Π∗-kernels of Lie Groups

We study the group of homotopy classes of self maps of compact Lie groups which induce the trivial homomorphism on homotopy groups. We completely determine the groups for SU(3) and Sp(2). We investigate these groups for simple Lie groups in the cases when Lie groups are p-regular or quasi p-regular. We apply our results to the groups of self homotopy equivalences. Introduction Let [X,Y ] be the set of based homotopy classes of maps from a space X to a space Y . We denote by Z(X,Y ) the subset of [X,Y ] consisting of all homotopy classes which induce the trivial homomorphism on homotopy groups in dimensions less than or equal to n. We denote by Z(X,Y ) the subset of [X,Y ] consisting of all homotopy classes which induce the trivial homomorphism on all homotopy groups. We write Z(X) and Z(X) ifX = Y for short. In stable theory the set Z(X,Y) has been previously considered by Christensen [2], where X,Y are spectra. He calls the elements of Z(X,Y) ghosts. Indeed there is a conjecture by Freyd [4] which states that Z(X,Y) is trivial for finite spectra. On the other hand, in the unstable case the situation is quite different. Z(X,Y ) is often nontrivial, even infinite for some spaces. Let us consider the case where Y is an H-space. If Y is an H-complex, [X,Y ] is an algebraic loop which is actually a group if Y is homotopy associative by a result of James [5]. In this case, Z(X,Y ) is a normal subgroup of [X,Y ] for any n. It is not commutative, but nilpotent in many cases. A main object of this paper is a study of the groups Z(X) for simple Lie groups. We will show that these groups have rather simple structure and can be computable in many cases. We obtain the following theorem. Theorem 3.3. Z(SU(3)) = Z(SU(3)) ∼= Z12 for n ≥ 5. (1) Z(Sp(2)) = Z(Sp(2)) ∼= Z120 for n ≥ 7. (2) When G is a compact connected Lie group it is known [8] that Z(G) is equal to Z(G) for some n. The smallest such n is written sz(G) and called the stability of a descending series {Z(G)}. We denote by lz(G) the length of {Z(G)}. We also define invariants of G by szp(G) = sz(Gp) and lzp(G) = lz(Gp) respectively. We will show that if G is p-regular or quasi p-regular, then szp(G) and lzp(G) are determined by its rational type: Theorem 5.1. Let G be a compact connected, simply connected simple Lie group, H(G;Q) ∼= Λ(x1, ..., xr) with degxi = ni, n1 ≤ · · · ≤ nr. If G is quasi p-regular, then szp(G) = nr for all G and 1