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\def\P{\bold P}
\def\L{\bold L}
\def\PL{\text{Poly GEM}}
\def\G{\text{GEM}}
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\document
\centerline{\bf Homotopy localization nearly preserves fibrations}
\bigskip
\centerline{\bf by}
\bigskip
\centerline{\bf E. Dror Farjoun and J. H. Smith}
\bigskip
\noindent
{\bf 0. Introduction}
\medskip
The object of the present paper is to show that short of a `small abelian
error term' the periodization functor $X\to \P_AX$ with respect to any
suspension $A=\Sigma A'$ preserves homotopy fibration sequences. More generally, the $f$-localization functor
$X\to \L_fX$ with respect to double suspensions $f=\Sigma^2f''$, preserves
homotopy fibration sequences up to a ``generalized Postnikov stage''. This
implies that Bousfield's homological localization functor $\L_E$ with respect
to a general $p$-local homology theory $E=E{\Bbb Z}/p{\Bbb Z}_*$
preserves fibrations of the form
$\Omega^2F\to\Omega^2E\to\Omega^2B$ up to GEM with at most two non-vanishing
homotopy groups. These
results are based on the following general `technical'
\proclaim{Theorem A}
Let $f:A\to D$ be any map and $X$ any space. If $\L_{\Sigma f}X\simeq *$ then
$\L_{\Sigma^2f}X$ is a weak product of possibly infinitely many Eilenberg
MacLane space (GEM). More generally if $F\to E\to B$ a fibration with
$\L_{\Sigma f}E\simeq \L_{\Sigma f}B\simeq *$ then $\L_{\Sigma f}F\simeq GEM$.
\endproclaim
The first part of Theorem A and Theorem D below generalize similar results of Bousfield [6]
where $D$ is assumed to be a point and $A$ is assumed to be a $p$-torsion
space with some restriction on its cohomology the bottom dimension, in that
case the GEM in question is an Eilenberg-MacLane space.
The control gained on the behavior of fibration sequences under localization
allows us to prove:
\proclaim{Theorem B}
Let $F\to E\to B$ be a fibration and $\L_{K\langle n\rangle}$ the homological
localization functor with respect to Morava $K$-theory $K\langle n\rangle$
where $n\ge 0$. Then up to an error term with at most two homotopy groups
in dimensions $n+1$, $n$, the functor $\L_{K\langle n\rangle}$ preserves the
double loop of the given fibration. [See 0.1 for definitions]. More generally
for any homology theory $E=E{\Bbb Z}/p{\Bbb Z}$, there exist an integer
$d(E)$ $1\le d\le\infty$
such that $\L_E$ preserves fibration up to an error term with none trivial
homotopy groups only in $\text{dim}\ d,d+1$.
\endproclaim
On the way to prove Theorem B we prove three basic theorems about the
behavior of loops space functor and general fibration sequences under
localizations of various types.
\demo{ 0.1 Definition }
We say that the co-augmented functor $X\to \L X$ {\it preserves the fibre
sequence} $F\to E\to B$ over the pointed space $B$ {\it up to a (poly) GEM}
if the homotopy fibre $J$ of the natural map
$$
\L F\to\text{fibre}(\L E\to \L B)
$$
is a (poly) GEM: Namely, the fibre $J$ is a possibly infinite
product of $E-M$-spaces $K(G,i)$ with $G$ abelian group (or can be
gotten from GEM's by a finite sequence of oriented fibrations.)
\enddemo
\demo{Example}
For the standard loop space fibration $\Omega X\to {*}\to X$ the above
homotopy fibre is the fibre of the natural map: $\L\Omega X\to\Omega\L X$.
\enddemo
Let us now formulate the three main consequences of Theorem A above
when coupled with the technique of [8],[9]:
\proclaim{Theorem C}
Let $A=\Sigma A'$ be any suspension space and $X$ any simply connected space.
Then the fibre $J$ of the natural map $\sigma$
$$
\sigma:\P_A\Omega X\to\Omega\P_AX
$$
is a GEM. In other words, $\P_A$ commutes with $\Omega$ up to a GEM.
Moreover $J$ is $A$-periodic and $\P_{A'}J\simeq\{*\}$.
\endproclaim
\demo{Remark}
For a given space $A'$ of finite type it is a simple matter to determine
the local $\Sigma A'$-GEM's which localize to $\{*\}$ under $\P_{A'}$.
\enddemo
\proclaim{Theorem D}
Let $A=\Sigma A'$ be any suspension. Then $\P_A$ preserves any fibration
sequence up to a GEM. Therefore $\P_{A'}$ preserves up to a GEM any loop
fibration $\Omega F\to\Omega E\to \Omega B$ for any space $A'$.
\endproclaim
And finally,
\proclaim{Theorem E}
Let $f=\Sigma^2g$ be any double suspension map $\Sigma^2g:\Sigma^2A\to
\Sigma^2D$. Then $\L_f$, the homotopy localization with respect to $f$,
preserves any fibration sequence up to a poly GEM $J$. Moreover $J$
is $f$ local and $\L_gJ\simeq *$. Therefore, for any map $g:A\to D$ the
functor $L_g$ preserves the double loops of any fibration sequence :
$\Omega^2F\to \Omega^2E\to\Omega^2B$, up to a polyGEM $\Omega^2J$ which
is $g$-local and satisfies $\L_gJ=*$.
\endproclaim
\proclaim{Corollary}
If one applies (E) to the standard loop space fibration one get that for
the localization with respect to any double suspension the difference between to localization of the loop of $X$ to the loop of the localization of $X$
is always a polyGEM.(Probably, it is in fact a GEM.)
