By Bruno Harris

This topic has been of serious curiosity either to topologists and to quantity theorists. the 1st a part of this publication describes a few of the paintings of Kuo-Tsai Chen on iterated integrals and the elemental team of a manifold. the writer makes an attempt to make his exposition obtainable to starting graduate scholars. He then proceeds to use Chen's structures to algebraic geometry, displaying how this ends up in a few effects on algebraic cycles and the Abel-Jacobi homomorphism. eventually, he offers a extra common viewpoint concerning Chen's integrals to a generalization of the concept that of linking numbers, and finally ends up with a brand new invariant of homology periods in a projective algebraic manifold. The ebook relies on a path given by means of the writer on the Nankai Institute of arithmetic within the fall of 2001.

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Then a(x) E U, for some x E Xb, where U = Z -

7 Fibrewise fibre bundles 37 The Milnor construction Let G be a fibrewise open group, over a base space B. Following Milnor [113j, in the ordinary theory, a countably numerable fibrewise G-bundle can be constructed as follows. We regard the cylinder A = B x [ as a fibrewise space under the first projection, and denote the second projection by a : A -+ [. ·· E Gb. ° Restrict attention to those sequences such that a(a n ) = for all but a finite number of indices n and such that E a(a n ) = 1. Impose on this fibrewise set the fibrewise equivalence relation in which two such sequences (al, gl , ...

Then Ec becomes a fibrewise space, such that for each fibrewise space K a fibrewise function f : K -+ Ec is continuous if and only if each function an 0 f and gn 0 (f I (a 0 an 0 f)-I (0,1)) is continuous. Now consider the fibrewise action rc : Ec XB G -+ Ec of G on Ec given by rc([al,gl,a2,g2, ... ]. We denote the fibrewise orbit space Ec/G by Xc and the natural projection by pc. The fibrewise map an is invariant and so induces a fibrewise map an 0 (PC)-1 : Xc -+ A. The open sets {an 0 Pc/ 0 a-I (0, I)} form a countably numerable covering of Xc, and Ec is fibrewise G-trivial over each member of the covering.