Localizations, Fibrations,
and Conic Structures
by
Emmanuel Dror Farjoun
*
0. Introduction
The aim of the present paper is to investigate some general properties of hom*
*otopy
localization (and co-localization) with respect to a map f : A ! B, often with *
*B ' *.
The results give some understanding of the behavior of fibre and cofibre sequen*
*ces under
localizations. In particular we see how to commute localizations with the loop*
* space
functor.
We give several applications in this paper and others in a separate one which*
* will deal
with the classification of localization functors for B ' *. This classification*
* turns out to
be closely related to the Bousfield type of A as classified by Hopkins-Smith [1*
*2].
An interesting application is a decomposition theorem analogous to both the u*
*sual Post-
nikov fibration with higher connected covers as fibres and Quillen's plus const*
*ruction with
acyclic space as a fibre: To state this result we recall for a moment the nota*
*tions for
(co)localization functors:
We denote by PA X the localization of X with respect to the trivial map on A.*
* Thus
intuitively PA X is X "modulo the A information of X". In fact PA X is the uni*
*versal
space among all spaces Y with a contractible mapping space map(A; Y ) ' *
For any spaces A, X one can also consider the "best approximation to X built *
*out of
A via cofibrations": CW AX ! X. The space CW AX is the "co-localization" o*
*f X
with respect to A [B-1] and is built out of suspension of A in a similar fashio*
*n to the
construction of a CW -approximation to a space out of spheres (suspensions of S*
*0). It is
the universal space among all"A-spaces" Y with map(A; Y ) ' map(A; X).
0.1. Theorem. For any CW -complexes A; X, with [A; X] = * there is a homotopy f*
*ibre
sequence:
CW AX ! X ! PA X
in which, moreover, the pointed function space from fibre to base is contractib*
*le.
0.2. Lemma. For any CW -complexes A; X the equivalence PA X ' * implies that
'
CW AX -! X is a homotopy equivalence.
Remarks: Using recent powerful unpublished results of Bousfield that shows how *
*to con-
trol the difference between PA and PA it can be shown that the assumption [A; *
*X] = *
in 0.1 is not necessary if one slightly modifies the statement of the result, s*
*ee section (6)
below. Both results are more or less trivial in a stable category of spectra.
_________________________
*Part of this work was done with the kind support of the Mathematical Research *
*Institute ETH-Z, Zurich,
in Spring 1991.
0.3. Corollary. If 7X is K-acyclic, X a 7-connected p-torsion space, then X can*
* be
built by successively gluing together suspensions of V (1), the mapping cone of*
* an Adams
map between two Moore spaces. Namely, with the above notation
K*7X 0 impliesX = CW V (1)X;
and so every 7-deloopable K-acyclic space can be built from the `minimal' K-acy*
*clic
space V (1) [6] . In particular it follows that homotopy equivalences between s*
*uch spaces
are detected by maps form V (1) and its suspensions.
Proof: (see (6) below) It follows from a recent result of Bousfield [B-3] and t*
*he results
below that for such X we have PV (1)X ' * and by (0.2) we get X = CW V (1)X.
Higher Morava K-theories: By a similar argument it follows that sufficiently *
*high
Eilenberg-MacLane spaces can be built from V (n)- spaces (unstably) and in part*
*icular are
K < n >-acyclic (7.5), a result of Ravenel and Wilson [13].
General topological spaces: Since it seems that several of the main technical*
* theo-
rems below might apply to general topological spaces, it may be interesting to *
*consider that
case, too. For example, P1T is gotten by systematically killing all the homotop*
*y groups of
a space T by attaching cells. Likewise, CW S0T is clearly the usual CW -approx*
*imation
of T . In view of (0.2) one is led to
0.4 Conjecture. Let T be any path connected topological space. If P1T ' * is*
* a
contractible space then T is homotopy equivalent to a CW -complex.
Acknowledgments: I would like to thank Zig Fiedorowicz and Pete Bousfield for s*
*everal
illuminating discussions pertaining to to the present paper. Thanks are also d*
*ue to the
University of Geneve and ETH-Z for their hospitality and support.
1. Notation and main results
Throughout the present paper we consider only pointed spaces with the homotop*
*y type
of CW -complexes.Thus function complexes are spaces of pointed maps. We denote *
*by Lf
the homotopy localization functor with respect to an arbitrary map f . In the c*
*ase f is
the trivial map on a space W we use a slightly different notation to suggest it*
*s similarity
to the Postnikov construction and its special good properties: We often but not*
* always
denote the homotopy localization by PW and refer to f-local spaces as W - peri*
*odic spaces.
The lack of control over fibration under localization and the non-commutativi*
*ty of taking
loop space with taking say K-localization is a major obstacle in understanding *
*them. But
K-localization does not necessarily commute with loops even for highly connecte*
*d space
and not only in the bottom dimensions. It turns out that although homotopy loca*
*lization
with respect to an arbitrary map does not commute with loops there is a simple *
*formula
for LX namely it is homotopy equivalent to L X where L denote the localization
with respect to the suspension of f if L = Lf. Moreover localization with respe*
*ct to f
behaves much nicer than Lf and this allows us to conclude for example:
2
Example: The fibre of X ! (X)E , where ( )E is the usual Bousfield localization*
* with
any homology theory E*, is a loop space on a E*-acyclic space.
In some way the basic reference to the present work is [3], especially sectio*
*n 5 and 6.
But we also use [7] as a main reference. We will not repeat much of the techni*
*cal work
involved in [3,7] that can clearly be adapted to our situation here. The result*
* (B) below
about commuting loops and localization can be seen as a destabilisation of [B,1*
*.1]; where
a special stable localization is proven to commute with 1 .
