Localizations,Fibrations,
and Conic Structures
by
EmmanuelDror 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 understandingof the behavior of fibre and cofibre sequenc*
*es under
localizations. In particular wesee 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 Aas classified by Hopkins-Smith [12*
*].
An interesting application is a decomp osition theorem analogous to both the *
*usual 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 recallfor a moment the notati*
*ons for
(co)localization functors:
We denote by PAX the localization of X with respect to thetrivial map on A. T*
*hus
intuitively PAX is X "modulo the A information of X". In fact PAX is the univer*
*sal
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 o*
*ut of
A via cofibrations": CW A X ! X. The space CW AX is the "co-localization" of*
* 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 (suspensionsof S 0*
*). It is
the universal space among all"A-spaces" Y withmap(A; Y ) ' map(A; X).
0.1. Theorem. For any CW-complexes A; X, with [A; X] = there is a homotopy fib*
*re
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!!A X! Xis a homotopy equivalence.
!
Remarks:!Using recent powerful unpublishedresults of Bousfield that shows how t*
*o 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,
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 j 0 impliesX = C W 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 su*
*ch 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 parti*
*cular 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 top ological 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,C W S0T is clearly the usual CW -approxi*
*mation
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: Iwould like to thank Zig Fiedorowicz and Pete Bousfield for se*
*veral
illuminating discussions pertaining to to thepresent paper. Thanks arealso due*
* 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 resp ect to an arbitrary map f . In the *
*case f is
the trivial map on a space W we usea slightly different notation to suggest its*
* similarity
to the Postnikov construction and its special good properties: We often but not*
* always
denote the homotopy localization byPW and refer to f-local spaces as W- period*
*ic spaces.
The lack of control overfibration under localization and the non-commutativit*
*y of taking
loop space with taking say K-localization is a major obstacle inunderstanding t*
*hem. 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 local*
*ization
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 Lfand this allows us to conclude for example:
Example: The fibre of X !(X)E ,where ( )E is the usual Bousfield localization w*
*ith
any homology theory E , is a loop spaceon 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 canclearly be adapted to our situation here. The result *
*(B) below
about commuting loops and localization can be seen asa destabilisation of [B,1.*
*1]; where
a special stable localization is provento 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 quoteseveral results to illustrate what we have in m*
*ind.
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-peri*
*odic
homotopy v11sss.
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 b eany p-local (or p-
complete) finite dimensional complex with ss1X = 0 and ss2X = torsion. Then the*
* Lf-
localization of the n-th connected cover X hni !X is homotopy equivalence to X,*
* i.e.
LfXhni '! X.
Using similar methods plus some control onthe behavior of fibration under loc*
*alization
we get an analogous result for Vk-torsion:
Theorem 2. Let X be anyp-local finite dimensional complex, 2-connected assume t*
*hat
for some n > 2 there exist a non-trivialv (1)-torsion, i.e. an element 0 6= ffw*
*ith 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 ofseveral 'technical identities' wh*
*ich we
now proceed to formulate. The bulk of the paper is devoted to this series oftec*
*hnical
results: Taken together (3-10) show that there is some hope that our understand*
*ing of the
behaviour of fibration under localization canb eimproved.
Theorem 3. Let L = Lfbe 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 Ljis a homotopy equivalenc*
*e.
If A ! X ! X [CA is a cofibration and L(i) is a homotopy equivalence then L(X*
* [
CA) .
Proof: See section 4.
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 eq*
*uivalence
of loop spaces
` : LfX AE Lf X : r
that is natural up to homotopy.
Corollary. There is an equivalence WkLf kX' Lkf X.
Theorem4. For any map f, Lfcommutes with finite products: The natural map Lf(X
Y) ! Lf(X) Lf(Y) is an homotopy equivalence.
Theorem 5. For any associative An-structure on X there is acorresponding natural
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 spaceW . Then the homotopy
fibre of X !PW X, localizes under PW to a contractible space.
Theorem 9. For anyf : A ! B, if EM is a generalized Eilenberg-Mac Lane space th*
*en
so is Lf(EM).
