A Topological Chern-Weil Theory by Anthony V. Phillips

By Anthony V. Phillips

This paintings develops a topological analogue of the classical Chern-Weil thought as a mode for computing the attribute sessions of critical bundles whose structural crew isn't really inevitably a Lie team, yet just a cohomologically finite topological team. Substitutes for the instruments of differential geometry, similar to the relationship and curvature types, 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, in an effort to spotlight the effect of neighborhood geometry on worldwide topology. not like the second one, calculations are performed on the small scale instead of the infinitesimal; in reality, this paintings might be seen as a scientific extension of the commentary that curvature is the infinitesimal kind of the disorder in parallel translation round a rectangle. This publication should be used as a textual content for a sophisticated graduate direction in algebraic topology.

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Define (real-valued) 3-forms x^ o n &A by *(1) = X<2> = ^ T ^ A A wA A w A ), 3^Tr(u; A Afi A ); thus x = X ^ + X ^ ls the Chern-Simons form of u>&. We denote by X^ and x the 3-cochains on £* given by integrating these forms over (singular) cells. It can be shown that (setting T = 70(71 | . . | 7r])- x (1) (r) = J0f (2) x (r) = dv in case r = 0, dim 70 = 3, otherwise; ' s £ /o /o Tr(W°(Y0(*o)) • ^[(s^dsods1 in case r = 1, dim 70 = dim7x = 1, 0 otherwise. A straightforward calculation shows that d\ = 7^*5^, where F is the 4-cocycle on B* defined by ra-yi 1 1

N} whereby IT = U[lHI2, U" = n'/un^'. )y Bn{V • H9) = Bni(V) • BU2{Ha). • H9) ® Bn(V -Ha) = (-l) ( f c - p + g ) e(n 1 ) £ (n 2 )Fn 1 (V). J Fh 3 (^)®5n 1 (V). J B n a (^). 15, and let j:Ae-+G be £ 71 an element of Qi. ) = £ ] T e ( n ) ( - i ) ^ ^ i=o n where II ranges over all two-fold partitions II7 U II" of the set of coordinates of C fc+n , and p = |IT|. Proof. Define the singular cube r : Cl —• G by Y = 7 o 7rc, where 7TC:C£ —>

R. ,r>. In particular, d i i ^ = v^ao* and ^if^ = V^. 4 Horizontal and vertical chains. 1). So an Ha is horizontal and a Va is vertical. 5 The twisting cochain. Let A* represent the chain complex generared by A. We are going to construct from (A, V) a twisting cochain

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