PHANTOM MAPS, SNT-THEORY, AND NATURAL
FILTRATIONS ON lim1 SETS
PIERRE GHIENNE
Abstract. We study the so-called Gray filtration on the set of phantom m*
*aps
between two spaces. Using both its algebraic characterization and the Su*
*llivan
completion approach to phantom maps, we generalize some of the recent re*
*sults
of Le, McGibbon and Strom. We particularly emphasize on the set of phant*
*om
maps with infinite Gray index, describing it in an original algebraic wa*
*y.
We furthermore introduce and study a natural filtration on SNT-sets (*
*that
is sets of homotopy types of spaces having the same n-type for all n), w*
*hich
appears to have the same algebraic characterization of the Gray one on p*
*han-
tom maps. For spaces whose rational homotopy type is that of an H-space
or a co-H-space, we establish criteria permitting to determinate those s*
*ubsets
of this filtration which are non trivial, generalizing work of McGibbon *
*and
Mfiller.
We finally describe algebraically the natural connection between phan*
*tom
maps and SNT-theory, associating to a phantom map its homotopy fiber or
cofiber. We use this description to show that this connection respect fi*
*ltra-
tions, and to find generic examples of spaces for which the filtration o*
*n the
corresponding SNT-set consists of infinitely many strict inclusions.
1.Introduction
In this paper we propose to study natural filtrations on the set of phantom *
*maps
between two spaces, as well as on sets of homotopy types of spaces having the s*
*ame
n-type for all n.
Unless specifically precised, all spaces have the homotopy type of CW-comple*
*xes,
are pointed, and maps between them are pointed maps.
Recall that a map f : X ! Y is said to be a F-phantom map, respectively a
FD-phantom map, if, for any CW-complex K which is finite, respectively finite
dimensional, and for any map g : K ! X, the composition f O g : K ! Y is
inessential.
Any FD-phantom map is in particular a F-phantom map, but the converse is
false in general (see the survey papers of McGibbon [12] or Roitberg [20] for e*
*xam-
ples). We write PhF (X, Y ), respectively PhFD (X, Y ), for the subset of [X, Y*
* ] con-
sisting of homotopy classes of F-phantom maps, respectively FD-phantom maps.
If X has finite type, which means that X admits finite skeleta, then the two no*
*tions
of phantom maps do coincide, and we can write simply Ph(X, Y ).
The filtration on FD-phantom maps we work with, namely the Gray filtration,
goes back to [4], and has been recently studied in [10] and [18]. In what foll*
*ows
we generalize the Gray filtration, and some of the results of [10], [18], to th*
*e case
of F-phantom maps. We also emphasize on and describe algebraically the set of
F-phantom maps with infinite Gray index.
____________
Date: March 11, 2002.
1991 Mathematics Subject Classification. 55Q05, 55S37, 55P15.
Key words and phrases. phantom maps, Gray index, lim1, SNT-theory.
1
2 PIERRE GHIENNE
Recall now that for a given space X, we denote SNT (X) the set of homotopy
types of spaces Y having the same n-type of X for all n. That is to say that X(*
*n)
and Y (n), the Postnikov sections of X and Y through dimension n, have the same
homotopy type for all n.
We introduce and study a natural filtration on this set, which appears to ha*
*ve
the same algebraic characterization of the Gray one on FD-phantom maps. Work-
ing algebraically, we show that the natural connection between FD-phantom maps
and SNT-theory (to a phantom map we associate its homotopy fiber or its homo-
topy cofiber) respect these filtrations. We use this result to find generic exa*
*mples
of spaces X for which the filtration on SNT (X) consists of infinitely many str*
*ict
inclusions.
We state our main results in Sections 2 and 3. The rest of the paper is devo*
*ted
to proofs and complementary results.
2. The Gray filtration on phantom maps
The definition of the Gray index of a phantom map we give below deserves some
comments. Originally [4], the Gray index is only define for FD-phantom maps,
by means of a particular cell-decomposition of the domain X. Our Definition 2.1
coincide with the usual one for FD-phantom maps [10], and has the advantage to
generalize naturally to the case of F-phantom maps.
Definition 2.1.Let f : X ! Y be a F-phantom map, respectively a FD-phantom
map. We say that the Gray index of f is greater than k, and write G(f) k, if
there exists a lift ~fof f through Y , the k-connected cover of Y ,
Y==_
___
~f____|__
______ |
____ fflffl|
X ___f__//_Y
such that ~fitself is a F-phantom map, respectively a FD-phantom map.
Remark however the following fact: the existence of a lift ~fis guaranted if*
* f is
FD-phantom, whereas it is not if f is only supposed to be F-phantom.
As an easy example, note that the Gray index of the trivial map is infinite:
G(*) = 1.
The Gray filtration on phantom maps is defined by the following rule: a phan*
*tom
map f in filtration k has Gray index G(f) k. More precisely, we define
PhkFD(X, Y ) := {f 2 PhFD (X, Y ) | G(f) k}
Ph kF(X, Y ) := {f 2 PhF (X, Y ) | G(f) k}
and we have two filtrations:
PhFD (X, Y ) . . . PhkFD(X, Y ) . . . Ph1FD(X, Y ) := Tk PhkFD(X, Y )
T | T | T |
PhF(X, Y ) . . . Ph kF(X, Y ) . . . Ph1F(X, Y ) := Tk PhkF(X, Y )
Notice that we have also an inclusion:
Ph 1F(X, Y ) PhFD (X, Y )
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 3
Indeed, a map f 2 Ph1F(X, Y ) admits in particular, for any k, a lift through Y*
* ,
and the composition X !f Y ! Y (k)is then inessential. The inclusion follows by
classical theory.
The following example, due to McGibbon-Strom, shows that the Gray filtration
can be highly non trivial:
Example 2.2. [18] Let X := CP (1) and Y := S2 _ S2. Then Ph (X, Y ) =
Ph 2(X, Y ), and for any k 1, we have Ph2k-1(X, Y ) = Ph2k(X, Y ) whereas the
inclusion Ph2k(X, Y ) Ph2k+1(X, Y ) is strict. Moreover, Ph1 (X, Y ) = {*}.
We list in the following theorem the results of [10] which are relevant to o*
*ur
present work.
Theorem 2.3. [10] Let X and Y be nilpotent spaces of finite type.
1) If Ph(X, Y ) = Ph1 (X, Y ), then in fact Ph(X, Y ) = {*}.
