Study of algebraic structures on homotopy sets

给定紧连通李群G，[G，G]表示G的自映射的同伦类集合，它继承了G.1的群结构。我们的第一个目标是以下猜想：如果G是简单的，那么[G，G]的幂零类大于等于G的秩。我们已经证明了小秩G的猜想。但是我们仍然不能给出这个猜想的证明或反例。我们的结果之一是下面的定理，它证明了这个猜想的一个弱形式。群[G，G]是可交换的当且仅当G同构于T^n （n【大于等于】0）、T^m×S^3（0【小于等于】m【小于等于】2）和SO(3)中的一个，其中T^n是n维环面。设ε_#(G)为可在同伦群上导出恒等的G的自同伦等价的同伦类群。我们证明了下列三个表述是等价的：(1)群ε_#(G)是平凡的；(2)左分配律aο(b+c)=aοb+aοc在[G，G]中成立，其中[G，G]中的群乘法用+表示；(3) G同构于T^n， S^3和SO(3)中的一个。设G为半单质，秩为2。我们已经确定了[G，G]的基团结构，除了G〓SO(3)×SO(3).5。设G_2为秩为2的例外李群，φ:S^3×S^<11>→G_2为正则映射。我们确定了奇素数p，使得定位φ_<(p)>:S^3_<(p)>×S^<11>_<(p)>→G_<2(p)>是一个h图。主要研究者Arkowitz和Strom研究了ε(X)的典型子群之间的关系，其中ε(X)是空间X的自同伦等价的同伦类群。

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.