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We begin by describing the model for yl = Map(I, Y). a. For the moment the reader may assume that AD = K. a obtained by adding free generators x" of positive degrees successively to AD such that the differential dx" is expressed as a polynomial (in the sense of graded 22 1. DE RHAM HOMOTOPY THEORY algebra) of elements which appeared in earlier stages with coefficients in Ao. In other words, A is described as the union of an increasing series Ao C Al C ... c At C ... 's such that the differential of any element XC" which was added to Ae-l to construct At, satisfies dx cx E At-I.

The connection form Wo of this bundle is given by Wo = q*O where q : M x G -+ G is the natural projection and () E Al(G; g) denotes the Maurer-Cartan form of G. 3. Let 7r : P -+ M be a fiat G-bundle and let f N -+ M be a Coo map. Then the pullback bundle f* P -+ N by becomes a fiat G-bundle. f By virtue of the Chern-Weil theory, which we recalled in the previous subsection, any real characteristic class of a fiat bundle vanishes. However, such bundle is not necessarily a trivial bundle as a principal bundle and furthermore, even if it were so, the fiat connection on it is not necessarily a trivial one.

Ii) p* : H*(M) -+ H*(A) is an isomorphism for * S i and injective for * = i + 1. If i = 00, it is simply called a minimal model. 36. The minimal model of the de Rham complex A*(sn) of the n-sphere sn is given by p : A(x) -+ A*(sn) for odd nand p : A(x,y) -+ A*(sn) for even n. Here degx = n,degy = 2n -l,dx = O,dy = x 2 and p(x) denotes the volume form of sn. a. version of the usual homotopy X x I -+ Y of a. pping X -+ Y, then we obtain the following definition. a. a. mapping H: A -+ B(t,dt) such that HII~o = fa, Hlt=l = h- Here B(t, dt) = B Q9 A(t, dt) and degt = 0, degdt = 1, d(t) = dt, d(dt) = O.

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