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K(V2, 2).

I) A = ukAk. (ii) Ak c Ak+l is a Hirsch extension for any k. (iii) For any k there exists m such that Ak cAm. If only conditions (i) and (ii) are satisfied, A is called generalized nilpotent. a. which is generated by elements of degree 1 is always nilpotent. 34. a. which is generated by elements of degree 1 is called a dual Lie algebra. It becomes automatically minimal. Let {Xdk be a basis of A(l) and suppose that the differential is given by dXk = z= afj Xi 1\ Xj' i,j Let us consider the dual of this.

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. A(t, dt) serves as a model for the de Rham complex of the unit interval 1= [0,1]. a. maps which corresponds to the adjoint map X -) Y I of the usual homotopy. He obtained this definition by constructing a model for the mapping space Y I = Map( I, Y). In this book we adopt this latter definition because it has some advantage over the former one, especially in its theoretical side.

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