$E_{\infty}$-Algebras and $p$-Adic Homotopy Theory
Michael A. Mandell
Let $\FP$ denote the field with $p$ elements and $\FPbar$ its
algebraic closure. We show that the singular cochain functor with
coefficients in $\FPbar$ induces a contravariant equivalence between
the homotopy category of connected $p$-complete nilpotent spaces of
finite $p$-type and a full subcategory of the homotopy category of
\einf $\FPbar$-algebras.
October 30, 1998.
This is a revision of the Jan 26, 1998 draft.
The major changes are the following:
The paper is now self-contained, and does not require the work from
"The Homotopy Theory of E-infty Algebras". The necessary results have
been included in this draft (pp. 5-8). Also, the proofs (pp. 28-35)
are entirly different and intrinsic (do not require shifting focus to
the ``linear isometries operad'').
More information on the analogue of the main theorem for finite fields
is included (Theorem A.2 stated on p. 35 proved pp. 37-39).
The connection with Bousfield-Kan p-completion (Remark 5.1 on p. 13)
and the p-pro-finite completion (Appendix B, pp. 39-43) is explained.
Results on the identification of the subcategory of E-infty algebras
in the image of the cochain functor is now included. Among other
things, the new ``Characterization Theorem'' (p. 2) and pp. 17-23.