By Anthony V. Phillips

This paintings develops a topological analogue of the classical Chern-Weil thought as a style for computing the attribute periods of central bundles whose structural workforce isn't unavoidably a Lie team, yet just a cohomologically finite topological team. Substitutes for the instruments of differential geometry, comparable 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 techniques to attribute classes.In distinction to the 1st method, particular cocycles are used, on the way to spotlight the impact of neighborhood geometry on international topology. not like the second one, calculations are conducted on the small scale instead of the infinitesimal; in reality, this paintings can be seen as a scientific extension of the commentary that curvature is the infinitesimal kind of the illness in parallel translation round a rectangle. This booklet should be used as a textual content for a complicated graduate path in algebraic topology.

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**Additional resources for A Topological Chern-Weil Theory**

**Example text**

We concentrate on the case 7 = e 6 C/0. 10 we subdivided # a to K*(Ha) = X^Tj in /C* . On each T/, S^(Qi) = e[K(/)]; and this singular cell in EG factors through Q(&). Thus there is a unique map h\ **: Ac*(i7a) —* Q{cr) such that this diagram commutes: K*(H0y vrr Q{°) EG. Now set h\ = hi K o /c*: EG —• Q(a). Finally, define h}1(7 = id A x h\. Then (5) holds, because S*(i - Ha) = 7 • S*(Ha). Lastly we must construct the homotopies A7^. In fact we need only define ha:Cr x J —> Q(^") because we can then set hlyG = id A x ^ : A x Cr x / —» A x Q(^) = Q(7> 0")- We construct /i a by induction on dim a.

6)/i7,a|(afcA)xC^x/ = ihdk^i(T, hllCr\AxdjCrxl = ^fyc k = 0, . . , d i m A j = 1,... y): A x V ( 0 , j ) —> G is to be regarded as a singular simplicial chain Y^l' by the standard subdivision procedure, and the right-hand side is short for I C ^ y W j ) (7) Whenever 7 " C 7 ' C 7 and

By induction on k. For k = 1, this is the definition of ft. 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 • # , ) = £ £