Theorem C. Given F! E ! B a fibration with all spaces connected and pointed. Th*
*en
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, thenLf F! E ! Lf B i*
*s 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 toknow whther a version of (C) holds*
* for
more general maps. Note the essential difference between 7 and 8: For maps of t*
*he type
f : W ! onecan prove much stronger results about Lf=PW . One would like to
generalize C(3.) to nilpotent fibration: When ss1E acts nilpotently on ssnF= s*
*sn+1(B;E).
Statement (1.) includes allof well known cases of localizing at subrings of the*
* rationals or
other rings. Since there,one localizes with respect to a suspension map between*
*2-spheres
or bouquet thereof. Theorem Breplaces general localization of a loop space by l*
*ocalizing
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 preserv*
*es
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, there*
* 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 AKX etc. is a notation for the fibre of the K-localization and where LK i*
*s the K-
localization and LK is given as follows. Let f= V Kff! V Lffbe the pointedunio*
*n of all
K-isomorphisms between countable CW-complex. According to Bousfield Lf = LK = t*
*he
unstable Bousfield localization with respect to K-theory. Now let LK = Lf as *
*a functor.
Then by theorem 2 the bottom map ` is ahomotopy equivalence and therefore so is*
* the top
map a. Now by theorem8, LK AK X . But f is a K-isomorphism and thereforefor
any space W the map W !LK W is a K-isomorphism. ThereforeAK X is K-acyclic,
i.e. has the K-theory of a point. By theorem 1 of [M-T] we deduce that v11sssAK*
* 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 ap*
*ply
the theorem about spectrum to AX and get our desired vanishing above. Now since*
* a
is a homotopy equivalence by theorem (B) we also have v11ssSAK X = 0. But now
consider v11sss an unconnective generalized homology theory gotten by tensoring*
* ssswith
the Z-graded ring Z=pZ[v;v1 ]. Then the corresponding Serre spectral sequence *
*implies
an isomorphism
v11 sssW X = v11 sssX !v11 sssW LK 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 Y ) ! LX LY givenby the product of the maps induced by projections. We
need to find an inverse to thismap
LX LY !L(X Y ):
By adjunction we need a map.
2.1.1: LX !map LY;L(X Y ) .
2.2 Lemma. Let W be any f-local space,and Z any space whatever,the function com-
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 any*
* mapping
space into it.
It follows that the range of 2.1.1 isan f -local space.Therefore by universal*
* property of
L = Lf it is enoughto construct a map u : X ! map LY; L(X Y) . Now by repeating
this argument of universality it is enough to find a map Y ! map X; L(X Y) or *
*again
a map X Y !L(X Y ). We take this last map is simply the universal coaugmentat*
*ion
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 ! LXas 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 mapsY (i) : Y1! Yigiven by i(0) = i 1, i(1*
*) = i
is a homotopy equivalence
Y (i) : Yn ! Y1 : :Y:1
between the i-th space Ynand the n-th power of Y1. If in such a simplicial spac*
*e ss0Y1 is
a group, then Segal's theorem says that jY j ' Y1 [ Adams]. Now,given X we pres*
*ent
X as a "Segal loop space" by taking a monoidic version of X, say X, theMo ore *
*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! Top *
*be given
by Gn =( X)n; this has the right prop erties so that jG j ' 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,satisf*
*ying the
said conditions.
Notice that in dimension one we have ss0LG =ss0L X which is clearly a group*
*, since
the equivalence L(X X) ' LX LX is natural. Therefore for jLG j ' LG1' L X,
presenting L X as a loop space. In fact, since G !LG is a simplicial map betw*
*een
simplicial spaces it induces jG j ! jLGj a map which gives a loop map
X ' jGj ! jLG j ! L X :
This equivalence are of loop space so they combine to show that X !L X is a m*
*ap 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 givenby the
appropriate universality conditions on localizations. This will imply thatthe *
*maps are
mutual inverses. The easier map LX ! Lf Xexists as an extension over LX of the
obvious loop map X! Lf X by the following lemma towhich we will refer repeatedly
below.
2.5.1 Lemma. Aspace X is f-local with respect to the suspension of f : A! B if *
*and
only if X is f-local.
Proof: This is immediate adjunction
map (f; X) = map(f; X)
between suspension on loops. The leftis a homotopy equivalence iff the right is.