2) If Hk(X; ßk+1(Y ) Q) ~=0, then Phk(X, Y ) = Phk+1(X, Y ).
3) If Hm (X; ßm+1 (Y ) Q) ~=0 for any m k, then Phk(X, Y ) = {*}.
The first result we wish to present is the following generalization of Theor*
*em 2.3.
Theorem 2.4.
1) Let X and Y be nilpotent spaces of finite type. If Phk(X, Y ) = Ph1 (X, Y ) *
*for
some k, then in fact Phk(X, Y ) = {*}.
2) Let X be any 0-connected space, and Y nilpotent of finite type.
If Hk(X; ßk+1(Y ) Q) ~=0, then PhkF(X, Y ) = Phk+1F(X, Y ).
3) Let X be any 0-connected space, and Y nilpotent of finite type.
If Hm (X; ßm+1 (Y ) Q) ~=0 for any m k, then PhkF(X, Y ) = {*}.
Remark 2.5. Point 1) of Theorem 2.4 is more generally true for FD-phantom
maps between any spaces X and Y such that the groups [X, Y (n)] are countable
for n large enough.
Points 2) and 3) of Theorem 2.4 appears as natural generalization of the same
Points in Theorem 2.3. Point 1) of Theorem 2.4 generalizes both Points 1) and
3) of Theorem 2.3, saying that if the filtration stabilizes at any stage and fo*
*r any
reason, then it stabilizes to the trivial set.
We emphasize now on the sets Ph1FD(X, Y ) and Ph1F(X, Y ) of phantom maps
with infinite Gray index. In his thesis [4], Gray claims that the only FD-phant*
*om
map with infinite Gray index, between any spaces, is the trivial map. Counter-
examples have been recently found by McGibbon-Strom [18]: to any space X, they
associate canonically a FD-phantom map X : X ! W (X) with infinite Gray
index, and show this map is essential under some additional hypothesis on X.
However, for such X, even though these X are finite type, the target W (X) is
definitively not. We quote the following conjecture arising from [10] and [18].
Conjecture. If X and Y are nilpotent of finite type, then Ph1 (X, Y ) = {*}.
Special cases of this conjecture appear already in the previous two theorems*
*, as
well as in the following result. Here bYdenotes the profinite completion of Y .
Theorem 2.6. [18] Let X and Y be nilpotent of finite type, and suppose that t*
*he
function space Map*(X, bY) is weakly contractible. Then Ph1 (X, Y ) = {*}.
In the context of F-phantom maps, we provide the following generalization of
Theorem 2.6.
4 PIERRE GHIENNE
Theorem 2.7. Let X be any 0-connected space, and Y nilpotent of finite type.
1) If [X, bY] is finite, then Ph1FD(X, Y ) = {*} = Ph1F(X, Y ).
2) If [X, bY] is countable, then the equality PhkF(X, Y ) = Phk+1F(X, Y ) hold*
*s if and
only if Hk(X; ßk+1(Y ) Q) ~=0.
For the sake of completeness, we list in Proposition 5.5 well known cases wh*
*ere
Theorem 2.7 do apply.
Point 2) of Theorem 2.7 may be its most striking part: the determination of *
*the
Gray filtration then amounts to an easy rational computation. It gives for exam*
*ple
an easy proof of Example 2.2.
Most of our results so far will be proved using the following characterizati*
*on of
F-phantom maps, following directly from Sullivan [22, Theorem 3.2]. Let be: Y !*
* bY
be the profinite completion of the space Y . If Y is nilpotent of finite type, *
*then,
for any space X,
PhF(X, Y ) = Ker be*: [X, Y ] ! [X, bY]
We have to interpret the Gray index in that context. For any n 1, we define
a space P nY as a pullback, together with a map pn : Y ! P nY , as shown in the
following diagram:
Y_____________________________________________________*
*________________________________________________________________@
_____________________be______________________________*
*________________________________________________________________@
____________________________________________________*
*_____________________________________________________pn__
____________________________________________________*
*________________________________________________
______________________%%___________________________*
*__________________________!!____
_______________________P/nY/_b
________________________yY
________________________||
____________________||
_ÆÆ____________________fflffl|fflffl|be(n)
Y (n)____//bY (n)
Notice that there might be several homotopy classes of maps pn : Y ! P nY
such that the above diagram commutes. Indeed, by classical theory, there is an
action of [Y, bY (n)] on the set [Y, P nY ], and if any map pn is chosen, all *
*maps in
the orbit of pn are also convenient. Working carefully, we can avoid this ambig*
*uity
(Corollary 5.10), and then construct an infinite tower
tY JJ
pn+1tttt |n JJpn-1JJ
tt p| JJJ
zzttt fflffl| J$$
. ._.___//P n+1Y____//P nY____//P n-1Y____//_. . .
approximating Y , that is to say that the maps pn induce an equivalence from Y *
*to
the homotopy inverse limit of the tower.
Theorem 2.8. Let Y be nilpotent of finite type, X any space, and fn2 PhF (X, *
*Y ).
1) The Gray index G(f) n if and only if the composition X -f! Y -p! P nY is
inessential.
2) We have a bijection Ph1F(X, Y ) ~=lim1[nX, P nY ].
We refer to [1, p. 251] for the definition of the lim1 of a tower of arbitr*
*ary
groups. Recall from [1, p. 254] that we have a bijection:
PhFD (X, Y ) ~=lim1[nX, Y (n)]
Thanks to Point 2) of Theorem 2.8, we can give an algebraic description of the *
*inclu-
sion Ph1F(X, Y ) PhFD (X, Y ): it is the lim1onf the homomorphisms [X, P nY *
*] !
[X, Y (n)] defined naturally by the maps P nY ! Y (n).
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 5
For other lim1expressions of the set Ph1F(X, Y ), see Corollary 5.11.
The spaces P nY permit the following characterization of nilpotent spaces of
finite type which can't be the target of a non-trivial F-phantom map.
___________
Theorem 2.9. Let Y be nilpotent of finite type, and n 1 P nY be a CW-appro-
ximation of the (possibly non CW-type) product n 1 P nY_._Then_Ph1F(X,_Y ) =
{*} for every space X if and only if Y is a retract of n 1 P nY.
It seems nevertheless difficult to see if a given space verify or not the hy*
*pothesis
of Theorem 2.9. Here follows a first but incomplete result in this direction.
