I am trying to read Dennis Sullivan's paper "Infinitesimal Computations in Topology", Publ. Math. IHES 47, pp.269-332, and I'm having some trouble with it.
Right now I am stuck in the proof of Theorem 2.2 on pp.282-283.
First of all, he does not define the notation /\(x,dx). The simplest thing for it to be is all gadgets of the form p(x)+q(x)dx, where p,q are polynomials in x and x anticommutes with dx. But maybe, since it is an algebra, there are also things like the square of dx floating around, i.e. maybe it has a basis consisting of things of the form x^i (dx)^j, where x and dx anticommute.
Second, in lines 14-15, he defines M_1 to be the space of all elements of degree 1 in A_1 whose differentials belong to the product of A_1 with itself. He lets M^1 be the algebra generated by M_1. Then he says, "Using d^2=0 and the freeness of A, one sees that M^1 is actually closed under d." Maybe it is easy, but for some reason I don't see it.
Next, in the proof of Prop.2.3, he says on p.283 that if we denote by I the ideal in A generated by the elements of positive degree in C, then I^k/I^(k+1) is isomorphic to M tensor (x^k,x^(k-1) dx), this under the assumption that C is /\(x,dx) and the tensor product has the product differential. I don't know his notation (x^k,x^(k-1) dx) and there is the old difficulty of not knowing what /\(x,dx) is.
Finally, I don't know how he defines the chain homotopy on the next line, although maybe it will become obvious after I know his notation.
If you have read this paper and can explain this details, please let me know. Thanks.