We apply this partial understanding of behavior of fibrations under localizat*
*ion to give a
proof of an extension of Neisendorfer's theorem about connected covers of finit*
*e dimensional
2-connected spaces. Let us quote several results to illustrate what we have in *
*mind.
Theorem A. Let X be any 2-connected pointed space and j : X ! (X)K the Bous-
field K-localization map. Then W j induces an isomorphism on the stable v1-per*
*iodic
homotopy v-11sss*.
Proof: This follows from the example above using [M-T], see section 2 below.
Theorem 1. (Neisendorfer): Let f : K(Z=pZ) ! * and let X be any p-local (or p-
complete) finite dimensional complex with ss1X = 0 and ss2X = torsion. Then th*
*e Lf-
localization of the n-th connected cover X ! X is homotopy equivalence to X,*
* i.e.
LfX -'! X.
Using similar methods plus some control on the behavior of fibration under lo*
*calization
we get an analogous result for Vk-torsion:
Theorem 2. Let X be any p-local finite dimensional complex, 2-connected assume *
*that
for some n > 2 there exist a non-trivial v(1)-torsion, i.e. an element 0 6= ff *
*with vt1ff = 0,
then there exist such elements in infinitely many dimensions. Same for v(k)-tor*
*sion for all
k; 0 k < 1.
Remark:: The above results are consequences of several 'technical identities' w*
*hich we
now proceed to formulate. The bulk of the paper is devoted to this series of t*
*echnical
results: Taken together (3-10) show that there is some hope that our understand*
*ing of the
behaviour of fibration under localization can be improved.
Theorem 3. Let L = Lf be a localization with respect to any f : A ! B.
If F ! E -p! B is a fibration and LF ~ * then L(p) is a homotopy equivalence.
If A -i! X -j! X [CA is a cofibration and LA ~ *, then Lj is a homotopy equiv*
*alence.
If A ! X ! X [ CA is a cofibration and L(i) is a homotopy equivalence then L(*
*X [
CA) ~ *.
Proof: See section 4.
3
Theorem B. Let f : A ! B be a map of connected pointed spaces, f its suspension,
Lf; Lf the associated localizations. Then for any space X one has a homotopy e*
*quivalence
of loop spaces
` : LfX AE Lf X : r
that is natural up to homotopy.
Corollary. There is an equivalence W kLfkX ' Lkf X.
Theorem 4. For any map f, Lf commutes with finite products: The natural map Lf(*
*Xx
Y ) -! Lf(X) x Lf(Y ) is an homotopy equivalence.
Theorem 5. For any associative An-structure on X there is a corresponding natur*
*al
structure on LfX, with X -! LfX on An-map.
Theorem 6. For any loop space X there is a natural loop map X -! LX, i.e. an
A1 -map of H-spaces.
Theorem 7. Let f : A ! B be a map. Then the homotopy fibre of X ! Lf X localizes
under Lf to a contractible space.
Corollary. If Lf X ' * then LfX ' *.
Theorem 8. Let f : W ! * be the trivial map on any space W . Then the homotopy
fibre of X ! PW X, localizes under PW to a contractible space.
Theorem 9. For any f : A ! B, if EM is a generalized Eilenberg-Mac Lane space t*
*hen
so is Lf(EM).
Theorem C. Given F ! E ! B a fibration with all spaces connected and pointed. T*
*hen
under the assumptions below the noted localizations of the fibre sequences are *
*again fibre
sequences where f : W ! * is the trivial map for any space W :
1.If F is f-local then Lf preserves the fibration.
2.If B is f-local then Lf preserve the fibration.
3.E is f-local, F a group and the fibration principal, then LfF ! E ! Lf B *
*is a
fibration.
Corollary 10. F ! E ! B is preserved under localization for any fibration if F *
*is
P -periodic.
Remarks: It would certainly be interesting to know whther a version of (C) hold*
*s for
more general maps. Note the essential difference between 7 and 8: For maps of t*
*he type
f : W ! * one can prove much stronger results about Lf=PW . One would like to
4
generalize C(3.) to nilpotent fibration: When ss1E acts nilpotently on ssnF ~=s*
*sn+1(B; E).
Statement (1.) includes all of well known cases of localizing at subrings of th*
*e rationals or
other rings. Since there, one localizes with respect to a suspension map betwee*
*n 2-spheres
or bouquet thereof. Theorem B replaces general localization of a loop space by *
*localizing
with respect to a suspension Lf . This last functor according to C(i)(iii) has *
*more pleasant
properties than Lf itself.
Recently Bousfield has proven a very strong result showing that often localiz*
*ation with
respect to W ! * behave just like the case W = Sn; namely it very nearly preser*
*ves
fibration, modulo, at most, a single homotopy group in the long exact sequence.
The long fibration sequence: One could summarize many of the results above in
the following concise way: Given any fibration with F a W -periodic fibre, ther*
*e exists a
ladder of fibrations in which all the vertical arrows are
localizations as indicated:
F ----! E ----! B ----! F ----! E ----! B
?? ? ? ? ? ?
y= ?y ?y ?y= ?y ?y
F ----! PW E ----! PW B ----! F ----! PW E ----! PW B
Notice the following corollaries of theorem B.
(i)PW W ~ * for all W .
(ii)PW W ~ * for all W .
2. Proof of theorem A and B.
We will prove (A) using other results, clearly we will not need (A) in the pr*
*oof of 2-9.
Consider the following diagram of fibrations and the claim of the example above
a
AK X ----! AK X
?? ?
y ?y
=
X ----! X
?? ?
y ?y
'`
LK X ----! LK X
Where AK X etc. is a notation for the fibre of the K-localization and where LK *
*is the K-
localization and LK is given as follows. Let f = V Kff! V Lffbe the pointed u*
*nion of all
K-isomorphisms between countable CW -complex. According to Bousfield Lf = LK = *
*the
unstable Bousfield localization with respect to K-theory. Now let LK = Lf as*
* a functor.