Thus we have a factorization through L = Lf since LfX, Lf X are certainly f -
local. To construct themap the other way we may proceed by using the result abo*
*ve 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 jLG j as a model for W LfX from (2.3) above. Thenthe
map in the other direction will be givenas 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 comp o*
*site
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 rffij2 =j1 or rffi`ffij1 =j1, butby uniqueness of factorization through t*
*he universal
LfX, we get r ffi ` is homotopic to the identity. Similarly one gets ` ffi 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 , whileLf 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 -lo cal.
Proof: Simply if fXffgffis any small diagram on which one takes holim,then we h*
*ave a
canonical homotopy of maps for any map f : A ! B map(f; holimffXff) holimff(f;*
*Xff).
Since the holimof equivalences is a homotopy equivalence, the right handside is*
* 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 ff2 N so that Xffis not acyclic. This is easy to construct on*
*ce 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 infiniteproduct, although it does commute *
*with a
finite one. Now if LZis the Bousfield localization with respect to integral hom*
*ology, 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.
3.3 Lemma. For any diagram (Xff)ffof space there exist a natural map colimffLXf*
*f!
LcolimXff,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(colim*
*Xff)
induced by the canonical maps to thecolimit.
3.4 Proof of theorem 3: Consider the cofibration A ,! X ! X=A. We have a
factorization of map into a colimitof the diagram CA - A ,! X given by X=A !
LX=LA ! L(X=A) with the composition b eing 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 theidempotency 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) ! LXwhich is easily seen to be a homotopy equivalence as requi*
*red.
Now to get to the first assertion,assume LfF = LF . We use a fibrewise local*
*ization
to localize all the fibres to a point,this gives us the space B. But the locali*
*zation of the
fibrewise localization is the localization of Eso we get a map B ! LE,inducing *
*a map
LB !LE which is easily seen by theuniversal properties to be a homotopy equival*
*ence.
The construction of fibrewise localization isa small variation on the constru*
*ction 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 eachp oint in B we*
* get
the usual localization. In fact the map E ! LBE gotten this way, can be identif*
*ied in
a natural way to the total spaceof the fibration induced from u : B ! Baut LF w*
*here
autLF is the monoid of all (unpointed) self equivalences of the space LF . Alte*
*rnatively,
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 istrue even for the simplest kind of
localization, say rationalization, i.e.Bousfield's localization with respect to*
* homology with
rational coefficients. This functor LQ preservesfibration if ss1B = 0 = ss1E = *
*ss1F, other
conditions are not known. Our result will give somewhatmore general conditions.*
* 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 isknown and I do not believe that t*
*here are
theorems which are very specific toK -theory asopp osed 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 loc*
*alization. In
other words theorem Bdescribes 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. Considerthe localization with respect t*
*o the map
Sn+1 ! . This is simply killing higher homotopy groups or taking the n-th Postn*
*ikov
stage. Amoment reflection will show no amount of connectivityassumption will ma*
*ke 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 ! Baut F
?? ?? flfl ??
y oey oe. fl y
LfF ! X ! Lf X ! BautLfF
`
By lemma 2.5.1 (L = Lf) we haveB autLF is f-local since autLF itself is f-local*
* 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 westill have a version of lemma 2.2 for unpointed function*
* spaces.
Notice that for any fibration F! E ! B we have ahomotopy pullback diagram
E ! Baut _F
?? ??
(4.1.1) y y
B ! Baut F
[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,the
map X ! X factors through it. This gives a map oe. But this means that the fibr*
*ation
X !Lf X has a cross-section. Notice that by uniqueness of factorization we sha*
*ll get
` oe =id from ` oe ` = `, butoe ` = b and`b = ` by construction (compare2.5). *
*Thus the
map LfF !X induces a one-to-one map onp ointed homotopy class [W;LfF ] ! [W;X ]
for any space W . In particular forW = F . Notice however thatthe map F ! X fac*
*tors
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 Lfgi*
*ves
that LfF , as claimed.
4.2 Remark: It is essentialin (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: S 2! S2. If X = S2 itself ss1F will be Z[1!p]=Z a p-torsio*
*n group.
But it is clear from the construction ofLf that ss1Lf F = ss1F 6=0. However, i*
*f 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 also
equivalent to .
