Homotopy localization nearly preserves fibrations
by
E. Dror Farjoun and J. H. Smith
0. Introduction
The object of the present paper is to show that short of a `small abelian e*
*rror term'
the periodization functor X ! PA X with respect to any suspension A = A0 preser*
*ves
homotopy fibration sequences. More generally, the f-localization functor X ! Lf*
*X with
respect to double suspensions f = 2f00, preserves homotopy fibration sequences *
*up to
a "generalized Postnikov stage". This implies that Bousfield's homological loc*
*alization
functor LE with respect to a general p-local homology theory E = EZ=pZ* preser*
*ves
fibrations of the form 2F ! 2E ! 2B up to GEM with at most two non-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 infinitely many Eilenberg MacLane space (GE*
*M).
More generally if F ! E ! B a fibration with Lf E ' Lf B ' * then Lf F ' GEM.
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 and A is assumed to be a p-torsion s*
*pace
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 localizatio*
*n allows us
to prove:
Theorem B. Let F ! E ! B be a fibration and LK the homological localizat*
*ion
functor with respect to Morava K-theory K where n 0. Then up to an error te*
*rm
with at most two homotopy groups in dimensions n + 1, n, the functor LK pres*
*erves the
double loop of the given fibration. [See 0.1 for definitions]. More generally f*
*or any homology
theory E = EZ=pZ, there exist an integer d(E) 1 d 1 such that LE preserves fi*
*bration
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 about the behav*
*ior of
loops space functor and general fibration sequences under localizations of vari*
*ous types.
0.1 Definition: We say that the co-augmented functor X ! LX preserves the f*
*ibre
sequence F ! E ! B over the pointed space B up to a (poly) GEM if the homotopy *
*fibre
J of the natural map
LF ! fibre(LE ! LB)
is a (poly) GEM: Namely, the fibre J is a possibly infinite product of E -M-spa*
*ces K(G; i)
with G abelian group (or can be gotten from GEM's by a finite sequence of orien*
*ted
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 = A0 be any suspension space and X any simply connected
space. Then the fibre J of the natural map oe
oe : PA X ! PA X
is a GEM. In other words, PA commutes with up to a GEM. Moreover J is A-period*
*ic
and PA0J ' {*}.
Remark: For a given space A0 of finite type it is a simple matter to determ*
*ine the
local A0-GEM's which localize to {*} under PA0.
Theorem D. Let A = A0 be any suspension. Then PA preserves any fibration
sequence up to a GEM. Therefore PA0 preserves up to a GEM any loop fibration F !
E ! B for any space A0.
And finally,
2
Theorem E. Let f = 2g be any double suspension map 2g : 2A ! 2D. Then
Lf, the homotopy localization with respect to f, preserves any fibration sequen*
*ce up to a
poly GEM J. Moreover J is f local and LgJ ' *. Therefore, for any map g : A ! D*
* the
functor Lg preserves the double loops of any fibration sequence : 2F ! 2E ! 2B,*
* up
to a polyGEM 2J which is g-local and satisfies LgJ = *.
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 t*
*o localization
of the loop of X to the loop of the localization of X is always a polyGEM.(Prob*
*ably, it is
in fact a GEM.)
Remarks: 1. Notice the non example for theorem A for non-suspensions: If A *
*is an
acyclic space then PA A ' A even if PA A ' *. As a simply-connected counter ex*
*ample
when A is not a suspension one can take A = X = BS3. By Zabrodsky's Theorem [14*
* ]
the null component of map (BS3; BS3) is contractible. Therefore BS3 is BS3-peri*
*odic:
So in this case PA X ' X, but PA X ' * and BS3 is not a GEM. So we cannot disp*
*ense
with some assumption on A in order to allow the deduction in Theorem A:
PA X ' * ) PA X ' GEM:
2. It is not hard to construct examples in which P2A (A) has non-trivial h*
*omotopy
in all dimensions: Take A to be the wedge of all Moore spaces Mp(Z=pZ) p 2 th*
*en
P2A A is the product of all K(Z=pZ; p) p 2 when p is a prime.