Proposition 2.10. Let Y be nilpotent of finite type. Suppose there exists a weak
homotopy equivalence from Y to some product ffLff, in which_each_Lffis_a fini*
*te
type, rationally elliptic space. Then Y is a retract of n 1 P nY.
W
Example 2.11. If Y = i2I Sni is any finite wedge of simply connected spheres,
then for every space X we have Ph1F(X, Y ) = {*}. This follows from the above
results and the Milnor-Moore Theorem.
We shall at present go back to the algebraic characterization of FD-phantom
maps. Frow now on and until we state Theorem 2.12 below, we follow Le-Strom [10*
*].
Consider {Gn}n any tower of groups:
. .!.Gn ! Gn-1 ! . .!.G1 ! *
and define Gnk:= Im(Gn ! Gk) if n k, and Gnk:= Gn if n < k. We obtain a
natural diagram of surjections between towers (left of the following picture), *
*which
induces a diagram of surjections between their lim1(nright of the following pic*
*ture):
{Gnk+1}n lim1Gnnk+1
m6666m pk+16666l
mmmm | lllll
| 1 |p
{Gn}n QQ | =) lim GnnRR ||k,k+1
QQQ((((fflfflfflffl||QQ pkRR((((fflfflfflffl|RRR
{Gnk}n lim1Gnnk
Set L := lim1Gnn, and define Lk := Kerpk. We then have a filtration:
"
( ) L = L0 L1 . . .Lk . . .L1 := Lk
k
We also define, for m k, an equivalence relation ~m on Lk. Two elements
x, y 2 Lk are said to be m-equivalent, which we denote by x ~m y, if and only if
they have the same image by pm : Lk ,! L ! lim1Gnnm.
Theorem 2.12. [10] Let Gn := [X, Y (n)]. The filtration ( ) defined just abo*
*ve
on Ph FD(X, Y ) ~= L := lim1Gnnis precisely the usual Gray filtration on FD-
phantom maps: for any k, we have Lk ~=PhkFD(X, Y ).
This result justify the terminology in the following definition.
Definition 2.13. The algebraic Gray filtration on a lim1set L := lim1Gnnis the
filtration ( ).
Remark 2.14. We draw attention to the following fact: it's possible to find t*
*owers
{Gn}n and {G0n}n admitting the same lim1, but with radically different algebraic
Gray filtration on it (see Proposition 5.13 or Example 6.2).
6 PIERRE GHIENNE
Notice that if X and Y are nilpotent spaces of finite type, then the groups
[X, Y (n)] are nilpotent finitely generated [17]. Then Point 1) of Theorem 2.4
follows from Theorem 2.12 and the following result.
Theorem 2.15. Let {Gn}n be a tower of countable groups, and consider the al-
gebraic Gray filtration on L := lim1Gnn. If L1 = Lk for some k, then in fact
Lk ~={*}.
This result stated in all its generality will also have applications in SNT-*
*theory.
3.A filtration on SNT (X)
In this section we suppose spaces to be 0-connected. We denote by Aut(X) the
group of homotopy classes of self-homotopy equivalences of a space X.
Recall from [24] that we have a bijection:
SNT (X) ~=lim1Anut(X(n))
This fundamental result allowed a deep study of the subject by McGibbon-
Møller [14, 15, 16], with in particular the calculation of SNT (X) for many spa*
*ces
X. Nevertheless, little is known on the general structure of these sets, beside*
*s the
following [14]: if X is a nilpotent space with finite type over some subring of*
* the
rationals, then either SNT (X) = {X}, or else it is uncountably large.
In this section we study the algebraic Gray filtration (Definition 2.13) inh*
*erited
by SNT (X) from its lim1description:
"
SNT (X) SNT 1(X) . . .SNT k(X) . . .SNT 1(X) := SNT k(X)
k
One may ask if this filtration could have been defined without any aid of the
algebra, and indeed it could, as shown by the following result.
Theorem 3.1. Let X be a 0-connected space, and k 0. Then SNT k(X) is the
set of homotopy types of spaces Y , such that there exists a collection of homo*
*topy
equivalences fn : X(n)! Y (n), n 1, which are compatible through range k. More
precisely, for any couple (n, k), the Postnikov sections f(min(n,k))kand f(min(*
*n,k))n
must be homotopy equivalent.
Let us explain what is behind the proof of Theorem 3.1. Let F be an homotopy
functor from spaces to spaces. We the define a subset SNT F(X) of SNT (X) by the
following rule: the homotopy type of a space Y belongs to SNT F(X) if and only *
*if
there exists a collection of homotopy equivalences fn : X(n)! Y (n), n 1, whi*
*ch
are compatible after applying the functor F . More precisely, for any n 1, we
must have a commutative diagram:
F(fn+1)
F (X(n+1))______//F (Y (n+1))
| |
| |
fflffl|F(fn) fflffl|
F (X(n))________//F (Y (n))
where vertical maps are obtained by applying F to the canonical X(n+1)! X(n)
and Y (n+1)! Y (n).
Let AutF (X) be the kernel of the group homomorphism Aut(X) ! Aut(F X),
and suppose now that F commutes with Postnikov sections. Then the inclusions
Aut F(X(n)) ,! Aut(X(n)) fit together to give a map:
jF : lim1AnutF(X(n)) ! lim1Anut(X(n)) ~=SNT (X).
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 7
Theorem 3.2. If F commutes with Postnikov sections, then SNT F(X) is exactly
the image of the map jF .
We owe the idea of defining SNT F(X) and the proof of Theorem 3.2 to a simil*
*ar
construction of F'elix-Thomas [3]. The proof of Theorem 3.1 amounts to identify
SNT k(X) with SNT F(X) when the functor F is the Postnikov section -(k).
In complete analogy with the Gray filtration on phantom maps, our filtration
on SNT (X) is endowed with the following stability properties. For a given space
X, we denote Autk(X) the subgroup of Aut(X) of maps inducing the identity on
X(k),
Autk(X) := Ker Aut(X) ! Aut(X(k)) .
Theorem 3.3. Let X be a nilpotent space with finite type over some subring of*
* the
rationals.
1) If SNT k(X) = SNT 1(X) for some k, then in fact SNT k(X) = {X}.
2) If Autk(X(k+1)) is a finite group, then SNT k(X) = SNT k+1(X). This holds in
particular if Hk+1(X; ßk+1(X) Q) ~=0.
3) If Hm+1 (X; ßm+1 (X) Q) ~=0 for any m k, then SNT k(X) = {X}.