5
Then by theorem 2 the bottom map ` is a homotopy equivalence and therefore so i*
*s the top
map a. Now by theorem 8, LK AK X ~ *. But f is a K-isomorphism and therefore *
*for
any space W the map W ! LK W is a K-isomorphism. Therefore AK X is K-acyclic,
i.e. has the K-theory of a point. By theorem 1 of [M-T] we deduce that v-11sss**
*AK X = 0.
Since this stable homology groups are the homotopy group of the connected spect*
*rum
AX associated to AK X, and since that spectrum AX is still K-acyclic, we can a*
*pply
the theorem about spectrum to AX and get our desired vanishing above. Now sinc*
*e a
is a homotopy equivalence by theorem (B) we also have v-11ssS*AK X ~= 0. But n*
*ow
consider v-11sss*an unconnective generalized homology theory gotten by tensorin*
*g sss*with
the Z-graded ring Z=pZ[v; v-1 ]. Then the corresponding Serre spectral sequence*
* implies
an isomorphism
v-11sss*WX = v-11sss*X -! v-11sss*WLK X
as needed.
In order to prove Theorem B we first turn to:
2.1 Proof of theorem 4: Let L = Lf be any localization. There is an obvious map
L(X x Y ) ! LX x LY given by the product of the maps induced by projections. *
*We
need to find an inverse to this map
LX x LY - ! L(X x Y ) :
By adjunction we need a map.
2.1.1: LX ! map LY; L(X x Y ) .
2.2 Lemma. Let W be any f-local space, and Z any space whatever, the function c*
*om-
plex map (Z; W ) is f-local.
Proof: This follows directly from the definition: we must show that map f; map*
*(Z; W )
is a homotopy equivalence. By adjointness that map is just the map map Z; map(*
*f; W ) .
By assumptions map (f; W ) is a homotopy equivalence, so the same is true for a*
*ny mapping
space into it.
It follows that the range of 2.1.1 is an f-local space. Therefore by universa*
*l property of
L = Lf it is enough to construct a map u : X ! map LY; L(X x Y ) . Now by repe*
*ating
this argument of universality it is enough to find a map Y ! map X; L(X x Y ) *
* or again
a map X x Y ! L(X x Y ). We take this last map is simply the universal coaugmen*
*tation
of L. It is routine to check by adjunction and universal properties that the ma*
*p we get is
in fact a homotopy equivalence.
2.3 Proof of theorem 6: We will use only the fact that L is coaugmented and pre*
*serves
products up to homotopy. To present the map X ! LX as a loop map we use Segal
characterization of loop space as a space Y1 in a `special' simplicial space Y**
* with Y0 ~ *
and in which the product of all the maps Y (i) : Y1 ! Yi given by i(0) = i - 1,*
* i(1) = i
is a homotopy equivalence
Y (i) : Yn -! Y1 x : :x:Y1
6
between the i-th space Yn and the n-th power of Y1. If in such a simplicial spa*
*ce ss0Y1 is
a group, then Segal's theorem says that |Y*| ' Y1 [ Adams]. Now, given X we pre*
*sent
X as a "Segal loop space" by taking a monoidic version of X, say X, the Moore *
*loop
space of maps [0; a] ! X, so that we get a precise simplicial space made out of*
* the monoid
operations and projections into factors. Let the simplicial space G* : op ! T o*
*p be given
by Gn = ( X)n; this has the right properties so that |G*| ' G1 = X. Now take *
*the
simplicial space LG* with (LG*)n = L(Gn), the localization of the product ( X)*
*n. Since
the homotopy equivalence
Gn = ( X)n -! (G1)n = ( X)n
in this case the identity map given by a product of projection, this same proje*
*ction i
gives on LG* a product map L ( X)n ! (L X)n which we know by theorem 4 above
to be a homotopy equivalence. Therefore LG* is a special simplicial space, sati*
*sfying the
said conditions.
Notice that in dimension one we have ss0LG* = ss0L X which is clearly a grou*
*p, since
the equivalence L(X x X) ' LX x LX is natural. Therefore for |LG*| ' LG1 ' L X,
presenting L X as a loop space. In fact, since G* ! LG* is a simplicial map b*
*etween
simplicial spaces it induces |G*| ! |LG*| a map which gives a loop map
X-' |G*| -! |LG*|-~! L X :
This equivalence are of loop space so they combine to show that X ! L X is a *
*map of
loop spaces, or rather that there is a Stasheff A1 -structure on L X for which*
* X ! L X
is a map of A1 -monoids.
2.5 Proof of theorem B: We prove the existence of two maps, each is given by the
appropriate universality conditions on localizations. This will imply that the*
* maps are
mutual inverses. The easier map LX ! Lf X exists as an extension over LX of the
obvious loop map X ! Lf X by the following lemma to which we will refer repeate*
*dly
below.
2.5.1 Lemma. A space X is f-local with respect to the suspension of f : A ! B i*
*f and
only if X is f-local.
Proof: This is immediate adjunction
map (f; X) = map (f; X)
between suspension on loops. The left is a homotopy equivalence iff the right i*
*s.
Thus we have a factorization through L = Lf since LfX, Lf X are certainly f-
local. To construct the map the other way we may proceed by using the result ab*
*ove that
X ! LfX is a loop map, in a natural way, and so we can classify it to get X ! W*
* X !
W LfX. In fact we may use |LG*| as a model for W LfX from (2.3) above. Then the
7
map in the other direction will be given as a loop map if we construct a map of*
* spaces
Lf X ! W LfX, by looping down and composing with the obvious maps.
Again we use lemma 2.5.1 to notice that W LX is f-local and therefore by univ*
*ersality
of Lf it is sufficient to find a map X ! W LX. This can be taken as the compos*
*ite
W `
X -! W X -! W LX :
Now we have a diagram of maps
X? -=! X?
