Homotopical properties of Lie groups and related spaces

1.设X是具有乘法的连通CWHopf空间，μ对于路径连通的点空间A，从A到X的连续映射的同伦集[A，X]具有由μ诱导的二元运算“+”。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是至多有三个胞格的连通CWHopf空间，则[A.X；μ]成为一个群，当A=X时，它的群是确定的；在这个项目中，我们已经确定了所有其他情形的群[A，X；μ]。设G是连通李群，μ_0是G的乘积，则代数环[A，G；μ_0]是一个群，它满足G.W.Whitehead证明的关系：nil[A，G；μ_0][小于等于]CAT(A)，其中nil表示幂零类，CAT表示CAT(*)=0的Lusternik-Schnirelmann范畴.我们对nil[A，G；μ_0]的估计感兴趣。首先，虽然这是最有趣的情况，但我们考虑了A=G的情况。我们有两个猜想：(1)如果G是单的，则零[G，G；μ_0][大于或等于]排名(G)。(2)如果G是单的，且排名(G)[大于或等于]2，则零[G，G；μ_0][大于或等于]。在不作“简单”假设的情况下，两个猜想是一般错误的。我们证明了(2)当G的泛覆盖群不是自旋(N)且n〓O(Mod 4)时是肯定的。我们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P，使得X的P-局部化XP上的每一个乘法都是X上乘法的P-局部化，当X是2阶或单李群时，已给出了完整的答案。

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.