3. It was shown by Casacuberta and Peschke [7] that the localization with r*
*espect 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 coeffic*
*ients.
4. We were unable to determine whether the polyGEM in (E) is in fact a GEM.
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. W*
*e then prove
3
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 func*
*tion
complexes are spaces of pointed maps.
Acknowledgement: This paper grew out of a communication with Pete Bousfield. We
are grateful for the exchange which was very useful for our progress.
1. A quick review of homotopy localization with respect to a map
We recall from [6,8,9,10] several basic properties of Lf and PA where A is *
*any space
and f : A ! D any map.
Given any spaces A; D; X and a map f : A ! D there exist co-augmented funct*
*ors
X ! LfX and X ! PA X (where we use PA as a special notation for L(A!*) in part
because its properties generalize the corresponding properties of the Postnikov*
* section
functor Pn). X ! LfX is initial among all maps X ! T to f-local spaces T i.e. s*
*paces
with the function complex map:
(1.1) map (f; T ) : map (D; T ) ! map (A; T )
being a weak equivalence. The co-augmentation X ! LfX is also terminal among a*
*ll
maps X ! T which become equivalences upon taking their function complexes to any
f-local space.
The functor Lf can be defined in either the pointed or unpointed category o*
*f spaces
and its value for connected A, D, X does not depend, up to homotopy, on the cho*
*ice of
the category in which one works. (S orS*).
Being defined on the unpointed category and being homotopy functor it has a*
*n asso-
ciated fibrewise localization functor that turns any fibration sequence F ! E !*
* B into a
fibration sequence LfF ! "E! B.
In addition Lf (or PA ) enjoys the following properties.
~=
1.2There is a natural equivalence Lf(X x Y ) -! Lf(X) x Lf(Y ).
4
1.3Every fibration sequence F ! E ! B with LfF ' * gives a homotopy equivalen*
*ce
LfE -'! LfB.
1.4The localization of a loop space is a loop space in a natural way and ther*
*e is a natural
equivalence of loop spaces: LfX ' Lf X.
__ __ *
*__
1.5If PA X is the homotopy fibre of X ! PA X then PA (P AX) ' *. Similarly Lf*
*L f X '
*. (Notice f).
1.6If F ! E ! B any fibre map and B is A-periodic (i.e. map*(A; B) ' *) or m*
*ore
generally PA B ' PA B then PA F ! PA E ! B is also a fibre sequence. Simi*
*larly
if F is A-periodic then F ! PA E ! PA B is a fibration sequence.
1.7Lf hocolimIX~= Lf hocolimILfX~, and in particular LfX ' * implies LfkX ' *
for all k.
1.8If LfX ~ * and LgY ' * (or PW Y ~ *) then Lf^g(X ^ Y ) ' * (or Lf^W X ^ Y *
*~ *).
1.9If PA B ' * and PB C ' * then PA C ' *.
1.10 If for all ff 2 I X(ff) is f-local where X(ff) is a member of an I-diagram*
* X~ indexed
by a small category I, then so is holimIX~. Moreover in any fibration with*
* A-periodic
base and fibre the total space is also A- periodic.
1.11 If Y is a n-connected GEM then so is LfY for any f : A ! D.
2. Proof of Theorem A
For the proof of A we consider of course the general localization with res*
*pect to a
map f : A ! D. Thus with a proper choice of f we can get Lf to be any homolog*
*ical
localization. For brevity of notation we fix our map f once for all and denote*
* Lf by L and
Lf by L etc. Recall (1.4) LX ' L X so our assumption in (A) is LX ' *.
5
2.1 Lemma. If LX ' *, then the natural map L2 (X _ X) ! L2 (X x X) in an
equivalence. Under this condition L2 X is an infinite loop space.
