Homotopical properties of Lie groups and related spaces

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1. Let X be a connected CW Hopf space with a multiplication.μ For a path connected and pointed space A, the homotopy set [A, X] of continuous maps from A to X has a binary operation "+" induced from μ. We write ([A, X], +) = [A. X ; μ] which is an algebraic loop. Few years ago, we proved that if A and X are connected CW Hopf spaces with at most three cells, then [A. X ; μ] becomes a group and its group is determined for the case A=X ; in this project, we have determined the group [A, X ; μ] for all other cases.2. Let G be a connected Lie group and μ_0 the multiplication of G. Then the algebraic loop [A,G ; μ_0] is a group and it satisfies the relation : nil[A, G ; μ_0] 【less than or equal】 cat(A) as proved by G. W. Whitehead, where nil denotes the nilpotency class and cat denotes the Lusternik-Schnirelmann category with cat(*) = 0. We are interested in estimation of nil[A, G ; μ_0] from below. In the first place, though it is the most interested case, we have consider the case A = G. We had two conjectures :(1) If G is simple, then nil[G, G ; μ_0] 【greater than or equal】 rank(G).(2) If G is simple and rank(G) 【greater than or equal】 2, then nil[G, G ; μ_0] 【greater than or equal】 2.Of course if (1) is affirmative then so is (2). Without the assumption "simple", two conjectures are in general false.We have proved (2) affirmative when the universal covering group of G is not Spin(n) with n 〓 O (mod 4).3. We have given a partial answer to the problem : given a connected finite CW Hopf space X, determine sets P of prime numbers such that, every multiplication on the P-localization Xp of X is a P-localization of a multiplication on X. A complete answer has been given when X is of rank 2 or a simple Lie group.