Download A Topological Chern-Weil Theory by Anthony V. Phillips PDF

By Anthony V. Phillips

This paintings develops a topological analogue of the classical Chern-Weil idea as a style for computing the attribute sessions of central bundles whose structural crew isn't unavoidably a Lie crew, yet just a cohomologically finite topological crew. Substitutes for the instruments of differential geometry, resembling 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 ways to attribute classes.In distinction to the 1st process, particular cocycles are used, in order to spotlight the impression of neighborhood geometry on worldwide topology. unlike the second one, calculations are conducted on the small scale instead of the infinitesimal; actually, this paintings will be considered as a scientific extension of the commentary that curvature is the infinitesimal type of the disorder in parallel translation round a rectangle. This ebook may be used as a textual content for a sophisticated graduate direction in algebraic topology.

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PHILLIPS and DAVID A. 2) so dQa =< 123 >, etc. We will evaluate dil(H0) = fl(dH0). ] do not contribute to d£l{Hc). This leaves -Hdo<7)-n(HdlC) + n(H^)-n(Hd3(7) S*(Hd0a — HQ1(7 + Hd2(T — Hd3<7)- To describe the relation between 0 and Du>, where D is defined as above, we need to introduce a new family of cochains. 11 For each n, let \n:Cn —• Tg n _i be the cochain The cochain An vanishes except on the horizontal chains C^\ In fact An is a lifting to C* of dn\ An —• A n _i, in the following sense.

R — 1}, say I = { i o , . . , i p } with 1 < io < «i < «2 < • • • < **> < r - Given 711 • • • € #*, set [[£/<7l,---,7r >]]* = [ [ £ < 7 i > - - - > 7 t i > l £ < 7 . ' i + i , - . - , 7 . ' 2 > | - " l v < 7 i P + i . ,7r>r. Y = $*iB(n + l). a. coalgebras. (2) Y is a cocycle on B* that represents y. 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*.

Representatives always exists. ,-) may be chosen arbitrarily within the homotopy class determined by X{. Proof. Let ££'* be the Eilenberg-Moore spectral sequence (see below for more details) for B*. * = £ ( R , # * G , R ) or more precisely E{« is generated by the set of [z\ | • • • \zp] with Z{ G H*{G\ R ) and J2 d i m Zi — q — p. Set yi = [xi\ G jB 1 , n , + , for i = 1 , . . , N. It is known [26] that the yt- persist to £Joo; in fact, H*(BG; R ) is the polynomial algebra R [ j / i , . . , J/AT].

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