Algebras of iterated path integrals and fundamental groups

Abstract:

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 ${C^\infty }$ manifold $\mathfrak {M}$ with a base point p, we construct an algebra ${\pi ^1} = {\pi ^1}(\mathfrak {M},p)$ consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus ${\pi ^1}$ can be taken as a commutative algebra of functions on the fundamental group ${\pi _1}(\mathfrak {M})$, whose multiplication induces a comultiplication ${\pi ^1} \to {\pi ^1} \otimes {\pi ^1}$, which makes ${\pi ^1}$ a Hopf algebra. The algebra ${\pi ^1}$ 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 ${\pi ^1}$ 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 ${C^\infty }$ functions on $\mathfrak {M}$; (b) the A-module M of ${C^\infty }$ 1-forms on $\mathfrak {M}$; (c) the usual differentiation $d:A \to M$; and (d) the evaluation map at the base point p, $\varepsilon :A \to K$, K being the real (or complex) number field.