\endproclaim
\demo{Remarks}
1. Notice the non example for theorem A for non-suspensions: If $A$ is an acyclic space then $\P_{\Sigma A}A\simeq A$ even if
$\P_A A\simeq *$. As a simply-connected counter example when $A$ is not a
suspension one can take $A=X=BS^3$. By Zabrodsky's Theorem \cite{14 \ }
the null component of
$\text{map}(BS^3,BS^3)$ is contractible. Therefore $BS^3$ is
$\Sigma BS^3$-periodic: So in this case $\P_{\Sigma A}X\simeq X$, but
$\P_AX\simeq *$ and $BS^3$ is not a GEM.
So we cannot dispense with some assumption on $A$ in order
to allow the deduction in Theorem A:
$$
\P_A X\simeq *\Rightarrow \P_{\Sigma A}X\simeq GEM.
$$
2. It is not hard to construct examples in which $\P_{\Sigma^2A}(\Sigma A)$
has non-trivial homotopy in all dimensions: Take $A$ to be the wedge of all
Moore spaces $M^p({\Bbb Z}/p{\Bbb Z})$ $p\ge 2$ then
$\P_{\Sigma^2A}\Sigma\ A$ is the product of all $K({\Bbb Z}/p{\Bbb Z},p)$
$p\ge 2$ when $p$ is a prime.
3. It was shown by Casacuberta and Peschke [7] that the localization with
respect to the degree p map between circles does not behave as nicely. that
map of course is not a suspension. Still it can be partly understood using
homology with local coefficients.
4. We were unable to determine whether the polyGEM in (E) is in fact a GEM.
\enddemo
\bigskip
\bigskip
\demo{Organization of the paper}
The rest of the paper is devoted to the proofs. In section (1) we review
the properties of localization that will be used in this paper. We then prove
Theorem A in the second section and (C),(D) and (E) in sections three and four.
The last section concludes with the proof of (B).
We work in the category of pointed $CW$-complexes and in particular
all function complexes are spaces of pointed maps.
\enddemo
\noindent
{\bf Acknowledgement:} This paper grew out of a communication with Pete
Bousfield. We are grateful for the exchange which was very useful for
our progress.
\bigskip
\noindent
{\bf 1. A quick review of homotopy localization with respect to a map}
We recall from [6,8,9,10] several basic
properties of $\L_f$ and $\P_A$ where $A$ is any space and $f:A\to D$ any map.
Given any spaces $A, D, X$ and a map $f:A\to D$ there exist co-augmented
functors $X\to\L_fX$ and $X\to \P_AX$ (where we use $\P_A$ as a special notation
for $\L_{(A\to*)}$ in part because its properties generalize the corresponding properties of the
Postnikov section functor $\P_n$). $X\to\L_fX$ is {\it initial} among all maps
$X\to T$ to $f$-local spaces $T$ i.e. spaces
with the function complex map:
$$
\text{map}(f,T):\text{ map}(D,T)\to\text{ map}(A,T)\tag1.1
$$
being a weak equivalence. The co-augmentation $X\to\L_fX$ is also terminal
among all maps $X\to T$ which become equivalences upon taking their function
complexes to any $f$-local space.
The functor $\L_f$ can be defined in either the pointed or unpointed category
of spaces and its value for connected $A$, $D$, $X$ does not depend, up
to homotopy, on the choice of the category in which one works. $(\script S\
\text{ or } \script S_*)$.
Being defined on the unpointed category and being homotopy functor it has
an associated {\it fibrewise localization functor} that turns any fibration
sequence $F\to E\to B$ into a fibration sequence $\L_fF\to\tilde E\to B$.
In addition $\L_f$ (or $\P_A$) enjoys the following properties.
\bigskip
\item{1.2}
There is a natural equivalence $\L_f(X\times Y)\overset\cong\to\longrightarrow
\L_f(X)\times \L_f(Y)$.
\bigskip
\item{1.3}
Every fibration sequence $F\to E\to B$ with $\L_fF\simeq *$ gives
a homotopy equivalence $\L_fE\overset\simeq\to\longrightarrow \L_f B$.
\bigskip
\item{1.4}
The localization of a loop space is a loop space in a natural way and there is a natural equivalence of loop spaces: $\L_f\Omega X\simeq \Omega\L_{\Sigma f}X$.
\bigskip
\item{1.5}
If $\overline{\P}_AX$ is the homotopy fibre of $X\to\P_AX$ then $\P_A(
\overline{\P}_AX)\simeq *$. Similarly $\L_f\overline{\L}_{\Sigma f}X\simeq *$.
(Notice $\Sigma f$).
\bigskip
\item{1.6}
If $F\to E\to B$ any fibre map and $B$ is $A$-periodic
(i.e. $\text{map}^*(A,B)
\simeq *)$ or more generally $\P_{\Sigma A}B\simeq \P_AB$
then $\P_AF\to\P_AE\to B$ is also a fibre sequence. Similarly
if $F$ is $A$-periodic then $F\to\P_{\Sigma A}E\to\P_{\Sigma A}B$ is a
fibration sequence.
\bigskip
\item{1.7}
$\L_f\ \underset I\to{\text{hocolim}}\ \underset\sim\to X=
\L_f\ \underset I\to{\text{hocolim}}\ \L_f \underset\sim\to X$, and in
particular $\L_fX\simeq *$ implies $\L_f\Sigma^k X\simeq *$ for all $k$.
\bigskip
\item{1.8}
If $\L_f X\sim *$ and $\L_g Y\simeq *$ (or $\P_W Y\sim *$) then $\L_{f\wedge g}
(X\wedge Y)\simeq *$ (or $\L_{f\wedge W}X\wedge Y\sim *$).
