Transactions of the American Mathematical Society

Author(s): Yutaka Hemmi; Yusuke Kawamoto
Journal: Trans. Amer. Math. Soc. 356 (2004), 3823-3839.
MSC (2000): Primary 55P45, 55P48; Secondary 55P15, 52B11
Posted: May 11, 2004
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Abstract: In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an

-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected

-space has the finitely generated mod

cohomology for a prime

and the multiplication of it is homotopy commutative of the

-th order, then it has the mod

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 $i\ge 1$ .

1.: J. Aguadé and L. Smith, On the mod torus theorem of John Hubbuck, Math. Z. 191 (1986), 325-326. MR 87e:57044
2.: A. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer-Verlag, 1972. MR 51:1825
3.: C. Broto and J. A. Crespo, -spaces with noetherian mod two cohomology algebra, Topology 38 (1999), 353-386. MR 99i:55013
4.: W. Browder, The cohomology of covering spaces of -spaces, Bull. Amer. Math. Soc. 65 (1959), 140-141. MR 22:1891
5.: Y. Hemmi, Higher homotopy commutativity of -spaces and the mod torus theorem, Pacific J. Math. 149 (1991), 95-111. MR 92a:55010
6.: J. R. Hubbuck, On homotopy commutative -spaces, Topology 8 (1969), 119-126. MR 38:6592
7.: R. M. Kane, The homology of Hopf spaces, North-Holland Math. Library 40, North-Holland, 1988. MR 90f:55018
8.: M. M. Kapranov, The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra 85 (1993), 119-142. MR 94b:52017
9.: Y. Kawamoto, Loop spaces of -spaces with finitely generated cohomology, Pacific J. Math. 190 (1999), 311-328. MR 2000i:55027
10.: -, Homotopy classification of higher homotopy commutative loop spaces with finitely generated cohomology, Hiroshima Math. J. 30 (2000), 317-344. MR 2001e:55011
11.: -, Higher homotopy commutativity of -spaces with finitely generated cohomology, Pacific J. Math. 204 (2002), 145-161. MR 2003d:55010
12.: Y. Kawamoto and J. P. Lin, Homotopy commutativity of -spaces with finitely generated cohomology, Trans. Amer. Math. Soc. 353 (2001), 4481-4496. MR 2002g:55017
13.: J. P. Lin, A cohomological proof of the torus theorem, Math. Z. 190 (1985), 469-476. MR 87c:55009
14.: -, Loops of -spaces with finitely generated cohomology rings, Topology Appl. 60 (1994), 131-152. MR 95m:55022
15.: -, -spaces with finiteness conditions, Handbook of Algebraic Topology, edited by I.M. James, North-Holland, 1995, 1095-1141. MR 97c:55017
16.: J. P. Lin and F. D. Williams, Homotopy-commutative -spaces, Proc. Amer. Math. Soc. 113 (1991), 857-865. MR 92b:55011
17.: C. A. McGibbon, Higher forms of homotopy commutativity and finite loop spaces, Math. Z. 201 (1989), 363-374. MR 90f:55019
18.: R. J. Milgram, Iterated loop spaces, Ann. of Math. 84 (1966), 386-403. MR 34:6767
19.: M. Mimura and H. Toda, Topology of Lie groups, I and II, Trans. Math. Monographs 91, Amer. Math. Soc., 1991. MR 92h:55001
20.: V. Reiner and G. M. Ziegler, Coxeter-associahedra, Mathematika 41 (1994), 364-393. MR 95m:52023
21.: M. Slack, A classification theorem for homotopy commutative mod -spaces with finitely generated cohomology rings, Mem. Amer. Math. Soc. 92 (1991). MR 92k:55015
22.: J. D. Stasheff, Homotopy associativity of -spaces, I and II, Trans. Amer. Math. Soc. 108 (1963), 275-292, 293-312. MR 28:1623
23.: -, -spaces from a homotopy point of view, Lecture Notes in Math. 161, Springer-Verlag, 1970. MR 42:5261
24.: M. Sugawara, A condition that a space is group-like, Math. J. Okayama Univ. 7 (1957), 123-149. MR 20:3546
25.: -, On the homotopy commutativity of groups and loop spaces, Mem. College Sci. Univ. Kyoto Ser. A 33 (1960), 257-269. MR 22:11394
26.: F. D. Williams, Higher homotopy-commutativity, Trans. Amer. Math. Soc. 139 (1969), 191-206. MR 39:2163
27.: -, Higher homotopy commutativity and extension of maps, Proc. Amer. Math. Soc. 26 (1970), 664-670. MR 42:8491
28.: -, Higher Samelson products, J. Pure Appl. Algebra 2 (1972), 249-260. MR 46:2670
29.: G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Math. 152, Springer-Verlag, 1994. MR 96a:52011

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P45, 55P48, 55P15, 52B11

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: 10.1090/S0002-9947-04-03647-5
PII: S 0002-9947(04)03647-5
Keywords: Higher homotopy commutativity, $H$-spaces, $A_n$-spaces, $AC_n$-spaces, permuto-associahedra
Received by editor(s): November 27, 2001
Posted: May 11, 2004
Dedicated: Dedicated to the memory of Professor Masahiro Sugawara
Copyright of article: Copyright 2004, American Mathematical Society

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