Proof : We start our proof with recalling [10] that the homotopy fibre of t*
*he natural
inclusion i : X _ X ! X x X is homotopy equivalent to the joint X * X ' (X ^ X)
for any pointed connected CW -complex X. Therefore by (1.3) in order to show th*
*at L2 (i)
is a homotopy equivalence it is sufficient to show that L2 (X * X) ' *. But by*
* 1.7
we always have the implication LfA ' * ) Lf A ' *. Since X * Y ' (X ^ Y ) it is
sufficient to show that L (X ^ X) ' *. For this we use 1.8 above.
Since LfX ' * by assumption so X is connected thus since both A and D are.
Thus PS1X ' *. Using 1.8 we get LS1^f(X ^ Y ) ' *, where S1 is the 1-sphere.
We proceed to show that the natural equivalence L2 (X _ X) ! L2 (X x X) imp*
*lies
that L2 X is an 1-loop space.
For this we use [1,14]. Namely we construct a special -space Xo with X1 ' X*
*. First
W
we construct a "non-special" space Xo by setting Xn = X, the point-sum of n-*
*copies
n
of X. Clearly for any map of finite pointed sets S ! T we have a corresponding*
* map
W W
X ! X so Xo is a functor from the category of finite sets to spaces with X*
*0 = pt.
S T
The only condition of -space that is not satisfied by Xo is that the map Xn ! X*
*1x. .x.X1
(n-times) is not a homotopy equivalence. But now we define Xn = L2 Xn since L*
*2 is
functorial we get a -space. It is special because we have the equivalence:
_ Y
L2 ( X) ~= L2 X
S S
by the first part of our lemma we get the desired equivalence from the multipli*
*cative
property of Lf (1.2) Lf(W x W 0) = LfW x LfW 0for any f and W , W 0.
This concludes the proof of 2.1.
We now proceed with proof of Theorem A.
By Lemma 2.1 we can write L2 X ' Y . We saw above that X is connected so X
is simply connected. Therefore so is L2 X, we conclude that Y is simply connect*
*ed.
6
Consider map *(2 X; Y ). We claim it is contractible. This is true since *
*any map
X ! Y = L2 X factors through L2 X. But since L X ~=*, the latter is equivale*
*nt
to a point by (1.7). Similarly kX ! Y must be null for all k 1. Therefore t*
*he
condition of Bousfield's key lemma (2.2) below are satisfied, Y being simply c*
*onnected.
Thus any map X ! Y = L2 X factors uniquely through SP kX for all k 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-a*
*ugmentation
on X:
SP 1X
??
y
X ----! L2 X
If we now apply L2 to this triangle, using L2 L2 ' L2 we get that L2 X is a *
*homotopy
retract of L2 (SP 1X). But (1.11) asserts that Lf 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 r*
*elative
version. (see [10].)
2.2 Bousfield's Key Lemma. Let X be a connected, Y a simply-connected space*
*s.
Assume map *(2X; Y ) ' *. Then map *(X; Y ) ~= map *(SP kX; Y ) for any k 1. [*
*5,
6.9].
2.3 Remark: A way to understand (2.2) is to interpret it as saying that the*
* space
SP kX can be built by successively glueing together copies of `X for ` 1 with *
*precisely
one copy for ` = 1. Since the higher suspension 2+jX (j 0) will not contribute*
* anything
to map *( SP `X; Y ) we are left with map *(X; Y ).
More precisely, it can be easily seen by adjunction that (2.2) is equivalen*
*t to the
following:
For any space X the suspension of the Thom-Dold map t:X ! SP kX induces a
homotopy equivalence P2X (t) upon localization with respect to the double susp*
*ension of
X.
In fact the same holds for the James functor JkX and other cases.
7
This is a correct reformulation because by universality (1.1) a map t induc*
*es a ho-
motopy equivalence on the f-localization iff it becomes an equivalence upon tak*
*ing the
function complex of t into any f-local space. In this form (2.2) can be verifie*
*d using (1.7)
and a homotopy colimit presentation of the Dold-Thom functors in [6,6.4],and us*
*ing the
fact discussed above that the inclusion (X _ X ! X x X) becomes an equivalence *
*after
localization with respect to the above double suspension.
