Self maps of Lie groups

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1. Let X be a connected CW Hopf space with a multiplication μ. For a pathconnected 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. Our first result is the following : If A and X are connected CW Hopf spaces with at most three cells, then [A, X ; μ] becomes a group and its group structure can be determined. By the way, there are fifteen connected CW Hopf spaces with at most three cells and they have in general many multiplications. Therefore there were many groups we should compute.2. Let G be a connected Lie group and μィイD20ィエD2 the multiplication of G. Then the algebraic loop [A, G ; μィイD20ィエD2] is a group and it satisfies the relation : nil[A, G ; μィイD20ィエD2] 【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ィイD20ィエD2] from below. In the first place, though it is the most interested case, we have consider the case A = G. We have two conjectures :(1) If G is simple, then nil[G, G ; μィイD20ィエD2] 【greater than or equal】 rank(G).(2) If G is simple and rank(G) 【greater than or equal】 2, then nil[G, G ; μィイD20ィエD2] 【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 proved (1) affirmative when G is one of SO(3), SU(3), SU(4), Sp(2), Spin(7) and GィイD22ィエD2 ; and (2) affirmative when G is one of SU(5), SU(6), Sp(3), Spin(8), EィイD26ィエD2, EィイD28ィエD2, and FィイD24ィエD2. We determined the group [G, G ; μィイD20ィエD2] completely for G = SO(3), SU(3), Sp(2) and almost completely for G = GィイD22ィエD2. We have several fragmental results for G not simply connected or not simple.