The direct computation of SNT (X) for a given space X stays in general prob-
lematic, and it is certainly the same for its subsets SNT k(X). We however offe*
*r the
following criteria for H0-spaces (whose rational homotopy type is an H-space) a*
*nd
co-H0-spaces (whose rational homotopy type is a bouquet of spheres).
We clarify some notation. Let ZP be the integers localized at some set of pr*
*imes
P, and let X be a P-local space. We denote by Autk(H n (X; ZP)) the subgroup
of those ring automorphisms of the graded ring H n (X; ZP), that preserve the
degrees of homogeneous elements, and induce identity on H k (X; ZP). Dually,
Aut k(ß n X) denotes the group of those automorphisms of the graded ZP-module
ß n X, that preserve the Whitehead product pairing, and induce identity on ß k *
*X.
Theorem 3.4.
1) Let X be a 1-connected H0-space with finite type over ZP. Then the following
statements are equivalent:
(i) SNT k(X) = {X};
(ii) For all n k, the image of Autk(X) ! Autk(X(n)) has finite index.
(iii) For all n k, the image of Autk(X) ! Autk(H n (X; ZP)) has finite index.
2) Let X be a 1-connected co-H0-space with finite type over ZP. Then the fol-
lowing statements are equivalent:
(i) SNT k(X) = {X};
(ii) For all n k, the image of Autk(X) ! Autk(X(n)) has finite index.
(iii) For all n k, the image of Autk(X) ! Autk(ß n X) has finite index.
Theorem 3.4 generalizes [14, Theorem 1] and [15, Theorem 1] which deal with
the case k = 0. Our methods of proof are similar.
Example 3.5. The set SNT 2m(BSU(m)) is trivial, whereas SNT 2m-1(BSU(m))
is not. Similarly, SNT 4m(BSp(m)) is trivial, whereas SNT 4m-1(BSp(m)) is not.
We shall now use phantom map theory to find examples of spaces Z for which
the filtration on SNT (Z) consists of many strict inclusions. The connection be*
*tween
phantom maps and SNT-theory is not new (it goes back to [5]), but is particular*
*ly
illuminating in the context of our work. It is given by the maps
SNT (Y _ X) Cof-PhFD (X, Y ) Fib-!SNT(X x Y )
associating to a FD-phantom map its homotopy cofiber or its homotopy fiber.
8 PIERRE GHIENNE
Theorem 3.6. The maps Fiband Cof respect filtrations. More precisely, we have:
1) The image of PhkFD(X, Y ) by Fib is included in SNT k-1(X x Y ).
2) The image of PhkFD(X, Y ) by Cof is included in SNT k(Y _ X).
The proof of Theorem 3.6 is based on an algebraic description of the maps Fib
and Cof (Theorem 7.1).
Theorem 3.6 is the key point to prove the following examples. The Example 3.7
is a particular case of a more general result (Proposition 7.2). The Example 3*
*.8
shows that, at least if we drop finiteness conditions on a space Z, then SNT 1 *
*(Z)
need not be trivial.
Example 3.7. (Example 2.2 continued) There are infinitely many strict inclusio*
*ns
in the filtration on SNT CP (1) x (S2 _ S2) .
Example 3.8. Let X : X ! W (X) be the canonical FD-phantom map with
infinite Gray index out of X [18]. Suppose X is finite type, and that its cohom*
*ology
is not locally finite as a module over the Steenrod Algebra for some prime p. T*
*hen
the cofiber W (X)=X of X belongs to SNT 1 (W (X) _ X), and is not homotopy
equivalent to W (X) _ X.
4. The algebraic approach to FD-phantom maps
In this section we focus on the algebraic Gray filtration we defined on any *
*lim1
set (Definition 2.13). We prove Theorem 2.15 as well as some complementary re-
sults on FD-phantom maps (Proposition 4.5).
Consider {Gn}n any tower of groups:
. .!.Gn ! Gn-1 ! . .!.G1 ! *
and define Gnk:= Im(Gn ! Gk) if n k, and Gnk:= Gn if n < k. We then obtain
naturally, for each k, another tower {Gnk}n:
. .,.! Gnk,! Gn-1k,! . .,.! Gk+1k,! Gk ! Gk-1 ! . .!.G1 ! *,
in which each map to the left of Gk is an inclusion.
By definition, the tower {Gn}n is said to be Mittag-Leffler if and only if, *
*for
each k, the tower {Gnk}n eventually stabilizes. Now, as almost all maps in the *
*tower
{Gnk}n are inclusions, it is clear that {Gnk}n stabilizes if and only if {Gnk}n*
* is itself
Mittag-Leffler. This observation permits to reformulate [14, Theorem 2] as foll*
*ows.
Lemma 4.1. Let {Gn}n be a tower of countable groups. Then lim1Gnn~=* if and
only if lim1Gnnk~=* for any k. Otherwise, lim1Gnnis an uncountable set.
Remark now that for any n, and m k, the map Gm ! Gk induces a surjection
Gnmi Gnk. All these surjections, when k and m are fixed and n varies, fit toget*
*her
to give a map of tower {Gnm}n ! {Gnk}n, inducing a surjection lim1Gnnmi lim1Gnn*
*k.
This gives a right meaning to the following definition.
Definition 4.2.
1) The index Ind{Gn}n of the tower {Gn}n is the greatest k such that lim1Gnnk~=*
**.
If no such k exists, we say that the index is infinite.
2) Fix k 1, and set Knk:= Ker(Gn i Gnk). The k-th index Indk{Gn}n of the
tower {Gn}n is the index of the tower {Knk}n.
Remark 4.3. If the groups Gn are countable, Lemma 4.1 implies that lim1Gnn~=*
if and only if Ind{Gn}n = 1.
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 9
We shall now state and prove the key result of that section, which in partic*
*ular
describe the algebraic Gray filtration with respect to the indices defined abov*
*e.
Theorem 4.4. Let {Gn}n be a tower of countable groups, and consider Lk the
subsets of L := lim1Gnndefining the algebraic Gray filtration.
1) For any k, we have Lk = * if and only if lim1Knnk= *. Otherwise, Lk is
uncountable.
2) For any m k, the set of m-equivalence classes in Lk, namely Lk=~m , is eih*
*er
*, or else uncountable. Moreover, if Ker(Gm ! Gk) is finite, then Lk=~m = *.
