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 thought as a style for computing the attribute sessions of central bundles whose structural workforce isn't inevitably a Lie team, yet just a cohomologically finite topological team. Substitutes for the instruments of differential geometry, corresponding to 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 process, particular cocycles are used, in an effort to spotlight the effect of neighborhood geometry on international topology. not like the second one, calculations are performed on the small scale instead of the infinitesimal; in reality, this paintings should be seen as a scientific extension of the commentary that curvature is the infinitesimal type of the illness in parallel translation round a rectangle. This e-book will be used as a textual content for a sophisticated graduate path in algebraic topology.

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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]. The part of Moore's argument that we need is that all the differentials dr(yi) are 0, for i = 1 , . . , n and r > 1. m. representation. m. m. , it follows tha t each cp\ can be chosen freely within its homotopy class.

Definition 5,22 A Tgm-valued cochain on C* is any operator a:C* —* Tg* that can be factored as S+af where a' G A. The Grassmann product of Tg*-valued cochains a and /? is defined to be a A/? ') where a = S*(aO,/? '). Let us calculate some examples. 23 1) u Aw = 5*;4 ! 2) Let t = S*p[ '; then w A i = 5*p2 J 3)«Au; = - 5 » ^ 1 ) + p ( 2 ) ] . Proof. As paradigmatic 2-cells we may take the following faces of H = ^ (see Fig. 2): K2 = d3H = # < 0 1 2 > G C<2), / ^ = d2H = V * #<23> G C ^ , and in d 3 # = V = 7id - 7(i2) the simplex K0 = 7id G C20) with vertices V<03>, Kcoi>V and VVV<23>.

44 ANTHONY V. PHILLIPS and DAVID A. STONE Now assume fe„#, and hence also Ayj<7/ for any 7', has been constructed whenever d i m a < r; we wish to construct ha. Now (4),(5), and (6) prescribe hG on <9(Cr x 7 ) ; and there is no obstruction to extending ha over CT x 7, since (3( x h0). These conditions follow from the construction of the classifying map FiE^EG. ,VJ> K\d>c and x hx). 11. It remains to check the self-consistency of (6). The most interesting case is for hli(T\dicrndJCr x I* We must show tha t th vcr)xi = ihv^Mij^dtidJc^xi' Say i < j .

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