Algebras of iterated path integrals and fundamental groups
Algebras of iterated path integrals and fundamental groups
复制标题
迭代路径积分和基本群的代数
DOI：
--
复制
发表时间：
1971
期刊：
影响因子：
--
通讯作者：
Kuo
中科院分区：
文献类型：
--
作者：
Kuo
.利用沿着路径迭代积分的方法，将deRham上同调理论推广到基本群水平上的同伦理论。对于每个连通的C°°流形3 JI，我们构造了一个由迭代积分组成的代数■ 7 r1 = 7 T1（2Si，p），它的值沿着每个环在p处只依赖于环的同伦类.因此，ir 1可以被看作是基本群^（SBi）上的函数的交换代数，其乘法诱导出余乘法w1 -*■ u-1 ® ir 1，这使得w1成为Hopf代数。代数W1将基本群与流形的分析联系起来，我们得到了一些分析条件，这些条件足以使基本群非交换或不可解。我们还表明，w1基本上只依赖于可微同伦类型的流形。本文的后半部分主要研究迭代路径积分代数的结构。证明了这类代数可以由下列数据代数地构造：（a）93）上C函数的交换代数A，（B）SD！(c)通常的微分d：A -> M;以及（d）在基点p，s处的求值映射：A -*■ K，K是真实的（或复数）数域。
. A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected C°° manifold 3JI with a base point p, we construct an algebra ■7r1 = 7T1(2Si,p) consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus ir1 can be taken as a commutative algebra of functions on the fundamental group ^(SBi), whose multiplication induces a comultiplication w1 -*■ u-1 ® ir1, which makes w1 a Hopf algebra. The algebra w1 relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that w1 depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra A of C functions on 93); (b) the A-modale M of C" 1-forms on SD!; (c) the usual differentiation d: A -> M; and (d) the evaluation map at the base point p, s : A -*■ K, K being the real (or complex) number field.