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Extra info for Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments

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On the other hand R = S = Zp R (and S) concentrated in degree zero then also has a free model ~ with degx = 1 P and dx = p; it follows that H(~ 0 Z) ~ Z . 18 Fields. some h-DGF. 19 h-DGF, Lemma. Proof: cf. 7),) R itself does not. Every field ~ Ho(R) (For example, Free extensions permit, us to replace of the same homotopy type, which does contain has a free model of the form R of R by k. We need the ~[Xo] . The lemma follows via a direct limit argument from the following three obser- vations: (i) I, dx =p is a free model of model for an algebraic extension b E R ~, although then R, dx =ab -I Now suppose p: ~ Z-~k R is an and lift k(a); ~p; if to and (ii) R =R is a free model for h-DGF, p (iii) is a free o R(i/b).

The following assertions are then equivalent: (i) The Serre spectral sequence collapses at (ii) The morphism (iii) H*(E;~) Proof: H*(E;~) ~ H*(F;~) is a free H~(B;~) (resp. module In the topological case denote H*(E) ~ H~(F) is identified with E2 . H(S) ~ TorR(s;~)) (resp. H(S) ApL(B) ~ ApL(E) is surjective. is a free also by H(R)-module). R ~ S. Then H(S) ~ TorR(s;k). In either case, this is one edge homomorphism for the spectral sequence; the other is H(R) ~ H(S). Now clearly (i) (ii) and (iii).

The composite map which in a sense does not change the (~ ~3 was used by Jacobsson [Ja] to disprove a conjecture by Lemaire. Properties of As mentioned, we are concerned in this paper with describing the set ~7~ . It seems unlikely that there is any easy analytic way to characterize the elements of ~ , for we shall see in theorem 5 that J rather complicated operations. is closed under some Of special interest, however, are the rates of " 34 growth of the sequences analytic properties exponentially, {rank(An)}n20 of the series , and these rates are reflected in the A(z) .

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