3. Commuting fibrations and localizations:
Consider the following diagram:
Y ----! X ----! L E ----! LB
?? ? ? ?
y ?y ?y ?y
(3.1) B ----! F ----! E ----! B
?? ? ? ?
y ?y ?y ?y
LB ----! X ----! LE ----! LB
Where L is Lf a localization functor for some map f, and L(-) denotes the homot*
*opy
fibre of the coaugmentation. Let X, X , Y be the appropriate homotopy fibres th*
*at render
every sequence of two colinear arrows a fibre sequence.
Our main interest in the rest of the paper will be the space LX which as w*
*ill be seen
is the fibre of the map in (0.1) and measures the deviation from preservation o*
*f fibration
by the functor L. We will show that often LX is "small".
If we now assume that the map LY ! LY is a homotopy equivalence, we get
Y ~=L X. Therefore LL X ' * by (1.5) and thus LX ~= LL E ~= * by the same
reference. This implies, looking at F , that LF ~=LX. But X is a homotopy fibre*
* of two
local spaces so it is local and LX ' X. Therefore we conclude:
3.2 Observation. If a localization functor L = PA commutes up to homotopy w*
*ith
~=
B i.e. LB -! LB, then it will preserve any fibration over B.
Proof : Under the assumption LY ' *. Therefore LX ' L E and since L(L E)*
* '
*(1:3) we get LX ' *. But by [1.6] the fibration X ! F ! X is preserved in suc*
*h case so
LF = LX. Since X is local being the fibre of local spaces we are done: LX ' X.
8
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).
Theorem C. For any simply connected space X, and A = A0 the homotopy fibre
J of
PA X ! PA X
is a GEM space which is A-periodic, with PA0J = *.
Proof : Using [1.4] we get that the map in the theorem is just the loop on *
*the following
map
PA X ! PA X:
Since PA PA ' PA , (by definition!) the latter map is just the A-periodizati*
*on map:
PA X ! PA PA X, on PA X. By (1.5) the homotopy fibre F (with J = F ) of the
map dies under PA ; namely PA F ' *. By Theorem A it follows that PA F is a GE*
*M.
(A = 2A0). Since it is a fibre of a map of A-periodic spaces, F itself is A-pe*
*riodic
GEM. But (1.4) and (1.5) applied to the fibration in the theorem implies that i*
*f PA F ' *
that PA0J ' * as needed.
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 = PA , L = PA . By (1.5) above we get PA PA*
* B '
PA PA E ' *. Therefore (A = A0) we can use Theorem A to deduce PA X ' GEM.
Notice that X ! F ! X in diagram 3.1 is a fibration with a A-periodic base*
* space
X so by (1.6), it is preserved under PA . Thus the fibre of PA F ! X = PA X is *
*PA X, an
A-periodic GEM.
3.3 Corollary. Notice the following generalization of theorem (A) and inver*
*se to
(1.3). In any fibration F ! E ! B that induces an equivalence PA E ! PA B the
localization of the fibre with respect to the same suspension is a GEM.
9
4. Localization with respect to 2f
In this section we apply the above material to the fibre of LfY ! LfY and p*
*rove
Theorem E for a general map f : A ! D and a double loop space Y = 2X. The
main observation is that the difference between Lf and LC(f) (where C(f) = D [ *
*CA =
the mapping cone off) is an f-local GEM. Such a GEM lives only in the transiti*
*onal
dimensions for K-theory and other Morava theories.
First we make the following simple observation about any cofibration:
f h
4.1 Proposition. If A!- D!- C is a cofibration with cofibre C = D [ CA, t*
*hen
any f-local space is C-periodic and any A-periodic space is h-local.
Proof : This is immediate from the definition and the fact that for any spa*
*ce X the
sequence
h f
map (C; X)!- map (D; X)!- map (A; X)
is a fibre sequence with map (C; X) the homotopy fibre over the null component.*
* So if
map (A; X) ' * then h is an equivalence. While if f is an equivalence then map *
*(C; X) ' *.
