Study of algebraic structures on homotopy sets

给定一个紧连通李群G，[G，G]表示G的自映射的同伦类的集合，它继承了G.1的一个群结构。我们的第一个目标是如下猜想：如果G是单的，则[G，G]的幂零类大于或等于G的秩.我们证明了小秩图G的猜想.但我们仍然既不能证明这一猜想，也不能给出反例。我们的结果之一是下面的定理，它证明了猜想的一个弱形式。群[G，G]是交换的当且仅当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)是G的自同伦等价的同伦类群，它们诱导同伦群上的单位元。我们证明了下列三个命题是等价的：(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]中的群乘以+表示；(3)G同构于T^n，S^3和SO(3)中的一个。设G是半单的且秩为2，除G〓SO(3)×SO(3)外，我们已确定了[G，G]的群结构。设G_2是秩2的例外李群，φ：S^3×S^&lt；11&gt；→G_2是标准映射。我们确定了奇素数p，使得局部化φ_&lt；(p)&gt；：S^3_&lt；(p)&gt；×S^&lt；11&gt；_&lt；(p)&gt；→G_&lt；2(p)&gt；是H-映射。首席研究员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的群。

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.