McGibbon, C. A. Higher forms of homotopy commutativity and finite loop spaces. (English) Zbl 0682.55006 Math. Z. 201, No. 3, 363-374 (1989). Let X be a loop space. Its classifying space BX is an H-space iff X is strongly homotopy commutative in the sense of Sugawara. Strong homotopy commutativity is defined by an infinite sequence of coherent homotopies \(C_ n: X^{2n}\times I^ n\to X\) starting with the commuting homotopy for \(n=1\), and is equivalent to the existence of an \(A_{\infty}\) structure on the multiplication map \(X\times X\to X\). A loop space X is called a \(C^ n\)-space if the coherent homotopies \(C_ k\) exist for \(1\leq k<n\). The author investigates finite \(C^ p\)-spaces, p a prime, and proves that a non-contractible, 1-connected, p-local finite loop space is not a \(C^ p\)-space. This extends a result of John Hubbuck who studied homotopy commutative finite H-spaces. The proof is inspired by Hubbuck’s argument: A \(C^ p\)-structure on X makes the k-fold multiplication \(X^ k\to X\) and the k-th power map \(X\to X\) an \(A_ p\)- map. The result is obtained by studying the induced map of a suitable power map on the K-theory of p-th projective space \(P_ p(X)\). In addition, the author compares \(C^ n\)-structures with the notion of higher homotopy commutativity due to Frank Williams. Reviewer: R.Vogt Cited in 1 ReviewCited in 10 Documents MSC: 55P45 \(H\)-spaces and duals 55P35 Loop spaces Keywords:loop space; classifying space; strongly homotopy commutative; coherent homotopies; \(A_{\infty }\) structure; \(C^ n\)-space; p-local finite loop space; homotopy commutative finite H-spaces; \(A_ p\)-map; K-theory of p-th projective space PDF BibTeX XML Cite \textit{C. A. McGibbon}, Math. Z. 201, No. 3, 363--374 (1989; Zbl 0682.55006) Full Text: DOI EuDML OpenURL References: [1] Adams, J.F.: Algebraic Topology, A student’s Guide, Cambridge: University Press · Zbl 0417.55001 [2] Aguade, J., Smith, L.: On the modp torus theorem of John Hubbuck. Math. Z.191, 325-326 (1986) · Zbl 0591.55003 [3] Araki, S., James, I.M., Thomas, E.: Homotopy abelian Lie groups. Bull. Am. Math. Soc.66, 324-326 (1960) · Zbl 0152.01103 [4] Atiyah, M.F.: Vector bundles and the Künneth formula. Topology1, 245-248 (1962) · Zbl 0108.17801 [5] Atiyah, M.F.:K-Theory. New York: W.A. Benjamin, Inc. 1967 [6] Browder, W.: Homotopy commutativeH-spaces. Ann. Math.75, 283-311 (1962) · Zbl 0138.18503 [7] Ganea, T.: Lusternik-Schnirelmann category and strong category, Ill. J. Math.11, 417-427 (1967) · Zbl 0149.40703 [8] Hubbuck, J.R.: On homotopy commutativeH-spaces. Topology8, 119-126 (1969) · Zbl 0176.21301 [9] Hubbuck, J.R.: Modp associativeH-spaces of given rank. Math. Proc. Camb. Philos. Soc.88, 153-160 (1980) · Zbl 0453.55008 [10] Iwase, N.: On theK-ring structure ofX-projectiven-space. Mem. Fac. Sci., Kyush Univ., Ser. A38, 285-297 (1984) · Zbl 0555.55003 [11] James, I.M.: The topology of Stiefel manifolds. Lond. Math. Soc. Lect. Note Ser., No. 24. Cambridge: University Press 1976 · Zbl 0337.55017 [12] McGibbon, C.A.: Multiplicative properties of power maps II. Trans. Am. Math. Soc.274, 479-508 (1982) · Zbl 0531.55009 [13] McGibbon, C.A.: Homotopy commutativity in localized groups. Am. J. Math.106, 665-687 (1984) · Zbl 0574.55004 [14] Milgram, R.J.: The bar construction and abelianH-spaces. Ill. J. Mat.11, 242-250 (1967) · Zbl 0152.40502 [15] Porter, G.J.: Higher order whitehead products. Topology3, 123-135 (1965) · Zbl 0149.20204 [16] Porter, G.J.: Spaces with vanishing Whitehead products. Q. J. Math., Oxf. II. Ser.16, 77-85 (1965) · Zbl 0151.31601 [17] Stasheff, J.D.:H-spaces from a homotopy point of view. (Lect. Notes Math., vol.161.) Berlin Heidelberg New York: Springer 1970 · Zbl 0205.27701 [18] Sugawara, M.: A condition that a space is group-like. Math. J. Okayama Univ.6, 109-129 (1957) · Zbl 0077.16702 [19] Sugawara, M.: On the homotopy-commutativity of groups and loop spaces. Mem. Coll. Sci. Univ. Kyoto Ser. A33, 257-269 (1960/61) · Zbl 0113.16903 [20] Toda, H.: Composition methods in homotopy groups of spheres. Ann. Math. Stud. no. 49. Princeton: University Press 1962 · Zbl 0101.40703 [21] Whitehead, G.: Elements of homotopy theory G.T.M. No. 61. Berlin Heidelberg New York: Springer 1978 · Zbl 0406.55001 [22] Williams, F.D.: Higher homotopy commutativity. Trans. Am. Math. Soc.139, 191-206 (1969) · Zbl 0185.27103 [23] Williams, F.D.: Higher Samelson products. J. Pure Appl. Algebra2, 249-260 (1972) · Zbl 0239.55017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.