Download e-book for iPad: A Topological Chern-Weil Theory by Anthony V. Phillips

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By Anthony V. Phillips

ISBN-10: 0821825666

ISBN-13: 9780821825662

This paintings develops a topological analogue of the classical Chern-Weil thought as a style for computing the attribute periods of primary bundles whose structural staff isn't really unavoidably a Lie team, yet just a cohomologically finite topological team. Substitutes for the instruments of differential geometry, reminiscent of the relationship and curvature varieties, 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 strategy, particular cocycles are used, as a way to spotlight the impact of neighborhood geometry on international topology. unlike the second one, calculations are performed on the small scale instead of the infinitesimal; in reality, this paintings could be considered as a scientific extension of the commentary that curvature is the infinitesimal kind of the disorder in parallel translation round a rectangle. This ebook will be used as a textual content for a complicated graduate direction in algebraic topology.

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Map (p: G —• K(R,n) such that the cocycle K(R, n) can 2 be desuspended to a map (p( ): GxG —> if (R, n—1). Since # is indecomposable, it is also primitive; so fj,*x = l ® x + a ; ® l £ Hn(GxG;Ii).

15 Let V: Ck —» G be a singular cube , and let Ha:Cn —> E be as above; so V • Ha: Ck+n —• JS. By means of the standard simplicial subdivision of Ck we can regard V = £ a e(a)7a &s a chain in (/*, and y • iif^ as a chain in C*. Then vc(v •#,) = £ e(n)Fn(v. ) ® J3n(v • H„\ n where II ranges over all two-fold partitions of the set of k+n coordinates on C* + n . Proof. Let V have coordinates s i , . . , Sk+n- The two-fold partitions II = ITU II" are in one-toone correspondence with pairs of partitions IIa = U[ U II" of { 1 , .

T p - i , t p + i , . . t r - l > V , S « l ? *••' Sip-ll 29 •••> 5 * r - l )> = S ip+1 ? •••? S ir-l)' 30 ANTHONY V. PHILLIPS and DAVID A. STONE For G = < io^i > a 1-simplex, we will abbreviate V<,-ot-1>(0) to - I n particular the compatibility conditions for a 2-simplex become v ^(l) V(0) = V = v. f. enters into our topological Chern-Weil theory via its canonical pseudosection. 4). 1) is a collection of maps Ha:Cra —> E&G, one for each a G A (r = d i m a ) , satisfying 1) if cr = < i > , Ha: C® —• E&G takes 0 to e,-, the identity in G{.

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