\bigskip
\item{1.9}
If $\P_AB\simeq *$ and $\P_B C\simeq *$ then $\P_A C\simeq *$.
\bigskip
\item{1.10}
If for all $\alpha\in I$ $X(\alpha)$ is $f$-local where $X(\alpha)$ is a member
of an $I$-diagram $\underset\sim\to X$ indexed by a small category $I$,
then so is $\underset I\to{\text{holim}}\ \underset\sim\to X$. Moreover in any
fibration with $A$-periodic base and fibre the total space is also $A$- periodic.
\bigskip
\item{1.11}
If $Y$ is a $n$-connected GEM then so is $\L_f Y$ for any $f:A\to D$.
\bigskip
\noindent
{\bf 2. Proof of Theorem A}
For the proof of A we consider of course the general localization with
respect to a map $f:A\to D$. Thus with a proper choice of $f$ we can get
$\L_f$ to be any homological localization. For brevity of notation we fix
our map $f$ once for all and denote $\L_f$ by $\L$ and $\L_{\Sigma f}$ by
$\L_\Sigma$ etc. Recall (1.4) $\L\Omega X\simeq\Omega \L_\Sigma X$ so our
assumption in (A) is $\L\Omega X\simeq *$.
\proclaim{2.1 Lemma }
If $\L\Omega X\simeq *$, then the natural map $\L_{\Sigma^2}(X\vee X)\to
\L_{\Sigma^2}(X\times X)$ in an equivalence. Under this condition
$\L_{\Sigma^2}X$ is an infinite loop space.
\endproclaim
\demo{Proof}
We start our proof with recalling [10] that
the homotopy fibre of the natural inclusion $i:X\vee X\to X\times X$ is
homotopy equivalent to the joint $\Omega X*\Omega X\simeq\Sigma(\Omega X\wedge
\Omega X)$ for any pointed connected $CW$-complex $X$. Therefore by
(1.3) in order to show that $\L_{\Sigma^2}(i)$ is a homotopy equivalence it is
sufficient to show that $\L_{\Sigma^2}(\Omega X*\Omega X)\simeq *$. But by
1.7 we always have the implication $\L_fA\simeq *\Rightarrow
\L_{\Sigma f}\Sigma A\simeq *$. Since $X*Y\simeq\Sigma(X\wedge Y)$ it is
sufficient to show that $\L_\Sigma(\Omega X\wedge \Omega X)\simeq *$. For
this we use 1.8 above.
\enddemo
Since $\L_f\Omega X\simeq *$ by assumption so $\Omega X$ is connected thus
since both $A$ and $D$ are. Thus $\P_{S^1}\Omega X\simeq *$. Using 1.8 we get
$\L_{S^1\wedge f}(\Omega X\wedge\Omega Y)\simeq *$, where $S^1$ is the
$1$-sphere.
We proceed to show that the natural equivalence
$\L_{\Sigma^2}(X\vee X)\to \L_{\Sigma^2}(X\times X)$ implies that $\L_{\Sigma^2}X$
is an $\infty$-loop space.
For this we use [1,14]. Namely we construct a special
$\Gamma$-space $X_\bullet$ with $X_1\simeq X$. First we construct a
``non-special'' $\Gamma$ space $\check X_\bullet$ by setting
$\check X_n=\underset n\to\bigvee\ X$, the point-sum of $n$-copies of $X$.
Clearly for any map of finite pointed sets $S\to T$ we have a corresponding map
$\underset S\to\bigvee\ X\to\underset T\to\bigvee\ X$ so $\check X_\bullet$ is
a functor from the category of finite sets to spaces with
$\check X_0=pt$. The only condition of $\Gamma$-space that is not satisfied
by $\check X_\bullet$ is that the map $\check X_n\to \check X_1\times\dots\times\check X_1$ ($n$-times) is not
a homotopy equivalence. But now we define $X_n=\L_{\Sigma^2}\check X_n$ since
$\L_{\Sigma^2}$ is functorial we get a $\Gamma$-space. It is special because
we have the equivalence:
$$
L_{\Sigma^2}(\underset S\to\bigvee\ X)\cong \underset S\to\prod\ L_{\Sigma^2}\
X
$$
by the first part of our lemma we get the desired equivalence from the
multiplicative property of $\L_f$ (1.2)
$\L_f(W\times W')=\L_fW\times \L_fW'$ for any $f$ and $W$, $W'$.
This concludes the proof of 2.1.
We now proceed with proof of Theorem A.
By Lemma 2.1 we can write $\L_{\Sigma^2}X\simeq\Omega Y$. We saw above that
$\Omega X$ is connected so $X$ is simply connected. Therefore so is
$\L_{\Sigma^2}X$, we conclude that $Y$ is simply connected.
Consider $\text{map}^*(\Sigma^2\ X,Y)$. We claim it is contractible. This is
true since any map $\Sigma\ X\to\Omega Y=\L_{\Sigma^2}X$ factors through
$\L_{\Sigma^2}\Sigma\ X$. But since $\L_\Sigma X\cong *$, the latter is
equivalent to a point by (1.7). Similarly $\Sigma^k X\to\Omega Y$ must be null for all
$k\ge 1$. Therefore the condition of Bousfield's key lemma (2.2) below are
satisfied, $Y$ being simply connected. Thus any map $X\to\Omega Y=
\L_{\Sigma^2}X$ factors uniquely through $SP^kX$ for all $k\ge 1$. Because of
the uniqueness of the factorization we can conclude that they are compatible.
