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Help w/ Sullivan's paper
Posted:
Jul 1, 1996 6:18 AM
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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.
Allan Adler
adler@pulsar.cs.wku.edu
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