3) For any m k, the equality Lk = Lm holds if and only if m Indk{Gn}n.
Proof. For any k, and n m, the map Gn ! Gm induces another map Knk! Kmk.
Set Knk,m:= Im(Knk! Kmk) if n m, and set Knk,m:= Knkif n < m. By definition,
Indk{Gn}n is the greatest m such that lim1Knnk,m~=*.
Fix integers m k. We claim that there is an action of the group limGnnkon
lim1Knnk, as well as on lim1Knnk,m, such that:
Lk ~=lim1Knnk= limGnnk and Lk=~m ~= lim1Knnk,m= limGnnk.
Let's assume that claim. From Lemma 4.1 we know that lim1Knnkand lim1Knnk,m
are either *, or else uncountable. We then see the same must be true then for Lk
and Lk=~m . Indeed, if not *, these sets are quotient of an uncountable set by *
*the
countable group limGnnk Gk. For the same reason, Lk=~m = * if and only if
lim1Knnk,m= *, that is to say if and only if m Indk{Gn}n. Clearly, Lk=~m = * *
*if
and only if Lk = Lm . Moreover, for n m, we have Knk,m Kmk= Ker(Gm ! Gk),
and if this kernel is finite, we have clearly lim1Knnk,m= *. The theorem follow*
*s.
We now show our claim. Consider the following first diagram:
Knk,m"
___ `
___
____
" fflffl____
KnmØ_____//_"G`n__////_Gnm
| || |
| || |
fflffl|" || fflfflfflffl|
KnkØ_____//Gn_____////_Gnk
|ffn|
fflfflfflffl|
Knk,m
where the rows are exact by definition, and ffn is the obvious map. The left co*
*lumn
is easily seen to be exact, and the snake lemma induces the dotted map such that
the right column is also exact.
10 PIERRE GHIENNE
Consider now the following second commutative diagram:
lll5Lk___~RvRRR555l
lllllll _______RRRRRRR
limGnk________//lim1Knkl_____________________RR((R//_Lpk___////_lim1Gnk
n n ____ | n
|| | ____ | ||
|| | fflffl____ | ||
|| 1 | 1 n n |pm ||
|| limffnn| limnKk,m= limGknu | ||
|| | 6666m ~PP | ||
|| fflffl|mmmmm PPPP fflffl| ||
|| fflffl|mm P''P fflffl|p ||
limGnnk______//lim1Knk,m____________________//_lim1Gnm__k,m////_lim1Gnk
n n n
where the dotted arrow _ is to be constructed.
The upper row is the six-terms lim-lim1exact sequence associated to the bott*
*om
row of the first diagram: it identifies Lk := Kerpk as the quotient of lim1Knnk*
*by
some action of the group limGnnk.
The bottom row is the six-terms lim- lim1exact sequence associated to the
right column of the first diagram: it identifies Kerpk,m as the quotient of lim*
*1Knnk,m
by some action of the group limGnnk.
Commutativity of the front face in the second diagram follows by naturality
from the first diagram. Moreover, the map lim1fnfn commutes with the action of
lim Gnnk. It then induces the dotted map _ between the orbit sets. The map _ is
easily seen to be a surjection. Also, for any x, y 2 Lk, we have _(x) = _(y) if*
* and
only if pm (x) = pm (y). This holds precisely, by definition, if and only if x *
*~m y.
We conclude that _ induces a bijection from Lk=~m onto its image, and the_claim
is proved. |__|
Proof of Theorem 2.15.Suppose Lk = L1 for some k. From Point 3) of Theo-
rem 4.4, we then deduce that Indk{Gn}n = 1. By Definition 4.2 and Remark 4.3,
this readily implies that lim1Knnk= *. The result follows then by Point 1) of T*
*heo-
rem 4.4. |___|
The rest of that section is devoted to the interpretation of Definition 4.2 *
*in the
context of FD-phantom maps (Proposition 4.7 below).
One application is the following. As Ph kFD(X, Y ) is the image of the map
Ph FD (X, Y ) ! PhFD (X, Y ), it is easy to see that if all maps in PhFD (X,*
* Y )
have Gray index at least m k, then PhkFD(X, Y ) = PhmFD(X, Y ). The following
result shows that, at least in the finite type case, the converse is true!
Proposition 4.5. Let X and Y be nilpotent spaces of finite type, and m k. Then
the following statements are equivalent:
(i) Phk(X, Y ) = Phm (X, Y );
(ii) For any i k, we have Phk(X, Y *) = Phm (X, Y **);
(iii) Phm (X, Y ) = Ph(X, Y ).
Therefore, if X and Y are finite type, the Gray filtration on Ph(X, Y ) comp*
*letely
determines the Gray filtration on Ph(X, Y ), for any k.
Lemma 4.6. Let 1 ! {An}n ! {Bn}n ! {Cn}n ! 1 be an exact sequence of
towers of groups. Suppose each map An+1 ! An in the first tower is a surjection.
Then Ind{Bn}n = Ind{Cn}n.
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 11
Proof. For each k we have an exact sequence of towers 1 ! {Ank}n ! {Bnk}n !
{Cnk}n ! 1. Indeed, if n < k, this is the original exact sequence 1 ! An ! Bn !
Cn ! 1. If n k, we have Ank= Ak, and the sequence 1 ! Ak ! Bnk! Cnk! 1
is easily seen to be exact.
We apply the lim1.nAs almost all maps in the tower {Ank}n are identity, we h*
*ave
lim1Annk= *. We deduce that the map lim1Bnnk! lim1Cnnkis a surjection with triv*
*ial
kernel. Therefore its domain is * if and only if its target is *. The result_fo*
*llows by
definition of the index. |__|
Proposition 4.7. Let X and Y be nilpotent of finite type, and consider the tower
{Gn}n where Gn := [X, Y (n)].
1) The index Ind{Gn}n is the minimum value of G(f) for f 2 Ph(X, Y ).
2) For any k 1, the k-th index Indk{Gn}n is the minimum value of G(f) for
f 2 Ph(X, Y ).
Proof. Point 1) follows directly from Point 3) of Theorem 4.4: indeed it shows *
*in
particular that Ph(X, Y ) = Phk(X, Y ) if and only if k Ind{Gn}n.
Point 2). Using Point 1), it suffices to prove that the k-th index of {Gn}n *
*is the
index Ind [X, Y (n)] n.