?yj1 ?yj2
` : LfX --! Lf X : r
which commutes on both sides since the bottom arrows were found by universality*
*. This
means rOj2 = j1 or rO`Oj1 = j1, but by uniqueness of factorization through the *
*universal
LfX, we get r O ` is homotopic to the identity. Similarly one gets ` O r ~ Id.
3. Localization and homotopy limits
In general of course we can not commute localizations with either direct or i*
*nverse
homotopy limits. In fact theorem 3 above can be read as a commutation rule sinc*
*e LfX
is the localization of the homotopy limit in the diagram . -! X - ., while Lf *
*X is
a homotopy limit of the localization with respect to f of the previous diagram.*
* It is
essential however to notice that
3.1 Proposition. The homotopy inverse limit of f-local spaces is f-local.
Proof: Simply if {Xff}ffis any small diagram on which one takes holim, then we *
*have a
canonical homotopy of maps for any map f : A ! B map (f; holimffXff) ~ holimff(*
*f; Xff).
Since the holim of equivalences is a homotopy equivalence, the right handside i*
*s an equiv-
alence, so that holimffXffis f-local.
3.2 Counter-example: In contrary to finite products (2.1 above) localization do*
*es not
commute with infinite products. Here is an easy counterexample. Let Affbe a col*
*lection
of acyclic spaces ff 2 N so that Xffis not acyclic. This is easy to construct o*
*nce we have
a large product of perfect groups which is not perfect. In fact all we are sayi*
*ng here is that
abelianization does not commute with infinite product, although it does commute*
* with a
finite one. Now if LZ is the Bousfield localization with respect to integral ho*
*mology, then
LZXff= * but LZssXff6= 0, since it is homologically non trivial.
Still theorem 3 allows us to commute localization and homotopy limits in spec*
*ial case.
The basic tool use is a canonical map from the colimit of localization to the l*
*ocalization
of the colimit.
8
3.3 Lemma. For any diagram (Xff)ffof space there exist a natural map colimffLXf*
*f!
L colimXff, commuting with the obvious map of colimXffinto both.
Proof: We have the indicated map because maps, in general, are definable on col*
*imits
as the colimit of maps. We take the colimit of the diagram of maps LXff! L(coli*
*mXff)
induced by the canonical maps to the colimit.
3.4 Proof of theorem 3: Consider the cofibration A ,! X ! X=A. We have a
factorization of map into a colimit of the diagram CA - A ,! X given by X=A !
LX=LA ! L(X=A) with the composition being the co-augmentation of L.
Now if we assume that LA ! LX is a homotopy equivalence so that LX=LA is
contractible and X=A ! L(X=A) is null homotopic. By the idempotency of L we get
L(X=A) ~ * as needed.
Now assume that LA ~ *. Then we have an equivalence LX '! LX=LA. Therefore the
natural map X=A ! L(X=A) factors through a new map X=A ! LX. By universality we
get a map L(X=A) ! LX which is easily seen to be a homotopy equivalence as requ*
*ired.
Now to get to the first assertion, assume LfF = LF ~ *. We use a fibrewise lo*
*calization
to localize all the fibres to a point, this gives us the space B. But the local*
*ization of the
fibrewise localization is the localization of E so we get a map B ! LE, inducin*
*g a map
LB ! LE which is easily seen by the universal properties to be a homotopy equiv*
*alence.
The construction of fibrewise localization is a small variation on the constr*
*uction of local-
ization: Recall the construction of LfX [F]. Now instead of attaching all maps *
*gffAff! E
we simply attach only map gffwith pgff= pt 2 B. Namely, over each point in B we*
* get
the usual localization. In fact the map E ! LB E gotten this way, can be ident*
*ified in
a natural way to the total space of the fibration induced from u : B ! B autLF *
*where
aut LF is the monoid of all (unpointed) self equivalences of the space LF . Alt*
*ernatively,
one can use [3, 5.5]. This completes the proof of (3).
4. Localizations of fibrations
It is clear that in general the localization of fibre sequence F ! E ! B, na*
*mely
LF ! LE ! LB is not a fibre sequence. This is true even for the simplest kind*
* of
localization, say rationalization, i.e. Bousfield's localization with respect t*
*o homology with
rational coefficients. This functor LQ preserves fibration if ss1B ~=0 ~=ss1E ~*
*=ss1F , other
conditions are not known. Our result will give somewhat more general condition*
*s. Our
main aim is to detect what is the situation with more general localization for *
*example with
respect to K-theory. On this nothing at all is known and I do not believe that *
*there are
theorems which are very specific to K-theory as opposed to other homology theor*
*ies and,
more generally, maps.
We have seen above a sample of the results that we have in mind: Namely, theo*
*rem 3 says
that any fibre sequence is preserved if the fibre localizes to a point. Theorem*
* B describes
the general localization of a loop space, given as a loop space of a related lo*
*calization. In
9
other words theorem B describes the following diagram of fibrations:
X ----! * ----! X
?? ? ?
y ?y ?y
LX ----! * ----! L X
and it says simply that the bottom raw is a fibration sequence. This is related*
*, of course,
to theorem C(3).
It is clear from the following example, however, that no amount of assumption*
* con-
cerning the connectivity of the spaces involved or the principality of the fibr*
*ation will
allow preservation under localization. Consider the localization with respect t*
*o the map
Sn+1 ! *. This is simply killing higher homotopy groups or taking the n-th Pos*
*tnikov
stage. A moment reflection will show no amount of connectivity assumption will *
*make the
Postnikov functor preserve fibration sequences.