Therefore we get a factorization through the infinite symmetric product of $X$ for the co-augmentation on $X$:
$$
\CD
@. SP^\infty X\\
@. @VVV\\
X @>>> \L_{\Sigma^2}X
\endCD
$$
If we now apply $\L_{\Sigma^2}$ to this triangle, using
$\L_{\Sigma^2} \L_{\Sigma^2}\simeq \L_{\Sigma^2}$ we get that $\L_{\Sigma^2}X$
is a homotopy retract of $\L_{\Sigma^2}(SP^\infty X)$. But (1.11) asserts
that $\L_f$ turns any GEM into a GEM and since a retract of
a GEM is a GEM we are done. In a similar fashion one proves the relative
version. (see [10].)
\proclaim{2.2 Bousfield's Key Lemma}
Let $X$ be a connected, $Y$ a simply-connected spaces. Assume $\text{map}^*
(\Sigma^2 X,Y)\simeq *$. Then $\text{map}^*(X,\Omega Y)\cong \text{ map}^*
(SP^kX,\Omega Y)$ for any $k\ge 1$. [5, 6.9].
\endproclaim
\demo{2.3 Remark}
A way to understand (2.2) is to interpret it as saying that the space
$\Sigma SP^kX$ can be built by successively glueing together copies of
$\Sigma^\ell X$ for $\ell\ge 1$ with precisely one copy for $\ell=1$.
Since the higher suspension $\Sigma^{2+j}X$ $(j\ge 0)$ will not contribute
anything to $\text{map}^*(\Sigma\ SP^\ell X,Y)$ we are left with
$\text{map}^*(\Sigma X,Y)$.
\smallskip
More precisely, it can be easily seen by adjunction that (2.2) is equivalent
to the following:
{\it For any space X the suspension
of the Thom-Dold map t:$\Sigma X\to\Sigma SP^kX$
induces a homotopy equivalence $ \P_{\Sigma^2X}(t)$ upon localization
with respect to the double suspension of $X$}.
\enddemo
\noindent
In fact the same holds for the James functor $J_kX$ and other cases.
This is a correct reformulation because by universality (1.1) a map t induces
a homotopy equivalence on the f-localization iff it becomes an equivalence
upon taking the function complex of t into any f-local space.
In this form (2.2) can be verified using (1.7) and a homotopy colimit presentation of the Dold-Thom functors in [6,6.4],and using the fact discussed above
that the inclusion $\Sigma (X\vee X\to X\times X)$ becomes an equivalence
after localization with respect to the above double suspension.
\noindent
{\bf 3. Commuting fibrations and localizations:}
Consider the following diagram:
$$
\CD
Y @>>> \bar X @>>> \bar{\L} E @>>> \bar {\L} B\\
@VVV @VVV @VVV @VVV\\
\Omega B @>>> F @>>> E @>>> B\\
@VVV @VVV @VVV @VVV\\
\Omega \L B @>>> X @>>> \L E @>>> \L B
\endCD\tag3.1
$$
Where $\L$ is $\L_f$ a localization functor for some map $f$, and
$\bar {\L}(-)$ denotes the homotopy fibre of the coaugmentation. Let $X$,
$\bar X$, $Y$ be the appropriate homotopy fibres that render every sequence
of two colinear arrows a fibre sequence.
Our main interest in the rest of the paper will be the space $\L\bar X$ which
as will be seen
is the fibre of the map in (0.1) and measures the deviation from preservation
of fibration by the functor $\L$. We will show that often $\L\bar X$ is
``small''.
If we now assume that the map $\L\Omega Y\to\Omega \L Y$ is a homotopy
equivalence, we get $Y\cong\bar {\L}\Omega X$. Therefore $\L\bar{\L}\Omega X
\simeq *$ by (1.5) and thus $\L\bar X\cong \L\bar {\L} E\cong *$ by the same
reference. This implies, looking at $F$, that $\L F\cong \L X$. But $X$ is a
homotopy fibre of two local spaces so it is local and $\L X\simeq X$. Therefore
we conclude:
\proclaim{3.2 Observation }
If a localization functor $\L=\P_A$ commutes up to homotopy with $\Omega B$
i.e. $\L\Omega B @>\cong>> \Omega \L B$, then it will preserve any fibration
over $B$.
\endproclaim
\demo{Proof}
Under the assumption $\L Y\simeq *$. Therefore $\L\bar X\simeq\bar {\L} E$ and
since $\L(\bar{\L} E)\simeq *(1.3)$ we get $\L\bar X\simeq *$. But by [1.6]
the fibration $\bar X\to F\to X$ is preserved in such case so $\L F=\L X$.
Since $X$ is local being the fibre of local spaces we are done: $\L X\simeq X$.
\enddemo
The following theorem C follows directly from (D) below. Still we provide a short proof that depends only on the first, weaker part of (A).
\proclaim{Theorem C}
For any simply connected space $X$, and $A=\Sigma\ A'$ the homotopy fibre $J$ of
$$
\P_A\Omega X\to \Omega\P_AX
$$
is a GEM space which is $A$-periodic, with $\P_{A'}J=*$.
\endproclaim
\demo{Proof}
Using [1.4] we get that the map in the theorem is just the loop on the
following map
$$
\P_{\Sigma A}\ X\to\P_AX.
$$
Since $\P_A\P_{\Sigma A}\simeq \P_A$, (by definition!)
the latter map is just the $A$-periodization map: $\P_{\Sigma A}X\to
\P_A\P_{\Sigma A}X$, on $\P_{\Sigma A}X$. By (1.5)
the homotopy fibre $F$ (with $J=\Omega F$)
of the map dies under $P_A$; namely $\P_AF\simeq *$.
By Theorem A it follows that $\P_{\Sigma A}F$ is a GEM. $(\Sigma A=
\Sigma^2A')$. Since it is a
fibre of a map of $A$-periodic spaces, $\Omega F$ itself is $A$-periodic GEM.