From the fibration sequence 2Y (k)! Y (n)! Y (n)! Y (k), we deduce
a surjection from [X, Y (n)] onto Knk:= Ker [X, Y (n)] ! [X, Y (k)] . Let
Hnkbe the kernel of that surjection. Using the fibration sequence once more, we
deduce a surjection from [X, 2Y (k)] onto Hnk.
Consider now the following diagram:
1 ____//_Hn+1k___//_[X,33Y3(n+1)]_//Kn+1k____//13fffff
ffffffff | | |
[X, 2Y (k)] | | |
XXXXXX | | |
XXXXXXX++|fflffl++X fflffl| fflffl|
1 _____//_Hnk_____//_[X, Y (n)]___//Knk_____//1
where the rows are exact by definition. We see that for any n the map Hn+1k! Hnk
is in fact a surjection. We then apply Lemma 4.6 and deduce that Ind{Knk}n =
Ind [X, Y (n)] n. By definition, Ind{Knk}n is the k-th index of {Gn}n,_and*
* the
result follows. |__|
Proof of Proposition 4.5.Consider the following natural commutative diagram:
Ph m(X,"Y`)oooo_Phm (X,"Y`**)oooPhom(X,"Y`)_
| | |
| | |
fflffl| |fflffl fflffl|
Ph k(X, Y )oooo_Phk(X, Y **)oooo_Ph(X, Y )
Statement (i) (resp. (ii), (iii)) means that the left (resp. middle, right) inc*
*lusion
is in fact a bijection. It's then clear that (iii) implies (ii), which itself i*
*mplies (i).
Suppose now that (i) is true. By Point 3) of Theorem 4.4, that implies that
m Indk{Gn}n. By Proposition 4.7, we deduce that all maps f 2 Ph(X, Y ) __
have Gray index G(f) m, that is that (iii) is true. |_*
*_|
5. The completion approach to F-phantom maps
This section is devoted to proofs of Section 2 results, excepted Point 1) of
Theorem 2.4, and Theorem 2.15, to be found in Section 4.
12 PIERRE GHIENNE
Lemma 5.1. For any spaces X and Y , we have PhF (X, Y ) = Ph1F(X, Y ). More-
over, for any k, the 1-connected cover Y <1> ! Y induces a bijection PhkF(X, Y *
*<1>) ~=
Ph kF(X, Y ).
Proof. This is principally Zabrodsky observation [26] that the map PhF(X, Y <1>*
*) !
Ph F(X, Y ) induced by the 1-connected cover is a bijection. It's easy to see t*
*hat_it
induces also the stated bijections for any k. |_*
*_|
For any nilpotent finite type space Y , let Yæ be the homotopy fiber of the
completion of Y . We then have a fibration:
bY- ffi!Yæ -j!Y -be!bY
It follows from Sullivan characterization that a map f : X ! Y is a F-phantom
map if and only if it factorizes through Yæ. From Definition 2.1, it's easy to*
* see
that the Gray index G(f) k if and only if f factorizes through the composite
(Y )æ -! Y -! Y
The following fundamental result of Roitberg-Touhey allows us a easy description
of the spaces (Y )æ. Here bZdenotes the profinite completion of the ring Z *
*of
integers.
Theorem 5.2. [21] Let Y be a nilpotent space of finite type, with torsion fun*
*da-
mental group. Then the space Yæ splits as an infinite product of Eilenberg-MacL*
*ane
spaces: Yæ ' m 1 K(ßm+1 (Y ) bZ=Z, m).
Lemma 5.3. Let X be any 0-connected space and Y nilpotent of finite type. Then
Hk(X; ßk+1(Y ) bZ=Z) ~=0 if and only if Hk(X; ßk+1(Y ) Q) ~=0. Otherwise,
Hk(X; ßk+1(Y ) bZ=Z) is uncountable.
Proof. Recall that bZ=Z is an uncountable rational group. The first assertion t*
*hen
follows from the obvious isomorphisms
Hk(X; ßk+1(Y ) bZ=Z) ~=Hk(X; Q) ßk+1(Y ) bZ=Z
~= Hk(X; Q) ßk+1(Y ) Q bZ=Z ~=Hk(X; ßk+1(Y ) Q) bZ=Z.
The choice of any non zero element of Hk(X; ßk+1(Y ) Q) defines an inclusi*
*on
Q ,! Hk(X; ßk+1(Y ) Q) which, once tensorized by bZ=Z, gives an inclusion bZ=Z *
*,!
Hk(X; ßk+1(Y ) bZ=Z). The second assertion follows. |__*
*_|
Proof of Points 2) and 3) in Theorem 2.4.By Lemma 5.1 we can always suppose
k 1. Consider the following diagram:
dddd(Y )æ
ddddddd `
X _f__//_qqddddddddddYllZZZZZZZ ||
ZZZZZZZZ fflffl|
(Y )æ ' (Y )æ x K(ßk+1(Y ) bZ=Z, k)
where the vertical map is merely the inclusion of the first factor. Here we use
Theorem 5.2 to describe the spaces (Y )æ.
If Hk(X, ßk+1(Y ) Zb=Z) ~=0, then a map f factors through (Y )æ if and on*
*ly
if it factors through (Y )æ, and PhkF(X, Y ) = Phk+1F(X, Y ).
If Hm (X, ßm+1 (Y ) bZ=Z) ~= 0 for any m k, then the set [X, (Y )æ] ~=
m k [X, K(ßm+1 (Y ) bZ=Z, m)] is the trivial set, and PhkF(X, Y ) = {*}. *
* |___|
The proof of Theorem 2.7 begins with the following lemma:
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 13
Lemma 5.4. Let f 2 PhF (X, Y ), where Y is a nilpotent space of finite type, *
*with
torsion fundamental group, and X is any space. Then the following statements are
equivalent:
(i) G(f) k;
(ii) There exists OE : X ! Yæ a lift of f such that X - OE!Yæ ! (Yæ)(k-1)is
inessential;
(iii) If OE : X ! Yæ is any lift of f, then there exists a map ~ making commuta*
*tive
the following diagram:
______f__________________________________________*
*________________________________________________________________@
_______________________________________________________*
*________________________________________________________________@
X_________OE_______//_________**_________Yæj//_Y
___ |
~_______ |ß
fflffl___ffiß |fflffl
bY_____//Yæ____//(Yæ)(k-1)
Proof. As Theorem 5.2 applies, we have easy identifications (Y (k))æ ' (Yæ)(k-1)
and (Y )æ ' (Yæ), and a (trivial) fibration (Y )æ i!Yæ ß!(Yæ)(k-1).
