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

Hemmi, Yutaka; Kawamoto, Yusuke

doi:10.1090/S0002-9947-04-03647-5

Higher homotopy commutativity of $H$-spaces and the permuto-associahedra
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by Yutaka Hemmi and Yusuke Kawamoto PDF

Trans. Amer. Math. Soc. 356 (2004), 3823-3839 Request permission

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 $i\ge 1$.

References

Jaume Aguadé and Larry Smith, On the mod$\,p$ torus theorem of John Hubbuck, Math. Z. 191 (1986), no. 2, 325–326. MR 818677, DOI 10.1007/BF01164037
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
Carlos Broto and Juan A. Crespo, $H$-spaces with Noetherian mod two cohomology algebra, Topology 38 (1999), no. 2, 353–386. MR 1660329, DOI 10.1016/S0040-9383(98)00017-2
William Browder, The cohomology of covering spaces of $H$-spaces, Bull. Amer. Math. Soc. 65 (1959), 140–141. MR 111024, DOI 10.1090/S0002-9904-1959-10299-8
Yutaka Hemmi, Higher homotopy commutativity of $H$-spaces and the mod $p$ torus theorem, Pacific J. Math. 149 (1991), no. 1, 95–111. MR 1099785, DOI 10.2140/pjm.1991.149.95
J. R. Hubbuck, On homotopy commutative $H$-spaces, Topology 8 (1969), 119–126. MR 238316, DOI 10.1016/0040-9383(69)90004-4
Richard M. Kane, The homology of Hopf spaces, North-Holland Mathematical Library, vol. 40, North-Holland Publishing Co., Amsterdam, 1988. MR 961257
Mikhail M. Kapranov, The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra 85 (1993), no. 2, 119–142. MR 1207505, DOI 10.1016/0022-4049(93)90049-Y
Yusuke Kawamoto, Loop spaces of $H$-spaces with finitely generated cohomology, Pacific J. Math. 190 (1999), no. 2, 311–328. MR 1722897, DOI 10.2140/pjm.1999.190.311
Yusuke Kawamoto, Homotopy classification of higher homotopy commutative loop spaces with finitely generated cohomology, Hiroshima Math. J. 30 (2000), no. 2, 317–344. MR 1777518
Yusuke Kawamoto, Higher homotopy commutativity of $H$-spaces with finitely generated cohomology, Pacific J. Math. 204 (2002), no. 1, 145–161. MR 1905195, DOI 10.2140/pjm.2002.204.145
Yusuke Kawamoto and James P. Lin, Homotopy commutativity of $H$-spaces with finitely generated cohomology, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4481–4496. MR 1851180, DOI 10.1090/S0002-9947-01-02791-X
James P. Lin, A cohomological proof of the torus theorem, Math. Z. 190 (1985), no. 4, 469–476. MR 808914, DOI 10.1007/BF01214746
James P. Lin, Loops of $H$-spaces with finitely generated cohomology rings, Topology Appl. 60 (1994), no. 2, 131–152. MR 1302469, DOI 10.1016/0166-8641(94)00007-7
James P. Lin, $H$-spaces with finiteness conditions, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 1095–1141. MR 1361907, DOI 10.1016/B978-044481779-2/50023-7
James P. Lin and Frank Williams, Homotopy-commutative $H$-spaces, Proc. Amer. Math. Soc. 113 (1991), no. 3, 857–865. MR 1047005, DOI 10.1090/S0002-9939-1991-1047005-2
C. A. McGibbon, Higher forms of homotopy commutativity and finite loop spaces, Math. Z. 201 (1989), no. 3, 363–374. MR 999733, DOI 10.1007/BF01214901
R. James Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386–403. MR 206951, DOI 10.2307/1970453
Mamoru Mimura and Hirosi Toda, Topology of Lie groups. I, II, Translations of Mathematical Monographs, vol. 91, American Mathematical Society, Providence, RI, 1991. Translated from the 1978 Japanese edition by the authors. MR 1122592, DOI 10.1090/mmono/091
Victor Reiner and Günter M. Ziegler, Coxeter-associahedra, Mathematika 41 (1994), no. 2, 364–393. MR 1316615, DOI 10.1112/S0025579300007452
Michael Slack, A classification theorem for homotopy commutative $H$-spaces with finitely generated mod $2$ cohomology rings, Mem. Amer. Math. Soc. 92 (1991), no. 449, iv+116. MR 1086565, DOI 10.1090/memo/0449
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
James Stasheff, $H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372, DOI 10.1007/BFb0065896
Masahiro Sugawara, A condition that a space is group-like, Math. J. Okayama Univ. 7 (1957), 123–149. MR 97066
Masahiro Sugawara, On the homotopy-commutativity of groups and loop spaces, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 257–269. MR 120645, DOI 10.1215/kjm/1250775911
Francis D. Williams, Higher homotopy-commutativity, Trans. Amer. Math. Soc. 139 (1969), 191–206. MR 240818, DOI 10.1090/S0002-9947-1969-0240818-9
F. D. Williams, Higher homotopy commutativity and extension of maps, Proc. Amer. Math. Soc. 26 (1970), 664–670. MR 273613, DOI 10.1090/S0002-9939-1970-0273613-9
F. D. Williams, Higher Samelson products, J. Pure Appl. Algebra 2 (1972), 249–260. MR 303533, DOI 10.1016/0022-4049(72)90005-9
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1