But (1.4) and (1.5) applied to the fibration in the theorem implies that if $\P_AF\simeq *$ that $\P_{A'}J\simeq *$
as needed.
\enddemo
\demo{Proof of Theorem D}
We argue with diagram (3.1) using the second, relative part of Theorem A.
We now read (3.1) with $\L=\P_A$, $\bar{\L}=\bar{\P}_A$. By (1.5) above
we get $\P_A\bar {\P}_AB\simeq \P_A\bar{\P}_AE\simeq *$. Therefore
$(A=\Sigma A')$ we can use Theorem A to deduce $\P_A\bar X\simeq GEM$.
Notice that $\bar X\to F\to X$ in diagram 3.1 is a fibration with
a $A$-periodic base space $X$ so by (1.6), it is preserved under
$\P_A$. Thus the fibre of $P_AF\to X=\P_AX$ is $\P_A\bar X$,
an $A$-periodic GEM.
\proclaim{3.3 Corollary} Notice the following generalization of theorem (A)
and inverse to (1.3). In any fibration $F\to E \to B$ that induces an equivalence $\P_{\Sigma A} E\to \P_{\Sigma A }B$ the localization of the fibre with respect to the same suspension is a GEM.
\endproclaim
\bigskip
\vfill\eject
\noindent{\bf 4. Localization with respect to $\Sigma^2f$}
In this section we apply the above material to the fibre of
$\L_f\Omega Y\to\Omega \L_fY$ and prove Theorem E
for a general map $f:A\to D$ and a double loop
space $Y=\Omega^2X$. The main observation is that the difference between
$\L_f$ and $\L_{C(f)}$ (where $C(f)=D\cup CA=\text{ the mapping cone of } f$)
is an $f$-local GEM. {\it Such a GEM lives only in the transitional dimensions for
$K$-theory and other Morava theories.}
First we make the following simple observation about any cofibration:
\proclaim{4.1 Proposition} If $A@>f>> D @>h>> C$ is a cofibration with cofibre
$C=D\cup CA$, then any $f$-local space is $C$-periodic and any $A$-periodic
space is $h$-local.
\endproclaim
\demo{Proof}
This is immediate from the definition and the fact that for any space $X$ the
sequence
$$
\text{map}(C,X)@>{\bar h}>> \text{ map}(D,X) @>{\bar f}>>
\text{map}(A,X)
$$
is a fibre sequence with $\text{map}(C,X)$ the homotopy fibre
{\it over the null component}. So if $\text{map}(A,X)\simeq *$ then
$\bar h$ is an equivalence. While if $\bar f$ is an equivalence then
$\text{map}(C,X)\simeq *$.
\enddemo
\demo{4.2 Remark}
So when we consider a Barratt-Puppe sequence:
$$
A @>f>> D @>h>> C@>>> \Sigma A @>{\Sigma f}>> \Sigma D @>>> \Sigma C @>>>
\Sigma^2A @>>> \Sigma^2D @>>>\dots
$$
As we pick maps and spaces more and more to the right: $A, h, \Sigma A,
\Sigma h\dots$ being local or periodic with respect to these maps and spaces
becomes a strictly weaker condition.
\enddemo
\proclaim{4.3 Corollary}
As a direct result of (4.1) we get $\P_A\L_h=\P_A$ and $\L_f\P_C=\L_f$ .
\endproclaim
We are now ready to prove the very useful
\proclaim{4.4 Lemma}
For any map $f:A\to D$ the fibre of $\L_{\Sigma^2f}X\to \P
_{\Sigma C}X$
is a $\Sigma^2f$-periodic GEM that localizes to a point under $\Sigma C$,
where $C$ is $C(f)$ the mapping cone of $f$.
\endproclaim
\demo{Proof}
The cofibration sequence depicted above together with 4.1, show that
any $\Sigma C$ periodic space is $\Sigma^2f$-local. Therefore (4.3):
$$
\P_{\Sigma C}\L_{\Sigma^2f}\simeq \P_{\Sigma C}.
$$
Therefore (1.5) the fibre $F$ of the periodization:
$$
\L_{\Sigma^2f}X\to \P_{\Sigma C}X.
$$
satisfies $\P_{\Sigma C}F\simeq *$ and $F$ is $\Sigma^2C$ periodic being
$\Sigma^2f$-local. But by Theorem A above $\P_{\Sigma^2C}F$ is a GEM.
So $F$ is a GEM which is $\Sigma^2C$-periodic, and $\Sigma^2f$ periodic.
\enddemo
\bigskip
This can be rewritten in a more appealing form:
\bigskip
\proclaim{4.5 Corollary}
There is a natural map for any $f$:
$$
\sigma: \L_f\Omega^2X\to\Omega \P_{C(f)}\Omega X
$$
whose fibre is an $f$-periodic GEM.
\endproclaim
\demo{Proof}
Consider the map
$$
\L_{\Sigma^2f}X\to \P_{\Sigma C}X
$$
given in the proof of the lemma above. Since (1.4)
$\L_f\Omega Y\cong \Omega \L_{\Sigma f}Y$, we can rewrite the map as
$$
\bar W^2\L_f\Omega^2X\to\bar W \P_C\Omega X
$$
with the fibre being $\Sigma^2C$-local, GEM.
Looping it down twice we get the desired map $\L_f\Omega^2X\to\Omega \P_C\Omega
X$ with $C=C(f)$ the mapping cone of $f$. But now the fibre is a $f$-periodic
GEM being the double loop of a $\Sigma^2f$-periodic space.
\enddemo
\bigskip
\demo{Remark}
The usefulness of the last corollary follows from the fact the $\P_W$, the
periodization functor behaves much nicer with respect to fibrations
(1.6),(1.5) and (D) then $\L_f$ for a general map $f$.
