(2012)11454-6
Keywords: Lusternik-Schnirelmann category, sectional category, topological complexity, $H$-space
Received by editor(s): June 16, 2011
Received by editor(s) in revised form: September 2, 2011
Posted: October 23, 2012
Additional Notes: The first author acknowledges the hospitality and support of EPFL and the support of the Cleveland State University FRD grant program.
The second author is partially supported by FEDER/MEC grant MTM2010-20692. Both authors acknowledge the support of the Swiss National Science Foundation (project IZK0Z2_133237).
Communicated by: Brooke Shipley
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|