4.1 Proof of theorem 7 and 8: We start with (8). Theorem 7 has essentially the *
*same
proof but uses (4.3) below. Consider the fibration sequence in which all the ma*
*ps are the
natural ones: `
F ----! X ----! Lf X ----! B autF
?? ? fl ?
y oe?y oe. flfl ?y
LfF ----! X ----! Lf X ----! B autLfF
`
By lemma 2.5.1 (L = Lf) we have B autLF is f-local since autLF itself is f-loca*
*l by
a version of lemma 2.2. Notice that we are using here unpointed self equivalenc*
*es but the
fibration aut:X ! autX ! X relating the pointed and unpointed mapping spaces, s*
*hows
that for connected X we still have a version of lemma 2.2 for unpointed functio*
*n spaces.
Notice that for any fibration F ! E ! B we have a homotopy pullback diagram
E ----! B autF_
? ?
(4.1.1) ?y ?y
B ----! B autF
[Dror-Zabrodsky], that in our case present X as a homotopy limit of f-local sp*
*aces.
Thus X is f-local. But now by the universality property of the map X ! Lf X, t*
*he
map X ! X factors through it. This gives a map oe. But this means that the fib*
*ration
X ! Lf X has a cross-section. Notice that by uniqueness of factorization we sh*
*all get
`. oe = id from `. oe . ` = `, but oe . ` = b and `b = ` by construction (compa*
*re 2.5). Thus the
map LfF ! X induces a one-to-one map on pointed homotopy class [W; LfF ] ! [W; *
*X]
for any space W . In particular for W = F . Notice however that the map F ! X f*
*actors
10
through the base of the fibration Lf X, and therefore it is a null homotopic ma*
*p. But
this means that F ! LfF is null homotopic, and therefore the idempotency of Lf *
*gives
that LfF ~ *, as claimed.
4.2 Remark: It is essential in (7) thatLfF ~ * rather than Lf F ~ * holds. The *
*latter
is, in general, not true as can be observed from taking f to be the degree p-ma*
*p between
two 2-spheres f = p : S2 ! S2. If X = S2 itself ss1F will be Z[_1p]=Z a p-torsi*
*on group.
But it is clear from the construction of Lf that ss1Lf F = ss1F 6= 0. However,*
* if we had
taken f = p : S1 ! S1, then with the same F we would have LfF = *.
Proof of theorem 8 is very similar, the only point to add is this:
4.3 Lemma. Let F ! E ! B be any fibration with F and B f-local with respect to a
map f : W ! * of any space W to a point. Then E is also f-local.
Proof: This follows since for any fibration if B ~ * and F ~ * then E must be a*
*lso
equivalent to *.
4.4 Remark: Lemma 4.3 is definitely not true for other maps, even of simple typ*
*e as
p : Sn ! Sn n 1.
4.5 Proof of theorem C: Let us start with C(2). Consider the diagram of fibrati*
*ons
L F ________LF ----! *
?? ? ?
y ?y ?y
F ----! E ----! B
?? ? ?
y ?y ?y
LF ----! E1 ----! B
In which LF are the fibres and E1 is the fibrewise localization of E over B. Th*
*en by the
argument above since B and LF are local with respect to W ! * so is E1 (see 4.3*
*). Now
by theorem 8 above (4.1) we know that L(L F ) ~ *. By theorem 3 applied to the *
*middle
vertical fibre sequence we know that LE !' LE1 is a homotopy equivalence. But *
*E1 is
f-local so LE1 = LE = E1 therefore LF ! LE ! B is a fibre sequence as needed.
Consider now C(1). By the argument of (4.1) applied to the diagram (4.8) bel*
*ow we
know that if L = Lf , then since F is f-local, E1 is f-local. Apply theorem (7)*
* above
to get that the fibre L B = Lf B must localize to a point under L :
* ----! L B ________L B
?? ? ?
y ?y ?y
(4.8) F ----! E ----! B
?? ? ?
y ?y ?y
L F ________F----! E1 ----! L B:
11
Now as above since E1 is f-local we get E1 ' Lf E as needed.
Consider now the case C(3). Consider the fibre sequence
G -! E -! B -! BG :
Since we assume E is f-local then for the fibration sequence over BG we are in *
*the situation
C(1) proven above. Therefore the f-localization sequence E ! Lf B ! Lf W G is a
fibration sequence. Therefore the fibre of E ! Lf B is Lf W G. By theorem (B) *
*that
we proved above this is the same as LfW G = LfG, as needed. This completes the *
*proof
of (C).
5. Proof of theorems (1) and (2)
Consider the space Bo = K(Z=pZ; 1). We give an independent proof of a result*
* of
J. Neisendorfer that gives a strong version of Serre's result that the finite d*
*imensional
2-connected complexes have homotopy in an infinite number of dimensions. Consid*
*er the
localization functor Lo with respect to the map K(Z=pZ; 1) ! *. H. Miller's res*
*ult, proving
the Sullivan conjecture, can be formulated by saying that for any finite dimens*
*ional space
K one has a homotopy equivalence K ! LoK or simply that K is (Bo ! *)-local.
5.1 Neisendorfer theorem. Let ss1K = 0 and ss2K < 1, and K is a p-local space.
Then LoK = K for any n-connected cover K of K.