4.2 Remark: So when we consider a Barratt-Puppe sequence:
f h f
A!- D!- C!- A --! D!- C!- 2A!- 2D!- : : :
As we pick maps and spaces more and more to the right: A; h; A; h : :b:eing loc*
*al or
periodic with respect to these maps and spaces becomes a strictly weaker condit*
*ion.
4.3 Corollary. As a direct result of (4.1) we get PA Lh = PA and LfPC = Lf .
We are now ready to prove the very useful
4.4 Lemma. For any map f : A ! D the fibre of L2f X ! PC X is a 2f-periodic
GEM that localizes to a point under C, where C is C(f) the mapping cone of f.
Proof : The cofibration sequence depicted above together with 4.1, show tha*
*t any C
periodic space is 2f-local. Therefore (4.3):
PC L2f ' PC :
10
Therefore (1.5) the fibre F of the periodization:
L2f X ! PC X:
satisfies PC F ' * and F is 2C periodic being 2f-local. But by Theorem A above
P2C F is a GEM. So F is a GEM which is 2C-periodic, and 2f periodic.
This can be rewritten in a more appealing form:
4.5 Corollary. There is a natural map for any f:
oe : Lf2X ! PC(f)X
whose fibre is an f-periodic GEM.
Proof : Consider the map
L2f X ! PC X
given in the proof of the lemma above. Since (1.4) LfY ~= Lf Y , we can rewrite*
* the
map as
W 2 Lf2X ! W PC X
with the fibre being 2C-local, GEM.
Looping it down twice we get the desired map Lf2X ! PC X with C = C(f)
the mapping cone of f. But now the fibre is a f-periodic GEM being the double l*
*oop of a
2f-periodic space.
Remark: The usefulness of the last corollary follows from the fact the PW ,*
* the peri-
odization functor behaves much nicer with respect to fibrations (1.6),(1.5) and*
* (D) then
Lf for a general map f.
We now turn to the proof of Theorem E, reformulated below for the convenien*
*ce of
the reader.
11
p
4.6 Theorem. Let g : A ! D be a map, and F!- E!- B be a fibration sequence
with B connected. Then the homotopy fibre of the natural map
LF ! fibre(LE ! LB)
where L = L2g , satisfies
1. is a poly GEM.
2. is 2g-local.
3.Lg ' *.
Remark: Conditions (1), (2) and (3) force to be `small' at each prime p. I*
*t 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 Lf.
Proof : We first deal with (3) by:
4.7 Lemma. Let be as in (4.6), then Lg() ' *.
Proof : In diagram (3.1) with L = L2f we get Y = fibre of(B ! Lg B) and
therefore by (1.5) above LgY ' *. Now by (1.3) and (1.5) above we get LgX = *.*
* Now
consider the diagram below which is derived from (3.1). Notice that X is 2g-loc*
*al being
a fibre of a map between such.
L2g F ________L2gF ----! *
?? ? ?
y ?y ?y
X ----! F ----! X
?? ? fl
y ?y flfl
----! L2g F ----! X = L2g X
By a similar argument one gets LgL2g F ' * and therefore Lg ' LgX ' * as need*
*ed.
12
4.8 Remark: Therefore by Theorem A, if we assume that f = f0 we get Lf () is
a GEM. However itself is not an f-local space so that we cannot conclude in ge*
*neral
for Lf that is a GEM even for L = L3f0 .
Proceeding with the proof we compare Pc(g) with L2f as follows using f Lem*
*ma 4.4
above.
F3 ----! (GEM)1 ----! (GEM)2
?? ? ?
y ?y ?y
(4.9) F2 ----! LfE ----! LfB (W = C(g); f = 2g)
?? ? ?
y ?y ?y
F1 ----! PW E ----! PW B
It follows that F3 is a poly GEM. On the other hand using Theorem D to compare *
*PW
(fibre) and fibre PW (E ! 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 Fi are from (4.9) above. (W = C(g))
(Poly GEM )2 ----! ----! (GEM )4
?? ? ?
y ?y ?y
(GEM )3 ----! LfF ----! PW F
(4.10) ?? ? ?
y ?y ?y
(Poly GEM ) = F3----! F2 ----! F1 = fibre(PW (E ! B))
By the above diagram F3 is a Poly GEM by Theorem D (GEM )4 is a GEM since F1*
* given
as fibre of PW (E ! B). By (4.4) we get that (GEM )3 is also a GEM , therefor*
*e is a
Poly GEM as needed.