4.4 Remark: Lemma 4.3 is definitely not true forother maps, even of simple type*
* as
p : Sn ! Sn n 1. !
4.5 Proof of theorem C: Let us start!with!C(2).Consider!the!diagram of fibratio*
*ns
LF !!! LF !
?? ?? ??
y y y
F ! E ! B
?? ?? ??
y y y
LF ! E1 ! B
In which LF are the fibres and E1is the fibrewise localization of E over B. The*
*n bythe
argument above since Band LF are local with respect to W ! so is E1 (see 4.3). *
*Now
by theorem 8 above (4.1) weknow that L(L F) . By theorem 3 applied to the midd*
*le
vertical fibre sequence we know that LE !' LE1 is a homotopy equivalence. But E*
*1 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) belo*
*w we
know that if L = Lf , then since F is f-local, E1is!f -local. Apply theorem (7)*
* above
to get that the fibre L B =Lf B must localize to!a!point underL :
!!!
! L B !! L B
?? ?? ??
y y y
(4.8) F ! E ! B
?? ?? ??
!!! y y y
!!!
L F !! F ! E1 ! L B:
Now as above since E1is 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 Eis f -lo cal 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 fibreof E ! Lf B is Lf W G. By theorem (B)th*
*at
we proved above this isthe same as LfW G = LfG, as needed. This completes thepr*
*o of
of (C).
5. Proof of theorems (1) and (2)
Consider the space B| = K(Z=pZ;1). We give an independent proof of a result *
*of
J. Neisendorfer that gives a strongversion of Serre's result that the finite di*
*mensional
2-connected complexes have homotopy in an infinite number of dimensions. Consid*
*er the
localization functor L|with respect to the map K(Z=pZ;1) ! . H. Miller's result*
*,proving
the Sullivan conjecture,can be formulated by saying that for any finite dimensi*
*onal space
K one has a homotopy equivalence K ! L|K or simplythat K is (B| ! )-local.
5.1 Neisendorfer theorem. Let ss1K = 0 and ss2K <1, and K is a p-local space.
Then L|Khni = K for any n-connectedcover Khni of K.
Proof: We first note that if K(ss;n) is p-local then L|K(ss;n) = . This follows*
* by the
induction from theorem (3) above ifwe show L|K(ss; 2) . But ssis p-local group*
*, since
LW W for any space L|B| . Now since ss is abelian and p-local so is L|K(ss;2)
and we have the map B|; L|K (ss; 2) = , this means by standard computation of t*
*his
function complex that ssiL|K(ss ;2) = 0. Now consider the fibration
PnK ! Khni ! K ! PnK :
Again since PnK is connected with ssfinite group we have L|PnX using inductively
theorem (3) we get L|PnK and therefore
L|Khni '! L|K 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 a*
*gain
we know L|Khni ! 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 thatff*
* 6= 0 by
ff ffi 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
ssr("v1) : ssrmap M3(p);K ! ssrmap M3+q(p);K
is not an isomorphism. If ssr("v1) is not injective we have a v1-torsion in ssr*
*+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(r1)+3(p) [ CM (r1)+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 Kis (B |*
*! )-
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 homo*
*topy
equivalence. It follows that for every n 0 the map "v1hni induced on the conne*
*cted
covers is not a homotopy equivalence, since L|"v1hni = "v1and L| 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 ssr1 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 rin any case.*
* This
completes the proof of theorem 2.
6. The splitting of X with respect to A
Recall that the localization map X ! PAX of X with respect to A ! turns in a
universal fashion the space Xinto a space "without any A-information": map(A; P*
*AX) '
. Thus in a way PAX is X modulo the A-information of X .One would expect that t*
*he
fibre of X !PAX will contain "all the A-information of X" and no more,in other *
*words
the fibre F of X ! PAX should be CW AX. As we claimed in (0.1) this is often t*
*rue,
but not always:
Example: If X = Mn(p2) and A = Mn(p) then LAX ' but X 6' CW A X, in fact
ssn1 CW A X= ssn1 A.
6.1. Conic subcategories: A subcatory C of (pointed) spaces willb e 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 of*
* any ful-
l subcategory of S . For any space A one can form the conic closure of A denote*
*d 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 equivalenceup ontaking function complexes from A, i.e. map(A; u)*
* is a
homotopy equivalence. It is not hard to prove that amap ff : W ! V betweenmem-
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 this is e*
*quivalent
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.
