Higher Homotopy-Commutativity

Higher homotopy commutativity of H-spaces with finitely generated cohomology

“…To describe an odd prime version of Theorem 1.1, we need to generalize the homotopy commutativity of H-spaces to the higher ones. Such notions were first considered by Sugawara [24] and Williams [25] in the case of loop spaces. Later Hemmi [8] introduced the higher homotopy commutativity of H-spaces.…”

“…Furthermore, in the case that X is a loop space, by [8, Thm. 2.2], quasi C n -space is the same condition as C n -space in the sense of Williams [25,Def. 5] (see also [20,Thm.…”

“…Theorem 2.1 shows that L A and CW A preserve the higher homotopy associativity of H-spaces up to homotopy.Kawamoto[14, Thm. 2.14] has shown that the nullification functor L A preserves the higher homotopy commutativity in the sense of Williams[25] of loop spaces. Since the proof of Theorem 2.1(1) is similar to the one of[14, Thm.…”

Higher homotopy commutativity and extension of maps

“…In [6], Samelson treated this case for n = 2 and Xi and X2 being spheres, by relating the extension problem to the Pontryagin product in the homology of OF, the loop space of Y. In this note, Samelson's method is extended to the general situation of arbitrary X\, ■ ■ • , Xn through the use of the higher homotopy commutativity introduced in [7]. The outline is as follows.…”

“…Observe that a C2-form for two maps/i and/2 is just a homotopy between the product maps /i-/2 and /j-/i. For further intuition into C-commutativity, the reader is referred to p. 193 of [7]. In that paper is defined the notion of C"-space: An associative 77-space is a C"-space provided that there is a C"-form for the maps /ii • • ■ ./n where each /¿ = 1 : G-*G. Clearly, if G is a C"-space, then any maps/,-:Z,->G, t = l, • • • , n, possess a C-form.…”