×

Higher homotopy commutativity in localized groups. (English) Zbl 0821.55009

We generalize the results of McGibbon on homotopy commutativity [C. A. McGibbon, Am. J. Math. 106, No. 3, 665-687 (1984; Zbl 0574.55004)] to higher homotopy commutativity in the sense of F. D. Williams [Trans. Am. Math. Soc. Math. 139, 191-206 (1969; Zbl 0185.271)]. The main results obtained are the following, where in both cases the type also means that the rationalization is homotopy equivalent to \((S^{2n_ 1 -1}\times \cdots\times S^{2n_ l -1} )_{(0)}\):
Theorem A. Let \(G\) be a 1-connected, compact, simple Lie group, different from \(G_ 2\) at \(p=5\), of type \((2n_ 1, \dots, 2n_ l)\) where \(n_ 1\leq \cdots\leq n_ l\), and let \(k\geq 2\) be an integer. Then (i) If \(p> kn_ l\) then \(G_{(p)}\) is a \(C_ k\)-space; (ii) If \(p< kn_ l\) when \(G_{(p)}\) is not a \(C_ k\)-space, except in the case \(\text{Sp} (2)\), or equivalently \(\text{Spin}(5)\) at \(p=3\) and \(k=2\).
And we also prove for finite loop spaces and loop multiplications:
Theorem B. Let \((X, \mu)\) be a loop space where \(X\) has the homotopy type of 1-connected, \(p\)-local CW-complex, with type \((2n_ 1,\dots, 2n_ l)\) where \(n_ 1\leq \dots \leq n_ l\), and let \(k\geq 2\) be an integer. Then (i) If \(p> kn_ l\) then \((X,\mu)\) is a \(C_ k\)-space; (ii) If \(n_ l<p <kn_ l\) then \((X, \mu)\) is not a \(C_ k\)-space.

MSC:

55P60 Localization and completion in homotopy theory
55P99 Homotopy theory
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ewing, J.: Some examples of sphere bundles over spheres which are loop spaces mod p, Bull. Amer. Math. Soc.80 (1974), p. 935–938 · Zbl 0288.55016 · doi:10.1090/S0002-9904-1974-13583-4
[2] Hemmi, Y.: Higher homotopy commutativity of H-spaces and the mod p torus theorem, Pacific. J. of Math.149(1) (1991). p. 95–111 · Zbl 0691.55007
[3] James, I.M. and Thomas E.: Homotopy abelian topological groups, Topology1 (1962), p. 237–240 · Zbl 0107.40602 · doi:10.1016/0040-9383(62)90105-2
[4] McCleary, J.: Mod p decompositions of H-spaces; Another approach, Amer. Lecture Notes in Math763 (1978), p. 70–87 · doi:10.1007/BFb0088078
[5] McGibbon, C.A.: Some properties of H-spaces of rank 2, Proc. Amer. Math. Soc.81 (1981), p. 121–124 · Zbl 0475.55007
[6] McGibbon, C.A.: Homotopy commutativity in localized groups, Amer J. Math274 (1984), p. 665–678 · Zbl 0574.55004 · doi:10.2307/2374290
[7] McGibbon, C.A.: Higher forms of homotopy commutativity and finite loop spaces, Math Z.201 (1989), p. 363–374 · Zbl 0682.55006 · doi:10.1007/BF01214901
[8] Mimura, M and Toda, H.: Cohomology operations and the homotopy of compact Lie groups I, Topology9 (1979), p. 317–336 · Zbl 0204.23803 · doi:10.1016/0040-9383(70)90056-X
[9] Porter, G.J.: Higher order Whitehead products, Topology3 (1965), p. 123–135 · Zbl 0149.20204 · doi:10.1016/0040-9383(65)90039-X
[10] Toda, H.: ”Composition methods in homotopy groups of spheres,” Ann. of Math. Studies n.49. Princenton University Press, 1962 · Zbl 0101.40703
[11] Williams, F.D.: Higher homotopy commutativity, Trans. Amer. Math. Soc.139 (1969), p. 191–206 · Zbl 0185.27103 · doi:10.1090/S0002-9947-1969-0240818-9
[12] Williams, F.D.: Higher Somelson products, J. Pure and Applied Alg.2 (1972), p. 249–260 · Zbl 0239.55017 · doi:10.1016/0022-4049(72)90005-9
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.