Homotopical properties of Lie groups and related spaces

1.设X是一个具有乘法的连通CW Hopf空间，μ是一个路连通的点空间A，从A到X的连续映射的同伦集[A，X]有一个由μ导出的二元运算“+”.我们写（[A，X]，+）= [A. X ; μ]，这是一个代数循环。几年前，我们证明了：如果A和X是至多有三个胞腔的连通CW Hopf空间，则[A. X ; μ]成为一个群，它的群是在A=X的情况下确定的;在这个项目中，我们已经确定了所有其他情况下的群[A，X ; μ]。2.设G是连通李群，μ 0是G的乘法.则代数圈[A，G ; μ_0]是一个群，它满足G证明的关系：nil[A，G ; μ_0] [小于或等于] cat（A）. W. Whitehead，其中nil表示幂零类，cat表示Lusternik-Schnirelmann范畴，cat（*）= 0。我们感兴趣的是从下面的零[A，G ; μ_0]的估计。首先，虽然这是最感兴趣的情形，但我们考虑了A = G的情形。（1）若G是单的，则nil[G，G ; μ_0] [大于或等于] rank（G）. (2)若G是单的且秩（G）[大于或等于] 2，则nil[G，G ; μ_0] [大于或等于] 2.当然，若（1）是肯定的，则（2）也是肯定的.在没有“简单”假设的情况下，这两个命题一般都是假的，我们证明了（2）当G的泛覆盖群不是Spin（n）且n <$O（mod 4）时，命题是肯定的。我们已经给出了这个问题的部分答案：给定一个连通的有限CW Hopf空间X，确定素数集合P，使得X的P-局部化Xp上的每个乘法都是X上乘法的P-局部化。当X是秩2或单李群时，给出了完整的答案。

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.