By Anthony V. Phillips

This paintings develops a topological analogue of the classical Chern-Weil thought as a style for computing the attribute sessions of vital bundles whose structural crew isn't really unavoidably a Lie team, yet just a cohomologically finite topological crew. Substitutes for the instruments of differential geometry, similar to the relationship and curvature kinds, are taken from algebraic topology, utilizing paintings of Adams, Brown, Eilenberg-Moore, Milgram, Milnor, and Stasheff. the result's a synthesis of the algebraic-topological and differential-geometric methods to attribute classes.In distinction to the 1st technique, particular cocycles are used, with the intention to spotlight the impression of neighborhood geometry on international topology. unlike the second one, calculations are performed on the small scale instead of the infinitesimal; in truth, this paintings will be considered as a scientific extension of the commentary that curvature is the infinitesimal kind of the illness in parallel translation round a rectangle. This publication will be used as a textual content for a complicated graduate path in algebraic topology.

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**Example text**

Proof. (1) is due to Clark [10] in this algebraic context, following a geometric construction due to Sugawara [37]. (2) By (1), $ respects the differentials in B* and B?. Since tB(n + l) is a cocycle on B*:, it follows that Y is a cocycle on B*. To show that Y represents j / , the transgression of x, we examine the Eilenberg-Moore spectral sequence (originally due, in this context, to Milnor [24]). This is the spectral sequence of the complex #*, filtered by {Blp)}. We first show that in El{n = Hn+l(B\,Bl), the cohomology class represented by the restriction of Y is [[x]].

T. Chen's theory of interated integrals and power-series connections ([7, 8] ; see also Gugenheim's explication [14]). In our Chern-Weil theory the singular complex Q* of G is taken as known, and representative cocycles of classes in H*(BG]R) are constructed. g. ; R ) ) . Thus Chen allows one to represent classes of H*(G; R ) in terms of differential forms on BG. It is intriguing to observe that while we use the bar construction on

Set ^ ) = W , 2 - S o W W =(«)^- 1 ,A S » U m ethe(,- 1 )- f o l a product ft*"1 = ft A • • • A ft = N(k - 1 ) & P S T 2 2 ) , and let 7 • #<, G C2k with dim 7 = p (so dim a = 2k — p). We apply ft Aft*""1to 7 • Ha. Since V C (7 • # , ) = £ £