Homotopy localization nearly preserves fibrations
by
E. Dror Farjoun and J. H. Smith
0. Introduction
The object of the present paper is to show thatshort of a `small abelian er*
*ror term'
the periodization functor X !PA X with resp ect to any suspension A = A0 preser*
*ves
homotopy fibration sequences. Moregenerally, the f-localization functor X !LfX *
*with
respect to double suspensions f = 2f00, preserves homotopy fibration sequences *
*up to
a "generalized Postnikov stage". This implies that Bousfield's homologicallo c*
*alization
functor LE with respect to a general p-local homology theory E = EZ=pZ preserv*
*es
fibrations of the form 2F ! 2E! 2B up to GEM with at most twonon-vanishing
homotopy groups. These results are based on the following general `technical'
Theorem A. Let f : A ! D be any map and X any space. If Lf X ' then
L2f X is a weak product of possibly infinitelymany Eilenberg MacLane space (GEM*
*).
More generally if F ! E ! Ba fibration with Lf E ' Lf B ' then Lf F 'GE M.
The first part of Theorem A and Theorem D below generalize similar results *
*of Bous-
field [6] where D is assumed to be a point andA is assumed to be a p-torsion sp*
*ace
with some restriction on its cohomologythe bottom dimension, in that case the G*
*EMin
question is an Eilenberg-MacLane space.
The control gained on the behavior of fibration sequences under localizatio*
*n allows us
to prove:
Theorem B. Let F !E ! B be a fibration and LKhnithe homological localization
functor with respect to Morava K-theory Khni where n 0. Then up to an error t*
*erm
with at most two homotopy groups in dimensions n + 1, n, the functor LKhniprese*
*rves the
theory E =EZ=pZ, there exist an integer d(E) 1 d 1 such that LE preserves fib*
*ration
up to an error term with none trivial homotopy groups only in dim d;d + 1.
On the way to prove Theorem B we prove three basic theorems ab out the beha*
*vior of
loops space functor and general fibration sequences under localizations of vari*
*ous types.
0.1 Definition: We say thatthe co-augmented functor X ! LX preserves the fi*
*bre
sequence F !E ! B over the pointed space B up to a (poly) GEMif the homotopy fi*
*bre
J of the natural map
LF !fibre(LE ! LB)
is a (poly) GEM: Namely,the fibre J is a possibly infinite pro duct of E M-spac*
*es K(G;i)
with G abelian group (or can be gotten from GEM'sby a finite sequence of orient*
*ed
fibrations.)
Example: For the standard loop space fibration X ! !X the above homotopy
fibre is the fibre of the natural map:LX ! LX.
Let us now formulate the three main consequences of Theorem A above when co*
*upled
with the technique of [8],[9]:
Theorem C. Let A = A0be any suspension space and X any simply connected
space. Then the fibre Jof the natural map oe
oe : PA X ! PAX
is a GEM. In other words,PA commutes with up to a GEM. Moreover J isA-perio dic
and PA0J ' fg.
Remark: For a given space A0of finite type it is a simple matter to determi*
*ne the
local A0-GEM's which localize to fg under PA0.
Theorem D. Let A = A0be any suspension. Then PA preserves any fibration
sequence up to a GEM. Therefore PA0preserves up to a GEM anylo opfibration F !
E !B for any space A0.
And finally,