\enddemo
We now turn to the proof of Theorem E, reformulated below for the convenience
of the reader.
\proclaim{4.6 Theorem}
Let $g:A\to D$ be a map, and $F @>>> E @>p>> B$ be a fibration
sequence with $B$ connected. Then the homotopy fibre $\Delta$ of the natural
map
$$
\L F\to\text{fibre}(\L E\to\L B)
$$
where $\L=\L_{\Sigma^2g}$, satisfies
\item{1.}
$\Delta$ is a poly GEM.
\item{2.}
$\Delta$ is $\Sigma^2g$-local.
\item{3.}
$\L_g\Delta\simeq *$.
\endproclaim
\demo{Remark}
Conditions (1), (2) and (3) force $\Delta$ to be `small' at each prime $p$.
It is likely that its $p$-completion has only two or three homotopy groups
just as in Theorem B but in order to prove that one might have it reformulate
the next section as well as [5] for the general localization functor $\L_f$.
\enddemo
\demo{Proof}
We first deal with (3) by:
\enddemo
\proclaim{4.7 Lemma}
Let $\Delta$ be as in (4.6), then $\L_g(\Delta)\simeq *$.
\endproclaim
\demo{Proof}
In diagram (3.1) with $\L=\L_{\Sigma^2f}$ we get $Y=\text{fibre of }
(\Omega B\to\L_{\Sigma g}\Omega B)$ and therefore by (1.5) above
$\L_gY\simeq *$.
Now by (1.3) and (1.5) above
we get $\L_g\bar X=*$. Now consider the diagram below which is derived from (3.1). Notice
that $X$ is $\Sigma^2g$-local being a fibre of a map between such.
$$
\CD
\bar{\L}_{\Sigma^2g}F @= \bar{\L}_{\Sigma^2g}F @>>> * @.\\
@VVV @VVV @VVV @.\\
\bar X @>>> F @>>> X @.\\
@VVV @VVV @\vert @.\\
\Delta @>>> \L_{\Sigma^2g}F @>>> X @. =\L_{\Sigma^2g}X
\endCD
$$
By a similar argument one gets $\L_g\bar{\L}_{\Sigma^2g}F\simeq *$ and
therefore $\L_g\Delta\simeq \L_g\bar X\simeq *$ as needed.
\enddemo
\demo{4.8 Remark}
Therefore by Theorem A, if we assume that $f=\Sigma f'$ we get
$\L_{\Sigma f}(\Delta)$ is a GEM. However $\Delta$ itself is not an
$\Sigma f$-local space so that we cannot conclude in general for $\L_f$ that
$\Delta$ is a GEM even for $\L=\L_{\Sigma^3f'}$.
\enddemo
Proceeding with the proof we compare $\P_{\Sigma c(g)}$ with $\L_{\Sigma^2f}$ as follows using
f
Lemma 4.4 above.
$$
\CD
F_3 @>>> (GEM)_1 @>>> (GEM)_2 @.\\
@VVV @VVV @VVV @.\\
F_2 @>>> \L_fE @>>> \L_fB @. (W=\Sigma C(g), f=\Sigma^2g)\\
@VVV @VVV @VVV @.\\
F_1 @>>> \P_WE @>>> \P_WB @.
\endCD\tag 4.9
$$
It follows that $F_3$ is a poly GEM. On the other hand using Theorem D
to compare $\P_W$ (fibre) and fibre $\P_W(E\to B)$ we get the following
diagram in which the central vertical sequence measures the difference between
the fibre of the localization and the localization of the fibre and $F_i$
are from (4.9) above.
$(W=\Sigma C(g)$)
$$
\CD
@. (\PL)_2 @>>> \Delta @>>> (\G)_4 @.\\
@. @VVV @VVV @VVV @.\\
@. (\G)_3 @>>> \L_fF @>>> \P_WF @.\\
@. @VVV @VVV @VVV @.\\
@. (\PL) = F_3 @>>> F_2 @>>> F_1 @. =\text{fibre}(\P_W(E\to B))\\
\endCD\tag4.10
$$
By the above diagram $F_3$ is a $\PL$ by Theorem D $(\G)_4$ is a $\G$
since $F_1$ given as fibre of $\P_W(E\to B)$. By (4.4) we get that
$(\G)_3$ is also a $\G$, therefore $\Delta$ is a $\PL$ as needed.
\bigskip
\noindent
{\bf 5. Example: Homological localizations}
\medskip
We now turn to homological localization $\L_E$ for generalized (non connected) homology theories and
in particular
to complex $K$-theory and higher Morava $K$-theories. In these
cases the general fibration theorem $E$ above specializes to yield rather small
``error term'' when localizing double loop spaces. It is very probable that
the same theorem holds even for single loop spaces.
\proclaim{5.1 Theorem}
Let $\L_K$ be the homological localization with respect to complex
$K$-theory. Let $F\to E\to B$ a fibration over a 2-connected pointed
space $B$. Then $\L_K$ nearly preserves the fibration
$\Omega^2F\to\Omega^2E\to\Omega^2B$ up to an error term
$J=K(F,2)\times K(G,1)$ where $F$ is torsion free and $G$ abelian.
Similarly if $\L_{K\langle n\rangle}$ is the homological localization
with respect to Morava $K\langle n\rangle$ at an odd prime then $\L_{K\langle n\rangle}$
nearly preserves the fibration up to an error term of the form $J=
K(F,n+1)\times K(G,n)$.