6.1.1. Example: If A = Sn+1 then itis not hard to see that CW AX = Xhn+ 1i, na*
*mely
the n-connected cover of X. In this particular case we have the fibration (0.1)*
* without
any assumption since PAX =PnX the n-th Postnikov section X. Clearly
Xhn + 1i ! 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 AD '.
6.2. Proof of 0.1: We start with asp ecial case (0.2) when PA X' .
'
6.2.2 Proposition. For any CW-complex A;X, if PA X ' then CW AX! X is a
homotopy equivalence.
Proof: Consider the fibre sequence:
(1) X ! F ! CW AX ! X:
In this sequence we now show that F .
Extra special case: Assume map (A;X) and PAX . In that case CW AX ! X is
a homotopy equivalence since X ' because the first equation says that X is A-p*
*eriodic
so X is preserved by PA .
By universal property of CW AX we have map (A;CW A X) map(A; X) it follows
that map(A; F) . Thus to show F by the extra-special case it is enough to show
PAF .
Claim: In the fibre sequence (1) above if PAX then PA F . This follows from
theorem 3 above: Since bythe same theorem it follows that PACW AX always, so*
* if
PA X we get PA F PA CW A X .
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 theorem*
* 8
above we know that PA F . Therefore we are in the special case of proposition *
*6.2.2
above and we have CW AF! F a h.e. Therefore by applying the function CW A *
*to
F !X we get a composition map
'
F CW AF ! CW AX:
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:
Lemma. For any map f : X ! Y with map(A; CW AX)! map (A;C W AY) h.e.
CW A f is a h.e.
Proof: This is clear by induction on the A-skeleton of, say CW AY (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] = fg then map (C; D) = map(C; D*
*).
We are given [A;X] = fg we claim that [A;F] = fg: this is so because we have ex*
*act
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 to*
*p 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 PAX PA X. Then we have a fibration se-
quence:
CW AX ! X ! PAX:
Proof: Consider the diagram
CW ?A X
?y & !j P
% X A X PAX
F
where F is the homotopy fibre of j. Clearly map (CW A X;PAX) since map (A; P*
*AX)
and CW AX is built by successive cofibres out of kA. Therefore there is a uni*
*quelift of
CW A X ! Xinto CW AX ! F .To find a homotopy inverse to this lift we use PA F
which implies by proposition 6.2.2 ab ove CW A F! F h.e.We get a composition:
F CW AF! CW A X:
Now using the universal properties of C W A and PA itis not hard to see that t*
*hese two
maps are homotopy inversesto each other: CW A X! X is universal map for `A-spa*
*ces'
while F !X is a universal map for space T with LAT :The triangle
T ! X
? f
t?y
PA X %
with PAX = F the fibre of X ! PAX becomes PAT = T ! PAX so the unique map t is
simply PAf .This completes our proof.
To see that sometimes PAX 6CW AX, consider the example following 6.4 below.
6.3. Proof of 0.3: Another example of applying 0.1 is:
0.3 Corollary. Let Xbe a 7-connected space with K (7X) ' 0. Then,up to homotop*
*y,
X can be constructed by successively gluing together suspensions of V (1),the m*
*apping
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*
*eremark,
that LCL2f X ' which implies LC L3f X ' whenever C is the mapping cone of f.
Now under our assumption 3Xis r-deloopable K-acyclic spaces and therefore by *
*[4]
we have LV13X ' but by Theorem Bthis implies L3V1 X ' ,and therefore by the
above remark (since L2V1 X= X) we have LV(1)X = . Now apply the (co)-localizati*
*on
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].
6.4 Theorem. Let A be a p-torsion (n 1)connected suspension space: A = B, with
non-vanishingHn(A; Z=pZ).
In the diagram:
FXx
P
`?? &% X ! PAX
j
C WAX
let F denote the homotopy fibre ofp. Then there is up to homotopy,a unique lift*
*ing ` of
j and the fibre of `has, at most, two non-vanishing homotopy groups whichare p-*
*torsion
in dimensions n, n 1.
