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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Topological complexity of $ H$-spaces


Authors: Gregory Lupton and Jérôme Scherer
Journal: Proc. Amer. Math. Soc. 141 (2013), 1827-1838
MSC (2010): Primary 55M30, 55S40, 57T99, 70Q05
Published electronically: October 23, 2012
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Abstract: Let $ X$ be a (not-necessarily homotopy-associative) $ H$-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 $ Y$ acting on $ X$.


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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
Published electronically: 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.