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Higher homotopy commutativity of $H$-spaces and the permuto-associahedra


Authors: Yutaka Hemmi and Yusuke Kawamoto
Journal: Trans. Amer. Math. Soc. 356 (2004), 3823-3839
MSC (2000): Primary 55P45, 55P48; Secondary 55P15, 52B11
DOI: https://doi.org/10.1090/S0002-9947-04-03647-5
Published electronically: May 11, 2004
MathSciNet review: 2058507
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Abstract: In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an $A_n$-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected $A_p$-space has the finitely generated mod $p$ cohomology for a prime $p$ and the multiplication of it is homotopy commutative of the $p$-th order, then it has the mod $p$ homotopy type of a finite product of Eilenberg-Mac Lane spaces $K(\mathbb{Z},1)$s, $K(\mathbb{Z},2)$s and $K(\mathbb{Z}/p^i,1)$s for $i\ge 1$.

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Additional Information

Yutaka Hemmi
Affiliation: Department of Mathematics, Faculty of Science, Kochi University, Kochi 780-8520, Japan
Email: hemmi@math.kochi-u.ac.jp

Yusuke Kawamoto
Affiliation: Department of Mathematics, National Defense Academy, Yokosuka 239-8686, Japan
Email: yusuke@nda.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-04-03647-5
Keywords: Higher homotopy commutativity, $H$-spaces, $A_n$-spaces, $AC_n$-spaces, permuto-associahedra
Received by editor(s): November 27, 2001
Published electronically: May 11, 2004
Dedicated: Dedicated to the memory of Professor Masahiro Sugawara
Article copyright: © Copyright 2004 American Mathematical Society