5. Example: Homological localizations
We now turn to homological localization LE for generalized (non connected) *
*homology
theories and in particular to complex K-theory and higher Morava K-theories. In*
* these
13
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 h*
*olds even
for single loop spaces.
5.1 Theorem. Let LK be the homological localization with respect to comple*
*x K-
theory. Let F ! E ! B a fibration over a 2-connected pointed space B. Then LK n*
*early
preserves the fibration 2F ! 2E ! 2B up to an error term J = K(F; 2) x K(G; 1)
where F is torsion free and G abelian.
Similarly if LK is the homological localization with respect to Morava K*
* at
an odd prime then LK nearly preserves the fibration up to an error term of t*
*he form
J = K(F; n + 1) x K(G; n).
Proof : We use (1.4) above to consider LK 2F etc. Namely let g : W1 ! W2 b*
*e a
map with LK = Lg. Thus g is the wedge of 'all' K-equivalences between countable*
* spaces
[8]. Since Lg2Y ~=2L2g Y we first consider L2g using Theorem E above and th*
*en
loop down the result.
We may assume that all the spaces involved are HR-local with respect to usu*
*al ho-
mology with the appropriate subring of the rationals since we work at a given p*
*rime p and
HR- localization preserves double looping of any fibration.
Remark & Notation: Condition 4.6 (3) leads us to consider E*-acyclic spaces*
* i.e.
spaces X with LE X ' *. It will be convenient to denote by PE the "periodizatio*
*n functor
with respect to E-acyclic spaces". Namely PE = PC(g)where C(g) is the mapping c*
*one
of g. In other words PE = PAcy(E)where Acy(E) is the wedge of all pointed E*-ac*
*yclic
spaces with cardinality not bigger than "E*S0). For example if E is integral ho*
*mology then
PE is the plus construction of Quillen. Notice that PE X ' * , LE X ' *.
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 er*
*ror term in
this case: Namely the fibre of
LK 2F ! fibre(LK 2E ! LK 2B)
14
is a K-local polyGEM, which is a double loop space. Therefore by lemma (5.2) be*
*low we
have only two homotopy groups and since it is a double loop space with no torsi*
*on 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 us*
*e also
condition (3) of (4.6).
Since by the calculations in [14, 12.1] "KK(Z; n+2) = 0 while "KK(Z=p*
*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"
LK2F ! fibre(LK2E ! LK2B)
fits into a fibration:
K(F; n + 1) ! Y ! 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 tra*
*nsitional
dimension is 1, namely it does not vanish on any p-complete Eilenberg-MacLane s*
*pace,
then by [5, 3.5] it has the same homology equivalences as HZ=pZ and therefore h*
*as the
same localization. Since HR localization preserves the loop of any fibration se*
*quence 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.
5.2 Lemma. (1) Let Y be a K-local poly GEM. Then Y fits into a fibration K(*
*F; 2) !
Y ! K(G; 1) where F is a torsion free group.
15
Proof : We use (5.4) and (5.5) below. Since by [2] reduced K-theory vanish *
*on K(Z; 3)
we get LK K(Z; 3) ~=*. Applying (5.5) we get that if Y our K-local polyGEM fits*
* into a
fibration:
K(G0; 2) ! Y ! K(G; 1):
We claim that G0 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 map *(K(T; 2); K(G; 1)) ' *. 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(G0; 2) must also be K-local and in particular K(T; 2)-periodic, so *
*it can admit
no maps from K(T; 2). Hence Tor G0~= 0 and G0 is torsion free.
