Topological complexity of 𝐻-spaces

Lupton, Gregory; Scherer, Jérôme

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

Topological complexity of $H$-spaces
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by Gregory Lupton and Jérôme Scherer PDF

Proc. Amer. Math. Soc. 141 (2013), 1827-1838 Request permission

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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