\endproclaim
\demo{Proof}
We use (1.4) above to consider $\L_K\Omega^2F$ etc. Namely let
$g:W_1\to W_2$ be a map with $\L_K=\L_g$. Thus $g$ is the wedge of 'all'
$K$-equivalences between countable spaces [8]. Since
$\L_g\Omega^2Y\cong \Omega^2\L_{\Sigma^2g}Y$ we first consider
$\L_{\Sigma^2g}$ using Theorem E above and then loop down the result.
We may assume that all the spaces involved are $HR$-local with respect
to usual homology with the appropriate subring of the rationals since
we work at a given prime p and $HR$- localization preserves
double looping of any fibration.
\demo{Remark \& Notation}
Condition 4.6 (3) leads us to consider $E_*$-acyclic spaces i.e. spaces $X$
with $\L_EX\simeq *$. It will be convenient to denote by $\P_E$ the
``periodization functor with respect to $E$-acyclic spaces''. Namely
$\P_E=\P_{C(g)}$ where $C(g)$ is the mapping cone of $g$. In other words
$\P_E=\P_{Acy(E)}$ where $Acy(E)$ is the wedge of all pointed $E_*$-acyclic
spaces with cardinality not bigger than $\tilde E_*S^0)$. For example if
$E$ is integral homology then $\P_E$ is the plus construction of Quillen. Notice that
$\P_EX\simeq *\Leftrightarrow \L_EX\simeq *$.
\enddemo
Let us first consider $K$-theory: Using the above remark and notation we
can apply Theorem E to the appropriate map $g$.
In this case we use only (1) and (2) of (4.6) above to conclude that the
error term in this case: Namely the fibre of
$$
\L_K\Omega^2F\to\text{fibre}(\L_K\Omega^2 E\to\L_K\Omega^2B)
$$
is a $K$-local polyGEM, which is a double loop space. Therefore by
lemma (5.2) below we have only two homotopy groups and since it is a double
loop space with no torsion in the second homotopy it is easy to check that
all possible $k$-invariants vanish.
This concludes the proof for the case of $K$-theory.
We now turn to Morava $K$-theories at a given odd prime $p$. Here we need to
use also condition (3) of (4.6).
Since by the calculations in [14, 12.1] $\tilde K\langle n\rangle K({\Bbb Z},
n+2)=0$ while $\tilde K\langle n\rangle K({\Bbb Z}/p{\Bbb Z},n+1)=0$ we can
apply (5.5) below as we did for $K$-theory. But here we need also (5.6)
below to conclude: If $Y$ is ``fibre of''
$$
\L_{K\langle n\rangle}\Omega^2F\to\text{fibre}(\L_{K\langle n\rangle}\Omega^2 E
\to \L_{K\langle n\rangle} \Omega^2 B)
$$
fits into a fibration:
$$
K(F,n+1)\to Y\to K(G,n)
$$
where $F$ is a torsion free group.
Again $Y$ being double loop space and $F$ torsion free we can conclude that
the possible $K$-invariants must vanish.
This concludes the proof for Morava $K$-theories at odd primes.
The general case follows similarly. Given any homology theory $E$
if its transitional dimension is $\infty$, namely it does not vanish on
any $p$-complete Eilenberg-MacLane space, then by [5, 3.5] it has the
same homology equivalences as $H{\Bbb Z}/p{\Bbb Z}$ and therefore has
the same localization. Since HR localization preserves the loop of any fibration sequence
there is nothing to prove in this case.
If the transitional dimension $d=d(E)$ [5, 8.1] is some finite integer one
uses (5.5) below to complete the proof.
\proclaim{5.2 Lemma}
(1) Let $Y$ be a $K$-local poly GEM. Then $Y$ fits into a fibration
$K(F,2)\to Y\to K(G,1)$ where $F$ is a torsion free group.
\endproclaim
\demo{Proof}
We use (5.4) and (5.5) below. Since by [2] reduced $K$-theory vanish on
$K({\Bbb Z},3)$ we get $\L_KK({\Bbb Z},3)\cong *$. Applying (5.5) we get
that if $Y$ our $K$-local polyGEM fits into a fibration:
$$
K(G',2)\to Y\to K(G,1).
$$
We claim that $G'$ is a torsion free group. To see this we observe that
both base and total space $Y$ are $K(T,2)$ periodic with respect to any
abelian torsion group $T$: For the base directly observe that
$\text{map}^*(K(T,2),K(G,1))\simeq *$. And the total space $Y$ is by
assumption $K$-local so it is periodic with respect to any $K$-acyclic space
such as $K(T,2)$ [2]. Therefore the fibre $K(G',2)$ must also be
$K$-local and in particular $K(T,2)$-periodic, so it can admit no maps from
$K(T,2)$. Hence $\text{Tor } G'\cong 0$ and $G'$ is torsion free.
\enddemo
\proclaim{5.4 Lemma}
Let $Y$ be a polyGEM. Then for all $i\ge n$ the homotopy groups vanish as follows.
$$
\pi_i \P_{K({\Bbb Z},n)}\Omega Y\cong 0.
$$
\endproclaim
\demo{Remark}
This is very likely to be true without looping down $Y$.
\enddemo
\demo{Proof}
For the duration of this proof denote $\P_{K({\Bbb Z},n)}$ by $\tilde{\P}_n$.
