Loop spaces of H-spaces with finitely generated cohomology

Homotopy commutativity of $H$-spaces with finitely generated cohomology

“…Theorem B generalizes his result to the case of C p -spaces with finitely generated mod p cohomology. We see that Theorem B also generalizes results of Lin [14] and Kawamoto [13], in which the same result was proved under the assumption that X is the loop space of an H-space. This paper is organized as follows: In §2 we recall the Dror Farjoun localization functor with respect to a space introduced in [9].…”

“…In [13], we proved that if X is a simply connected mod p H-space such that H * (ΩX) is a finitely generated algebra, then ΩX is homotopy equivalent to a finite product of Eilenberg-Mac Lane spaces of dimensions 1 and 2 (see also [14]). By combining a result of Aguadé-Smith [3] with the above argument, we have a simple proof of [13, Thm.…”

“…At present, only a few results are known for such Hspaces (cf. [5], [8], [13], [14], [16], [22], [23]). Recently, Broto and Crespo [5], [8] gave an important result for H-spaces with finitely generated cohomology.…”

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

“…Besides, since the universal covering of an AC n -space is also an AC n -space (see Lemma 3.9), we have the following stronger version: The above results are considered as mod p versions of the torus theorems by Hubbuck [6], Lin [13], Slack [21], Lin-Williams [16] and Broto-Crespo [3]. For the details of the mod p torus theorems, see Aguadé-Smith [1], Hemmi [5], Kawamoto [9], [10], [11], Kawamoto-Lin [12], Lin [14] and McGibbon [17]. In particular, since the loop space of an H-space is an AC n -space for any n ≥ 1 (see Example 3.2 (3)), we have the following result: For the rest of this paper, all spaces are assumed to be completed at a prime p in the sense of Bousfield-Kan [2].…”

Higher homotopy commutativity of H-spaces and homotopy localizations

“…Corollary 1.2 is a generalization of results from [Aguadé and Smith 1986;Kawamoto 1999;Lin 1994]. Bousfield [2001] studied the K (n) * -localizations of Postnikov H -spaces, where K (n) * denotes the Morava K -homology theory for n ≥ 1.…”