Study of algebraic structures on homotopy sets

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Given a compact connected Lie group G,[G,G] denotes the set of homotopy classes of self maps of G, and it inherits a group structure from G.1. Our first target is the following Conjecture : If G is simple, then the nilpotency class of [G,G] is greater than or equal to the rank of G. We have proved the conjecture for G with small rank. But we can still give neither a proof nor a counter example of the conjecture.2. One of our results is the following theorem which proves a weak form of the conjecture.Theorem. The group [G,G] is commutative if and only if G is isomorphic to one of T^n (n【greater than or equal】0), T^m×S^3 (0【less than or equal】m【less than or equal】2) and SO(3), where T^n is the n-dimensional torus.3. Let ε_#(G) be the group of homotopy classes of self homotopy equivalences of G which induce the identity on homotopy groups. We have proved that the following three statements are equivalent : (1) The group ε_#(G) is trivial ; (2) the left distributive law aο(b+c)=aοb+aοc holds in [G,G], where the group multiplication in [G,G] is denoted by + ; and (3) G is isomorphic to one of T^n, S^3 and SO(3).4. Let G be semi-simple and of rank 2. We have determined the group structure of [G,G] except for the case G〓SO(3)×SO(3).5. Let G_2 be the exceptional Lie group of rank 2 and φ:S^3×S^<11>→G_2 the canonical map. We have determined odd prime numbers p such that the localization φ_<(p)>:S^3_<(p)>×S^<11>_<(p)>→G_<2(p)> is an H-map.6. The head investigator, Arkowitz and Strom have studied relations between typical subgroups of ε(X), where ε(X) is the group of homotopy classes of self homotopy equivalences of the space X.