Proof: We first note that if K(ss; n) is p-local then LoK(ss; n) = *. This foll*
*ows by the
induction from theorem (3) above if we show LoK(ss; 2) ~ *. But ss is p-local g*
*roup, since
LW W ~ * for any space LoBo ~ *. Now since ss is abelian and p-local so is LoK(*
*ss; 2)
and we have the map Bo; LoK(ss; 2) = *, this means by standard computation of*
* this
function complex that ssiLoK(ss; 2) = 0. Now consider the fibration
PnK -! K -! K -! PnK :
Again since PnK is connected with ss finite group we have LoPnX using inductive*
*ly
theorem (3) we get LoPnK ~ * and therefore
LoK -'! LoK ~ K :
5.2 Proof of theorem 2: Assume we have a 3-connected p-local space, finite dime*
*n-
sional and simply-connected, so its loop space is connected with ss2K < 1. Then*
* again
we know LoK ! K is a homotopy equivalence. Assume we have p-torsion element
ff : Mn (p) ! K with vt1ff = 0. So ff is a vt1-torsion. Then we can assume that*
* ff 6= 0 by
ff O v1 = 0. Consider the map "v1: map M3(p); K 0 ! map M3+q(p); K 0, q = 2p*
* + 2,
n 0 there exist an r n 0 for which
3 3+q
ssr("v1) : ssrmap M (p); K -! ssrmap M (p); K
12
is not an isomorphism. If ssr("v1) is not injective we have a v1-torsion in ss*
*r+3(K; Z=pZ)
as required. These homotopy groups are all finite p-torsion by the nature of M*
*k(p).
Then it follows from the exact sequence of the fibration function complexes cor*
*responding
to M(r-1)+3(p) [ CM(r-1)+3+q(p) ! Mr+3+q (p) ! Mr+3(p) where the range is the
component hit by the domain that, by assumption, is connected. Now since K is (*
*Bo ! *)-
local so are the two mapping spaces (2.2). If ff 2 ss`(K; Z=pZ) is given by an*
* essential
map m3(p) ! K then by assumption, ("v1)# (ff) = 0, so "v1is certainly not a hom*
*otopy
equivalence. It follows that for every n 0 the map "v1 induced on the conn*
*ected
covers is not a homotopy equivalence, since Lo"v1 = "v1and Lo sends an equiv*
*alence
to an equivalence. Now this means that for any integer n there exist a r > n + *
*1 and an
element ff2 ssr-1 map M3(p); K , which is not trivial and is clearly v1-torsion*
*, sitting in
the mapping cone of v1, thus we got v1-torsion in dimension r - 1 or r in any c*
*ase. This
completes the proof of theorem 2.
6. The splitting of X with respect to A
Recall that the localization map X ! PA X of X with respect to A ! * turns in*
* a
universal fashion the space X into a space "without any A-information": map (A;*
* PA X) '
*. Thus in a way PA X is X modulo the A-information of X. One would expect that*
* the
fibre of X ! PA X will contain "all the A-information of X" and no more, in oth*
*er words
the fibre F of X ! PA X should be CW A X. As we claimed in (0.1) this is often*
* true,
but not always:
Example: If X = Mn (p2) and A = Mn (p) then LA X ' * but X 6' CW A X, in fact
ssn-1CW AX ~=ssn-1A.
6.1. Conic subcategories: A subcatory C of (pointed) spaces will be called coni*
*c if it
is a full subcategory of (pointed) spaces closed under arbitrary homotopy colim*
*its.
Thus if the (n + 1)-sphere Sn+1 is in any conic subcategory then this conic c*
*ategory
contains all n-connected spaces. There is an obvious notion of conic closure o*
*f any ful-
l subcategory of S*. For any space A one can form the conic closure of A denot*
*ed by
u
C(A). The CW A -construction gives for any space X a map CW AX!- X which *
*is
u
universal for maps from members of C(A) into X [B-1]. The map CW AX!- X in-
duces homotopy equivalence upon taking function complexes from A, i.e. map (A; *
*u) is a
homotopy equivalence. It is not hard to prove that a map ff : W ! V between m*
*em-
bers of C(A) is a homotopy equivalence if and only if it induces a homotopy equ*
*ivalence
map (A; ff) : map (A; W ) ! map (A; V ). If e.g. [A; W ] ~=[A; V ] = * then thi*
*s is equivalent
to the map ff inducing isomorphism and the A-homotopy groups for all k 1.
~=
[kA; W ] = ssk(W ; A) -! ssk(V ; A) = [kA; V ]:
In other words spaces in C(A) are detected and classified by A-information in*
* a similar
fashion to the way that homotopy type of CW -spaces are detected and classified*
* by the
spheres.
13
6.1.1. Example: If A = Sn+1 then it is not hard to see that CW AX = X, na*
*mely
the n-connected cover of X. In this particular case we have the fibration (0.1*
*) without
any assumption since PA X = PnX the n-th Postnikov section X. Clearly
X ! X ! PnX
is a fibre sequence for all X.
6.1.2. Example: If A = K(; n) for = Z=pZ, n 1, then by Miller's theorem map
(A; D) ' * for any finite dimensional D. Therefore CW A D ' *.
6.2. Proof of 0.1: We start with a special case (0.2) when PA X ' *.
'
6.2.2 Proposition. For any CW -complex A; X, if PA X ' * then CW A X -! X is a
homotopy equivalence.
Proof: Consider the fibre sequence:
(1) X ! F ! CW A X ! X:
In this sequence we now show that F ~ *.
Extra special case: Assume map (A; X) ~ * and PA X ~ *. In that case CW AX !*
* X is
a homotopy equivalence since X ' * because the first equation says that X is A-*
*periodic
so X is preserved by PA .
By universal property of CW AX we have map (A; CW AX) ~ map (A; X) it foll*
*ows
that map (A; F ) ~ *. Thus to show F ~ * by the extra-special case it is enough*
* to show
PA F ~ *.
Claim: In the fibre sequence (1) above if PA X ~ * then PA F ~ *. This follow*
*s from
theorem 3 above: Since by the same theorem it follows that PA CW AX ~ * always*
*, so if
PA X ~ * we get PA F ~ PA CW AX ~ *.
Now by our assumption (the special case) PA X ~ * but we have an equivalence
PA X ~ W PA X so W PA X ~ * so PA X ~ *, so we are done by claim 2.