Moreover, we have a commutative diagram:
_________ffi_______// ______j______//____be__//
bY Yæ Y Yb
ß0|| ß|| || ||
fflffl| ffi(k-1) fflffl| fflffl| fflffl|
( bY)(k-1)' bY (k)__//_(Yæ)(k-1)' (Y (k))æ_//Y (k)___//bY (k)
where the rows are fibrations and all vertical maps are Postnikov sections. By
naturality [8, Proposition 11.5], we also have a commutative diagram linking the
actions A and Ak-1 induced by these fibrations, namely, for any space X:
[X, bY] x [X, Yæ]___A______//_[X, Yæ]
ß0*xß*|| ß*||
fflffl| Ak-1 fflffl|
[X, ( bY)(k-1)] x [X, (Yæ)(k-1)]//_[X, (Yæ)(k-1)]
Suppose G(f) k. Choose _ : X ! (Y )æ any lift of f. Then i O _ : X ! Yæ
is a lift of f projecting to * in (Yæ)(k-1), and (ii) follows. Let now OE : X !*
* Yæ be
a lift of f. We have j O i O _ ' f ' j O OE. Then, by classical theory, there e*
*xists
~ : X ! bYsuch that OE = A(~, i O _). We deduce ß O OE ' Ak-1(ß0O ~, ß O i O _*
*) '
Ak-1(ß0O ~, *) ' ffi(k-1)O ß0O ~ ' ß O ffi O ~, and (iii) follows.
Suppose (iii) is true. Choose OE and ~ such that ß O OE ' ß O ffi O ~. Set*
* ~OE:=
A(-~, OE) : X ! Yæ. Then ~OEis also a lift of f. We have ß O ~OE' Ak-1(ß0O
(-~), ß O OE) ' Ak-1(-(ß0O ~), ffi(k-1)O ß0O ~) ' ffi(k-1)O (-(ß0O ~) + ß0O ~) *
*' * and
(ii) follows (the signs + and - above refer to the group structure on [X, bY] *
*and
[X, ( bY)(k-1)]). This shows moreover that there exists _ : X ! (Y )æ such t*
*hat
i O _ = ~OE. In particular _ is a lift of f, and G(f) k. *
* |___|
Proof of Theorem 2.7.By hypothesis, the domain X is 0-connected, and then we
have bijections [X, bY<1>] ~= [ X, bY<1>] ~= [ X, bY] ~= [X, bY]. Together w*
*ith
Lemma 5.1, this implies we can always suppose that Y is 1-connected. We can
then use Theorem 5.2 to describe Yæ.
14 PIERRE GHIENNE
Point 1). Let f 2 Ph1F(X, Y ), and choose OE : X ! Yæ any lift of f. By Theo-
rem 5.2, we can identify OE with the collection (OEk)k, where OEk is the compos*
*ition
X OE!Yæ ! (Yæ)(k).
Let now ~1, . .,.~s, be the finitely many elements of [X, bY]. Similarly, f*
*or any
1 m s, we identify ffi O ~m : X ! Yæ with the collection (~km)k, where ~kmi*
*s the
composition X ~m-! bY- ffi!Yæ -! (Yæ)(k).
Suppose f is essential. Then for any 1 m s, the maps OE and ffi O ~m a*
*re
non homotopic. Then there exists some integer k(m), depending on m, such that
OEk(m) and ~k(m)mare non homotopic. Take now any integer k - 1 greater than all
the k(m). Then for any 1 m s, the maps OEk-1 and ~k-1mare non homotopic.
By Lemma 5.4, we deduce G(f) < k, a contradiction. Therefore f is inessential
and the result follows.
For a distinct proof of the same result, see Remark 5.12.
Point 2). Suppose Hk(X; ßk+1(Y ) Q) Æ 0. To show that the inclusion
Ph kF(X, Y ) Phk+1F(X, Y ) is strict, we will produce a F-phantom map f : X !*
* Y
with Gray index exactly k.
Let K := K(ßk+1(Y ) bZ=Z, k). By Lemma 5.3, we see that [X, K] is uncount-
able. By hypothesis, [X, bY] is countable, and we can choose a map OE : X ! K
which does not factor through bY ! Yæ proj.-!K. Let f be the composition
X -OE!K ,! Yæ ! Y .
The composition X -OE!K ,! Yæ -ß! (Yæ)(k-1)is clearly trivial, and by
Lemma 5.4 we deduce that G(f) k.
Suppose now that G(f) k + 1. By Lemma 5.4, there exists then a map ~
making commutative the following diagram:______________________________________*
*________________________________________________________________@
_______________________________________________*
*________________________________________________________________@
X____OE_//_K"Ø____//Yæ______//K
___ | ||
~_______ |ß ||||
fflffl___ffiß fflffl| ||
bY_____//Yæ____//__________66____________________________*
*________________________________________________________________@
_________________________________________*
*_______________________________________________________________
proj.
This clearly contradicts our choice of the map OE, and then G(f) = k. *
*|___|
We list below some already known cases where Theorem 2.7 do apply.
Proposition 5.5. In each of the following cases, we have [X, bY] ~=*:
1) The spaces X and Y are finite type, with torsion fundamental group, and X '
iZ(m) whereas Y ' jW , for some integers m and i, j 0, for some space Z and
some finite CW-complex W .
2) The space Y is a one-connected finite CW-complex, and X ' BG, the classi-
fying space of some 0-connected Lie group G [12, Theorem 5.6.(ii)].
3) The space Y is a finite CW-complex, and X is a 0-connected infinite loop sp*
*ace
with torsion fundamental group [13, Theorem 3].
Proof. As far as we know, Point 1) is not clearly stated in the litterature, al*
*though
undoubtely known by experts. Here is a quick proof. Let Xø be the homotopy
fiber of the rationalization X ! X(0). Then [ Xø, Y ] ~=*, by [26, Theorem D].
But [ Xø, Y ] ~=[X, bY] by [23, Proposition 34], and Point 1) follows. *
* |___|
Remark 5.6. In the particular case where [X, bY] is trivial, we have an easy
description of the Gray filtration on PhF (X, Y ). Indeed, by Theorem 5.2, we h*
*ave
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 15
a bijection:
Ph F(X, Y ) ~=[X, Yæ] ~= m 1 Hm (X; ßm+1 (Y ) bZ=Z).
