On maps of H-spaces

Arkowitz, Martin; Curjel, C. R.

doi:10.1016/0040-9383(67)90030-4

Cited by 33 publications

(24 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Because Auti (F) c Autj. (π*(X)) is clearly a subgroup of finite index, we conclude from Proposition 3 that the group π*(X)) is finitely-presented, (b) It is shown in [1] that the natural map has finite kernel, and that the image of p (see p. 146 of [1]) is a subgroup of finite index. It is here that the assumptions on X are used.…”

mentioning

confidence: 95%

A note onH-equivalences

Kahn¹

1972

Pacific J. Math.

View full text Add to dashboard Cite

mentioning

confidence: 95%

A note onH-equivalences

Kahn¹

1972

Pacific J. Math.

View full text Add to dashboard Cite

“…The number N measures the lack of homotopy commutativity in X; in particular, N is a multiple of the order of the Samelson product of the identity map on X with itself. However, the number N is known in only a few special cases when the rank of X is 1 or 2 (see [AC,M1]). As a non-local example, N = 24 for X = S 3 , where the multiplication is from regarding S 3 as the unit quaternions.…”

Section: −→ Bg Has the Following Properties: (A) There Is A Homotopy mentioning

confidence: 99%

Power maps on $p$-regular Lie groups

Theriault

2013

Homology, Homotopy and Applications

View full text Add to dashboard Cite

A simple, simply-connected, compact Lie group G is pregular if it is homotopy equivalent to a product of spheres when localized at p. If A is the corresponding wedge of spheres, then it is well known that there is a p-local retraction of G off ΩΣA. We show that that complementary factor is very well behaved, and this allows us to deduce properties of G from those of ΩΣA. We apply this to show that, localized at p, the p th -power map on G is an H-map. This is a significant step forward in ArkowitzCurjel and McGibbon's programme for identifying which power maps between finite H-spaces are H-maps.

show abstract

“…To date this conjecture has been verified only in the four cases; « = 1,2,3 and oo [4,16,30]. The last case is especially interesting.…”

mentioning

confidence: 98%

Self-maps of projective spaces

McGibbon¹

1982

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The classical projective «-spaces (real, complex, and quaternionic) are studied in terms of their self maps, from a homotopy point of view. Self maps of iterated suspensions of these spaces are also considered. The goal in both cases is to classify, up to homology, all such maps. This goal is achieved in the stable case. Some partial results are obtained in the unstable case. The results from both cases are used to compute the genus groups and the stable genus groups of the classical projective spaces. Applications to other spaces are also given.1. Summary. Let P" denote a projective «-space (either real, complex, or quaternionic). This paper deals with the classification, up to homology, of self maps /: P" -» P" and g: 2*P" -* 2*P".Here HkP" denotes the zc-fold suspension of P". We shall assume that k is large with respect to « in order to make the second classification a stable one.It is clear that these two problems are related. Indeed for any space X and self maps /, i = l,...,m, one can use the track addition of maps on 1kX to form a linear combination of /i-fold suspensions nxzkfx + ---+nmzkfm.zkx^zkx.Under what conditions can every self map of ^kX be described, up to homology, in this manner? The following result answers this question for the classical projective spaces. In its statement, two maps are called homologous if they induce the same homomorphism in integral homology. Theorem 1. If k is large with respect to n, then every self map of 2,kP" is homologous to a linear combination of k-fold suspensions, if and only if (i) P" = RP" andn¥=3 or 1, or (u)Pn = CPnforalln,or (iii) P" = HP" and n< 4. □ This theorem is a corollary of some of the results that we obtain in the next two sections. In §2, the unstable classification problem is considered. There the quaternionic case receives special attention. Unfortunately, the classification of self maps

show abstract

On maps of H-spaces

Cited by 33 publications

References 7 publications

A note onH-equivalences

A note onH-equivalences

Power maps on $p$-regular Lie groups

Self-maps of projective spaces

Contact Info

Product

Resources

About