We can now proceed to prove 0.1. Let F be the fibre of X ! PA X. By theore*
*m 8
above we know that PA F ~ *.~Therefore we are in the special case of propositi*
*on 6.2.2
above and we have CW A F -! F a h.e. Therefore by applying the function CW *
*A to
F ! X we get a composition map
'
F - CW AF ! CW A X:
To show that this composition is a homotopy equivalence it suffices to show
~
map (A; CW AF ) -! map (A; CW AX)
is a homotopy equivalence since as we observed above:
14
~
Lemma. For any map f : X ! Y with map (A; CW AX) -! map (A; CW AY ) h.e.
CW Af is a h.e.
Proof: This is clear by induction on the A-skeleton of, say CW A Y (see [D-Z]).
~
Since we have by definition map (A; CW AX) -! map A(A; X) h.e. we will argue*
* in the
j
fibre sequence F!- X ! PA X to show that j induces an equivalence map (A; j).
Notice that for any two spaces C, D if [C; D] = {*} then map (C; D) = map(C;*
* D).
We are given [A; X] = {*} we claim that [A; F ] = {*}: this is so because we ha*
*ve exact
sequence of homotopy classes:
[A; PA X] ! [A; F ] ! [A; X]
with both ends singletons. Therefore
map (A; X) ~ map(A; X);
map (A; F ) ~ map(A; F ):
But since
F ----! X
?? ?
y ?y
. ----! PA X
j
is a homotopy pullback diagram so is
~
map (F; F ) ----! map (A; X)
?? ?
y ?y
. ----! map (A; PA X) ~ *
a homotopy pullback diagram. Since the bottom right space is contractible the t*
*op arrow
is a homotopy equivalence. Therefore by the two equivalences above
~
map(A; F ) -! map(A; X) h.e.
~
so map (A; F ) -! map (A; X) is a homotopy equivalence as needed.
Corollary. Assume A; X as above and PA X ~ PA X. Then we have a fibration se-
quence:
CW A X ! X ! PA X:
15
Proof: Consider the diagram
CW ?AX
?y & X !-j P X ~ P X
% A A
F
where F is the homotopy fibre of j. Clearly map (CW AX; PA X) ~ * since map (*
*A; PA X) ~
* and CW A X is built by successive cofibres out of kA. Therefore there is a u*
*nique lift of
CW AX ! X into CW AX ! F . To find a homotopy inverse~to this lift we use PA*
* F ~ *
which implies by proposition 6.2.2 above CW A F -! F h.e. We get a composition:
~
F - CW AF!- CW A X:
Now using the universal properties of CW A and PA it is not hard to see that t*
*hese two
maps are homotopy inverses to each other: CW AX ! X is universal map for `A-s*
*paces'
while F ! X is a universal map for space T with LA T ~ *: The triangle
T --! X
? f
t?y
PA X %
with PA X = F the fibre of X ! PA X becomes PA T = T ! PA X so the unique map t*
* is
simply PA f. This completes our proof.
To see that sometimes PA X 6~ CW A X, consider the example following 6.4 bel*
*ow.
6.3. Proof of 0.3: Another example of applying 0.1 is:
0.3 Corollary. Let X be a 7-connected space with K*(7X) ' 0. Then, up to homoto*
*py,
X can be constructed by successively gluing together suspensions of V (1), the *
*mapping
cone of Adam's map between Moore spaces [3].
Proof: By the fibration theorem 0.1 it is sufficient to show PV (1)X ' *.
We first note that for any map f : A ! B if C is the mapping cone of f then i*
*f a
space X is C-periodic then it is f-local. This follows immediately from the fac*
*t that the
mapping spaces of X into a homotopy pushout square form always a homotopy pullb*
*ack
square.
It now follows by the technique of (4.1), where (4.3) is replaced by the abov*
*e remark,
that LC L2f X ' * which implies LC L3f X ' * whenever C is the mapping cone of*
* f.
Now under our assumption 3X is r-deloopable K-acyclic spaces and therefore by*
* [4]
we have LV13X ' * but by Theorem B this implies L3V1 X ' *, and therefore by the
above remark (since L2V1 X = X) we have LV (1)X ~=*. Now apply the (co)-localiz*
*ation
fibration [0.1] to get X = CW V (1)X as needed.
We would like to conclude this section by giving a version of 0.1 that does n*
*ot assume
[A; X]=*. This uses crucially a recent result of Bousfield [5].
16
6.4 Theorem. Let A be a p-torsion (n - 1) connected suspension space: A = B, wi*
*th
non-vanishing Hn(A; Z=pZ).
In the diagram:
FXx
P
`?? &% X -! PA X
j
CWA X
let F denote the homotopy fibre of p. Then there is up to homotopy, a unique li*
*fting ` of
j and the fibre of ` has, at most, two non-vanishing homotopy groups which are *
*p-torsion
in dimensions n, n - 1.
Example: If X = Mn+1 (p2), A = Mn+1 (p) then PA X ' * and there is a fibration
CWA X ! F ! K(Z=pZ; n) with F = X.
Remark: In fact, it is natural to suspect as in the example above that the fibr*
*e of ` in 6.
has at most one non-vanishing homotopy group, but we cannot prove this at this *
*time.
Proof: We first consider the case PA X ' *.
Lemma. If PA X ' * then the fibre of CW AX ! X has at most two non-trivial hom*
*otopy
groups in dimension, n, n + 1 where n = connA.
Proof: By the basic property of CWA X ! X the fibre F satisfies map (A; F ) ' **
* and
thus F is A-periodic. Therefore by theorem C-1 above PA preserves the fibre se*
*quence
F ! CWA X ! X. In other words the sequence F ! PA CWA X ! PA X is a homotopy
fibre sequence: By a theorem of Bousfield [B-5] it follows from the assumption *
*on A that
the homotopy fibre of PA X ! PA X for any X is a K(; n). But in the present c*
*ase
PA CWA X ' PA X ' *.Thus F is homotopy equivalent to the fibre of a map between*
* two
K(; n)'s for different 's.