Example: If X = Mn+1 (p2), A = Mn+1 (p) then PAX ' and there is a fibration
CWAX ! F ! K(Z=pZ;n) with F = X.
Remark: In fact, it isnatural to suspect as in the example above that the fibre*
* of ` in 6.
has at most one non-vanishing homotopy group, but we cannot prove this at this *
*time.
Proof: We first consider the casePA X' .
Lemma. If PAX ' then the fibre of CW AX ! X has at most two non-trivial homoto*
*py
groups in dimension, n, n + 1 where n =conn A.
Proof: By the basic property of CWAX ! X the fibre F satisfies map(A; F) ' and
thus F is A-periodic. Therefore by theorem C-1 above PA preserves the fibre s*
*equence
F ! CWAX ! X. In other words the sequence F ! PA CWAX ! PA Xis 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 ! PAX for any X is a K(;n). But in the present case
PA CWA X ' PA X '.Thus F is homotopy equivalent to the fibre of a map between t*
*wo
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 arethe obvious ones. The map f is the unique liftin*
*g of j
where uniqueness follows from map(CWA X; PAX ) ' .
Now since by construction of PA Xmap (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 co*
*n-
struction map (A; CWAF) ' map (A; F). Therefore CWA (i) induces an isomorphism *
*on
map (A;). It follows as usual for A-conicspaces that CWA (i) is a homotopyequiv*
*alence.
Now by theorem 8 above PAF ' therefore by the lemma above the fibre of CWAF !F
has at most two homotopy groups. Therefore so does the fibre of f as needed.
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+2M 3(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 gr*
*oup
M weclaim:
7.1 Theorem. There is a homotopyequivalence 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 thehomotopy
type EM. And let Zn be a topological abelian monoidof type K (Z; n). Then we kn*
*ow
[Adams], that a space is equivalent to a product of Eilenberg-Mac Lane spaces i*
*fand only
if there is a weak module structure of the spectra fZng on V . Namely map Zn V *
*!V
in satisfying associativity and with a unit. Now because we have a natural equi*
*valence
L(X Y ) ! L(X) L(Y )
and a natural map X L(Y ) ! L(X Y) it is not hard to chase the required diagram
and check that there is an action of fZng on LV ,in the spirit of the proof of *
*theorem 6 in
2.3 above. This completes theproof 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 bec*
*ausein
a product of ssKithese are no non-trivial compositions. This is equivalent to s*
*aying that
for each Kiwe have ssn(Ki;Z=pZ) = 0 for all i 1. This means that ssnKiis uniqu*
*ely
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 existsome 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)Kis Vn1 periodic i.e. map(Vn1 ;LV(n)K) is h.e.,this is possible only if *
*the domain
and target are null since composition inGE Mmust vanish,at least for some power*
* of Vn1 .
It follows that map(V (n1);LV (n)K) = and by inductionmap (M`(p); LV(n)K (; i)*
*) '
for ` = connV (n) + 1,since M ae(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 localization*
* LV(n)is
a p-torsion space LV(n)K (;`) = .
As an immediate corollary we have the following well known:
7.5 Theorem. (Ravenel and Wilson) For every Morava K-theory Khni there exists an
integer k with KhniK(;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.A*
*lg. (1975).
3.!!! ,K -localizations and K-equivalences of infinite loop spaces, P*
*roc. London Math.
Soc.!3!(1982),!291-311.
4.!!! , On thev1-localization for highly connected spaces, (prelimina*
*ry manuscript 1991).
5.! , Private communication, (1991).
6.F. Cohen and J. Neisendorfer,Note on desuspension of the Adams map,Math. Pr*
*oceeding Comb.
Phil. Soc. 99 (1986), 59-64.
7.E. Dror Farjoun,The localization with respect to a mapand v1-periodicity, P*
*roceedingBarcelona
Conference 1990 (to appear).
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(1985).
9.M. Mahowald and R. Thompson, The K-theory localization of an unstable spher*
*e,.
10.G. Mislin, Localization with respect toK -theory, Journal of Pure and Appl.*
* Alg. 10 (1977), 201-213.
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12.D.Ravenel, Thenilpotency and periodicity theorems in stable homotopy theory*
*, Seminar Bourbaki
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