First let us consider $Y=K(G,n)$ for an arbitrary abelian group. Since
$K(G_\alpha,n)$ where $G_\alpha$ is finitely generated, one uses (1.3)
and (1.7) to deduce $\tilde{\P}_nK(G_\alpha,n)\simeq *$ and thus
$\tilde{\P}_n(G,n)\simeq 0$. Now (1.3) again implies
$\tilde{\P}_nK(G,n+j)\simeq *$ for all $j\ge 0$. Given any
$K({\Bbb Z},n)$-periodic GEM say $Y=\Pi\ K(G_i, i)$ each $K(G_i,i)$ is a
retract of $Y$ and thus also $K({\Bbb Z},n)$-periodic and by the above
argument $G_i=0$ for $i\ge 0$. Now we proceed by induction on the
construction of the polyGEM $Y$. Given a fibration
$$
Y\to Y_1\to J
$$
where $J$ is a GEM, and $Y_1$ is a polyGEM with $\pi_i\tilde{\P}_n\Omega Y_1\cong 0$
for $i\ge n$. Consider
$$
\tilde{\P}_n\Omega Y\to \tilde{\P}_n\Omega Y_1\to\tilde{\P}_n\Omega J.
$$
By Theorem D above this is a fibration up to a $K({\Bbb Z},n)$-periodic GEM
say $\Delta$, where $\Delta$ is the homotopy fibre of
$\tilde{\P}_n\Omega Y\to F$, and $F$ the homotopy fibre of
$\tilde{\P}_n\Omega Y_1\to\tilde{\P}_n\Omega J$. An easy exact sequence
argument now gives $\pi_i\tilde{\P}_n\Omega Y\simeq 0$ for $i\le n$ as
needed.
\enddemo
\proclaim{5.5 Corollary}
If $\L_fK({\Bbb Z},n)\simeq *$ for any $f$ then $\pi_i\L_f\Omega^2 Y\simeq 0$
for $i\ge n$ for any polyGEM $Y$.
\endproclaim
\demo{Remark}
Again this is very likely to be true without the double looping.
\enddemo
\demo{Proof}
By Theorem E $\L_f\Omega^2Y$ is again a polyGEM. However since
$\L_fK({\Bbb Z},n)\simeq *$ we have $\L_f=\P_{K({\Bbb Z},n)}\L_f$ or
$\L_f\Omega^2Y$ is a $K({\Bbb Z},n)$-periodic loop space (1.4) and the
result follows by (5.4) above.
\enddemo
\proclaim{5.6 Lemma}
Let $\Delta$ be an $H{\Bbb Z}/p{\Bbb Z}_*$ local, $E{\Bbb Z}/p{\Bbb Z}_*$
acyclic polyGEM with $\Omega^2\Delta$ $E{\Bbb Z}/p{\Bbb Z}_*$-local. Then
$\Delta$ has non-trivial homotopy group in at most two dimensions $d+2$ and
$d+3$.
\endproclaim
\demo{Proof}
We use \cite{5} to characterize $E=E{\Bbb Z}/p{\Bbb Z}_*$-local Eilenberg-
MacLane spaces and GEM spaces. If $E=E{\Bbb Z}/p{\Bbb Z}$ does not kill
any $K({\Bbb Z}/p{\Bbb Z},n)$ then [5, 3.5] $E_*$ equivalences are the
same as $H{\Bbb Z}/p{\Bbb Z}$-equivalences and so
$\L_E=\L_{H{\Bbb Z}/p\Bbb Z}$. In that case $\Omega^2\Delta$ being
$E$-local implies $\Delta$ is $E$-local and so we get $\Delta\simeq *$.
Otherwise consider with Bousfield [5] the transitional dimension
$d=d(E)$ as the greatest integer $n$ for which
$K({\Bbb Z}^{\hat{\phantom m}}_{p}, n+1)$ is still
$E{\Bbb Z}/p{\Bbb Z}$-local. For Morava $K\langle n\rangle$
we have $d=n$.
Now since $\Omega^2\Delta$ is $E_*$-local and $H{\Bbb Z}/p{\Bbb Z}$-local
it has vanishing homotopy groups above dimension $d+2$ by lemma (5.5) above:
Since we assume that for some $1\le d<\infty$ one has
$\L_EK({\Bbb Z}_{p}^{\hat{\phantom m}},d+2)\simeq *$.
This implies as in (5.5) that
$
\pi_i\L_E\Omega^2\Delta\simeq 0
$
for all $i>d+2$.
Therefore we conclude that for our $\Delta$ that satisfies $\L_{\Sigma^2E}
\Delta\simeq\Delta$ or $\L_E\Omega^2\Delta=\Omega^2\Delta$ one has
$\pi_i\Delta\cong 0$ for all $i\ge d+4$.
On the other hand it follows directly from [5, 3.5, 6.1-6.4] that the
lower Postnikov section $\P_{d+1}\Delta$ is $\P_E$-local since it is a
repeated extension of $\P_E$-local spaces by $\P_E$-local Eilenberg-MacLane
spaces (1.10).
Therefore in the fibration
$$
\tilde\Delta\to\Delta\to \P_{d+1}\Delta.
$$
The base is $\P_E$-local and the fibre has homotopy groups in at most two
dimensions $d+2$ and $d+3$. Now using (1.6) to localize the fibration:
we apply $\P_E$ to get
$$
\P_E(\tilde\Delta)\to\P_E\Delta\to\P_E(\P_{d+1}\Delta)=\P_{d+1}\Delta
$$
The base being local this is still a fibration. But since $\L_E\Delta\simeq *$
we get $\P_E\Delta\simeq *$. Moreover by our assumption and, $\P_E$ kills
the two $K(\pi,n)$'s involved in $\tilde\Delta$ and by (1.3)
$\P_E\tilde\Delta=*$. Therefore $\P_E\P_{d+1}\Delta=\P_{d+1}\Delta\simeq *$
and thus $\pi_i\Delta=0$ for $i\le d+1$. This combines with the upper
bound above on $\pi_i\Delta$ to complete the proof.
\enddemo
\bigskip
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\bigskip
\baselineskip=10pt
\hfill Hebrew University,
\hfill Jerusalem
\medskip
\hfill Purdue University
\hfill W. Lafayette
\enddocument