baselineskip 16pt .
Fibrewise Localization
and
Unstable Adams Spectral Sequences
by
William Dwyer
Haynes Miller
and Joseph Neisendorfer
0. Introduction.
LetLX be localization with respect to some homology theory and
let X ! LX be the lo calization map. If E ! B is afibration, then
fibrewise localization KE is defined to be a homotopy pullback
KE ! LE
# #
B ! LB :
This definition of fibrewise localization is completely general but, to be
useful, it should have threeprop erties.
First, we do not know in general that LE !LB has LF as its homo-
topy theoretic fibre, where F is the fibre of E !B . Hence, we do not
know that the homotopy theortic fibre of K E !B is LF and this, of
course, should be true of anything called fibrewise localization. Suppose
L is localization with respect to homology with coefficients in a ring R
where R is either Z=nZ or a subringof Q. Thena satisfactory solution
is given by,among others, Bousfield-Kan [2]. They construct a natural
transformation X !R1 X such that, at least for nilpotent X, R1 X is
the localization of X with respect to H( ; R). In addition, if ss1B acts
nilpotently on Hi(F;R) for all i, then the homotopy theoretic fibre of
R1 E !R1 B is R1 F . Forexample, this is the case if p is a prime, ss1
is a p-group, and R= Fp = the field with p elements.
Second, if Y is a space over B, we would like to get some hold
on the homotopy groups of the space (Y ;KE) of maps Y ! K E
over B. The Bousfield-Kan construction of Fp1 X gives an unstable
Adams spectral sequence for computing the homotopy groups of the s-
pace map (Y; Fp1 X) of pointed maps. We are led to imitate this in
order to compute the homotopy groups of (Y ;KE). Furthermore, the
E2 term of the Bousfield-Kan unstableAdams spectral sequence can be
identified by nonabelian homological algebra. Bousfield-Kan do give a
construction of fibrewise localization, denoted F_p1 E, and their construc-
tion gives an unstable Adams spectral sequence. But their construction
does not seem to permit an algebraic description ofthe E2 term. We
give in this paper a new construction offibrewise localization, denot-
ed BR1 (E), which gives when R = Fp a spectral sequence with an
algebraically identifiable E2 term.
Third, this spectral sequence gives information about (Y;BFp1 (E ))
but most often one really wants to know about (Y; E). We have not
resolved the question of what are the properties of the natural maps