Now we proceed with the proof of theorem 6.4.
Consider the diagram:
i
Fx !- X !- PA X
% ??f %
j
'
CWA F -! CWA X
In which the unnamed arrows are the obvious ones. The map f is the unique lifti*
*ng of j
where uniqueness follows from map (CWA X; PA X) ' *.
Now since by construction of PA X map (A; PA X) ' * we have map (A; F ) ' map*
* (A; X) '
map (A; CWA X), the equivalences being induced by the above maps. But again by *
*con-
struction map (A; CWA F ) ' map (A; F ). Therefore CWA (i) induces an isomorphi*
*sm on
map (A; -). It follows as usual for A-conic spaces that CWA (i) is a homotopy e*
*quivalence.
Now by theorem 8 above PA F ' * therefore by the lemma above the fibre of CWA*
* F ! F
has at most two homotopy groups. Therefore so does the fibre of f as needed.
17
7. Example. The localization with respect to the Adams map
We would like to compare the unstable localization with respect to the Adams *
*map
vi : 2p+2M3(p) ! M3(p) for an odd prime, with the usual KZ=pZ localization with
respect to mod p K-theory and higher Marava's K-theory. Consider any abelian g*
*roup
M we claim:
7.1 Theorem. There is a homotopy equivalence Lv1K(M; n) ! LKZp K(M; n) for all *
*n,
M.
7.2 A proof of theorem 9: Let V be a topological abelian monoid of the homotopy
type EM. And let Zn be a topological abelian monoid of type K(Z; n). Then we k*
*now
[Adams], that a space is equivalent to a product of Eilenberg-Mac Lane spaces i*
*f and only
if there is a weak module structure of the spectra {Zn} on V . Namely map Zn x *
*V ! V
in satisfying associativity and with a unit. Now because we have a natural equi*
*valence
L(X x Y ) -! L(X) x L(Y )
and a natural map X x L(Y ) ! L(X x Y ) it is not hard to chase the required di*
*agram
and check that there is an action of {Zn} on LV , in the spirit of the proof of*
* theorem 6 in
2.3 above. This completes the proof of theorem 9.
7.3 Proof of 7.1: To continue with the proof of 6.1, we conclude from theorem 9*
* that
Lv1K(M; n) ~ Ki where each Ki is an Eilenberg-Mac Lane space. Now we know that
LvK(M; n) is v1-periodic. But clearly the map map v1; Lv1K(M; n) is trivial b*
*ecause in
a product of ssKi these are no non-trivial compositions. This is equivalent to *
*saying that
for each Ki we have ssn(Ki; Z=pZ) = 0 for all i 1. This means that ssnKi is u*
*niquely
p-divisible for all n, i. Since Lv1 preserves the K-theory of the space, by th*
*eorems B
and 7 its fibre in loops on K-acyclic space, we can easily calculate that the m*
*ap is an
isomorphism on all homotopy groups by [Mislin] and therefore a homotopy equival*
*ence
as needed.
7.4. Proposition. For every n 0 there exist some k 0 such that
LV (n)K(; i) ' *
for all i 0.
Proof: LV (n)K is a product of EM-spaces. But map (V (n); LV (n)K) ' *, so that
LV (n)K is Vn-1 periodic i.e. map (Vn-1; LV (n)K) is h.e., this is possible onl*
*y if the domain
and target are null since composition in GEM must vanish, at least for some pow*
*er of Vn-1.
It follows that map (V (n-1); LV (n)K) ~=* and by induction map (M`(p); LV (n)K*
*(; i)) '
* for ` = conn V (n) + 1, since Mae(p) is the bottom Moore space in V (n). But*
* by the
conn. of V (n) if i `, there is no non-trivial map of Mj(p) ! LV (n)k(; i) for*
* i < `.
Therefore ss*(LV (n)K(; i), Z(pZ) ~=0 in all dimension and since the localizati*
*on LV (n)is
a p-torsion space LV (n)K(; `) = *.
As an immediate corollary we have the following well known:
18
7.5 Theorem. (Ravenel and Wilson) For every Morava K-theory K there exists an
integer k with KK(; k + i) = 0 for all i:
References
1.J.F. Adams, infinite loop spaces, Princeton U press..
2.A. K. Bousfield, Factorization systems in categories, J. of Pure and Appl. *
*Alg. (1975).
3.____________, K-localizations and K-equivalences of infinite loop spaces, P*
*roc. London Math.
Soc. 3 (1982), 291-311.
4.____________, On the v1-localization for highly connected spaces, (prelimin*
*ary manuscript 1991).
5.____________, Private communication, (1991).
6.F. Cohen and J. Neisendorfer, Note on desuspension of the Adams map, Math. *
*Proceeding Comb.
Phil. Soc. 99 (1986), 59-64.
7.E. Dror Farjoun, The localization with respect to a map and v1-periodicity,*
* Proceeding Barcelona
Conference 1990 (to appear).
8.E. Dror Farjoun and A. Zabrodsky, Unipotency and nilpotency in homotopy the*
*ory, Topology 19
(1985).
9.M. Mahowald and R. Thompson, The K-theory localization of an unstable spher*
*e,.
10.G. Mislin, Localization with respect to K-theory, Journal of Pure and Appl*
*. Alg. 10 (1977), 201-213.
11.J. Neisendorfer, Private communication.
12.D.Ravenel, The nilpotency and periodicity theorems in stable homotopy theo*
*ry, Seminar Bourbaki
Ex. No.728 June 1990 p. 728.
13.D. Ravenel and S. Wilson, Morava K-theory of Eilenberg MacLane Spaces and *
*the conner-floyd
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19