5.4 Lemma. Let Y be a polyGEM. Then for all i n the homotopy groups vanish
as follows.
ssiPK(Z;n)Y ~=0:
Remark: This is very likely to be true without looping down Y .
Proof : For the duration of this proof denote PK(Z;n)by "Pn. First let us *
*consider
Y = K(G; n) for an arbitrary abelian group. Since K(Gff; n) where Gffis finitel*
*y generated,
one uses (1.3) and (1.7) to deduce "PnK(Gff; n) ' * and thus "Pn(G; n) ' 0. No*
*w (1.3)
again implies "PnK(G; n + j) ' * for all j 0. Given any K(Z; n)-periodic GEM *
*say
Y = K(Gi; i) each K(Gi; i) is a retract of Y and thus also K(Z; n)-periodic an*
*d by the
above argument Gi= 0 for i 0. Now we proceed by induction on the construction *
*of the
polyGEM Y . Given a fibration
Y ! Y1 ! J
where J is a GEM, and Y1 is a polyGEM with ssi"PnY1 ~=0 for i n. Consider
"PnY ! "PnY1 ! "PnJ:
By Theorem D above this is a fibration up to a K(Z; n)-periodic GEM say , where
is the homotopy fibre of "PnY ! F , and F the homotopy fibre of "PnY1 ! "PnJ. An
easy exact sequence argument now gives ssi"PnY ' 0 for i n as needed.
16
5.5 Corollary. If LfK(Z; n) ' * for any f then ssiLf2Y ' 0 for i n for any
polyGEM Y .
Remark: Again this is very likely to be true without the double looping.
Proof : By Theorem E Lf2Y is again a polyGEM. However since LfK(Z; n) ' *
we have Lf = PK(Z;n)Lf or Lf2Y is a K(Z; n)-periodic loop space (1.4) and the r*
*esult
follows by (5.4) above.
5.6 Lemma. Let be an HZ=pZ* local, EZ=pZ* acyclic polyGEM with 2 EZ=pZ*-
local. Then has non-trivial homotopy group in at most two dimensions d + 2 and*
* d + 3.
Proof : We use [5] to characterize E = EZ=pZ*-local Eilenberg- MacLane spac*
*es and
GEM spaces. If E = EZ=pZ does not kill any K(Z=pZ; n) then [5, 3.5] E* equivale*
*nces are
the same as HZ=pZ-equivalences and so LE = LHZ=pZ. In that case 2 being E-local
implies is E-local and so we get ' *.
Otherwise consider with Bousfield [5] the transitional dimension d = d(E) a*
*s the
greatest integer n for which K(Zp^; n + 1) is still EZ=pZ-local. For Morava K we have
d = n.
Now since 2 is E*-local and HZ=pZ-local it has vanishing homotopy groups ab*
*ove
dimension d + 2 by lemma (5.5) above: Since we assume that for some 1 d < 1 on*
*e has
LE K(Zp^; d + 2) ' *.
This implies as in (5.5) that ssiLE 2 ' 0 for all i > d + 2.
Therefore we conclude that for our that satisfies L2E ' or LE 2 = 2
one has ssi ~=0 for all i d + 4.
On the other hand it follows directly from [5, 3.5, 6.1-6.4] that the lower*
* Postnikov
section Pd+1 is PE -local since it is a repeated extension of PE -local spaces *
*by PE -local
Eilenberg-MacLane spaces (1.10).
Therefore in the fibration
" ! ! Pd+1:
17
The base is PE -local and the fibre has homotopy groups in at most two dimensio*
*ns d + 2
and d + 3. Now using (1.6) to localize the fibration: we apply PE to get
PE ( ") ! PE ! PE (Pd+1) = Pd+1
The base being local this is still a fibration. But since LE ' * we get PE ' *
**. Moreover
by our assumption and, PE kills the two K(ss; n)'s involved in " and by (1.3) P*
*E " = *.
Therefore PE Pd+1 = Pd+1 ' * and thus ssi = 0 for i d + 1. This combines with
the upper bound above on ssi to complete the proof.
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