Through this product description, we see thanks to Lemma 5.4 that a phantom
map has Gray index k if and only if its (k - 1)-th first coordinates are zero.
We shall now go towards the proof of Theorem 2.8. Recall that we have defined
spaces P nY as homotopy pullbacks:
Y_____________________________________________________*
*________________________________________________________________@
_____________________be______________________________*
*________________________________________________________________@
____________________________________________________*
*_____________________________________________________pn__
____________________________________________________*
*________________________________________________
______________________%%___________________________*
*__________________________!!____
_______________________P/nY/_b
________________________yY
________________________||
____________________||
_ÆÆ____________________fflffl|fflffl|be(n)
Y (n)____//bY (n)
There might exist several homotopy classes of maps pn making the above dia-
gram commutative. To avoid this ambiguity, we shall state the following lemma.
Lemma 5.7. For any n 0, there exists a functor P n from spaces to spaces,
together with a coaugmentation (a natural transformation) pn : Id ! P n, such
that, for any space Y , the space P nY is the homotopy pullback associated to *
*the
(n)
maps Y (n)be-!bY (n)- bY.
Proof. Let F be a functor from spaces to spaces with coaugmentation j : Id ! F .
Such a functor associates, to any map f : X ! Y , a square
f
X ______//_Y
j|| |j|
fflffl|Fffflffl|
F X ____//_F Y
which is strictly commutative, and not just commutative up to homotopy.
The technical key point of our proof is the fact that Postnikov section p(n):
Y ! Y (n), as well as completion be: Y ! bY, can be chosen as such functors. Th*
*is
comes from the theory of localization with respect to a map, see for example the
lines following Theorem 6.3. of [2].
Having chosen these functors (Postnikov section and completion), we then def*
*ine
P nY , as usual, as the set of triples (x, !, z) where x 2 Y (n), z 2 bY, and !*
* is a
path in bY (n), starting at be(n)(x), ending at p(n)(z). We also define pn : Y*
* !
P nY by y 7! p(n)(y), cy, be(y) , where cy denotes the constant path at the po*
*int
be(n)p(n)(y) = p(n)be(y). For any map f : X ! Y , strict commutativity shows th*
*at
the assignement
x, t 7! !(t), z 7! f(n)(y), t 7! bf(n)(!(t)), bf(z)
well defines a map P nf : P nX ! P nY such that the following square strictly
commutes:
f
X ________//Y
pn || |pn|
fflffl|Pnffflffl|
P nX ____//_P nY
|___|
Remark 5.8. More generally, to any pair of maps ff : A ! B and fi : C ! D, we
can associate a functor Lff,fifrom spaces to spaces, together with a coaugmenta*
*tion
16 PIERRE GHIENNE
jff,fi: Id ! Lff,fi, as follows. Let Lffand Lfibe the localization functors wit*
*h respect
to the maps ff, fi, together with their coaugmentation jff, jfi. Then the funct*
*or Lff,fi
takes a space Y into the homotopy pullback associated to the diagram:
LffY Lff(jfi)-!Lff(LfiY ) jff-LfiY
In the proof above ff is Postnikov section and fi is completion.
Lemma 5.9. 1) The map pn*: ßk(Y ) ! ßk(P nY ) induced in homotopy by pn is
an isomorphism if k n. If k > n, it is, up to natural isomorphism of its targ*
*et,
the completion be*: ßk(Y ) ! ßk(bY).
2) For any space Y and k n, the map pk : P nY ! P k(P nY ) is an equivalence.
3) For any map f : X ! P nY and k n, there exists, up to homotopy, a natural
factorization of f as shown in the following diagram:
f
X ______//P;nY;____
_____
pk|| ________
fflffl|_____
P kX
Proof. Point 1) follows from the Mayer-Vietoris sequence in homotopy of the pul*
*l-
back involving P nY , and implies easily Point 2). In the following diagram,
f
X ________//P nY
pk|| ' |pk|
fflffl|Pkf fflffl|
P kX ____//_P k(P nY )
the right vertical map pk is then an equivalence when k n. If (pk)-1 denotes
its homotopy inverse, the composition (pk)-1 O P kf is the desired factorizatio*
*n_of
f. |__|
Corollary 5.10. There exists a commutative diagram
uY SSSS
pn+1uuuu |n SSSp0=beSSS
uu p| SSSSSS
zzuuu fflffl| S))
. ._.___//P n+1Y____//P nY____//_______33__________________________*
*_________________________././._P 0Y = bY
_______________________________________*
*________________________________________________________________@
ffln
and the maps pn induce a equivalence from Y to the homotopy inverse limit of the
tower P nY n.
Proof of Theorem 2.8.Construct the following diagram, where all vertical sequen*
*ces
are fibrations:
Y __be_//bY____bY
_==_y___
f~____|___ | |
______| | |
___ fflffl| fflffl| fflffl|
X ___f__//_Y_pn__//_PynY____//_bY
| | |
| | |
fflffl| fflffl| fflffl|
Y (n)______Y (n)___//_bY (n)
The left upper square is then a pullback, by construction.
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 17
Let f 2 PhF (X, Y ). If there exists a lift ~fsuch that beO ~f' *, then cert*
*ainly
pn O f ' *. But, by pullback property, the converse is also true. As beO ~f' * *
*if and
only if ~f2 PhF (X, Y ), Point 1) follows.
We deduce the existence of an exact sequence of pointed sets
* -! Ph1F(X, Y ) -! [X, Y ] -! lim[Xn, P nY ] -! *
the base point being the trivial map. Then Point 2) follows from [1, Ch. _IX,_
Corollary 3.2.]. |__|
Corollary 5.11. Let Y be a nilpotent space of finite type, and X be any space. *
*Set
n (n) n
Hn := Im [X, P Y ] ! [X, Y ] and Hbn := Im [X, P Y ] ! [X, bY]
1) The surjections [X, P nY ] i Hn induce a bijection Ph1F(X, Y ) ~=lim1Hnn.
2) The surjections [X, P nY ] i bHninduce a bijection Ph1F(X, Y ) ~=lim1bnHn.
Remark 5.12. Point 2) of Corollary 5.11 gives an other proof of Point 1) of
Theorem 2.7. Indeed, if [X, bY] is finite, so are its subgroups Hbn. We dedu*
*ce
lim1Hnbn~=*, and the result follows.
Proof of Corollary 5.11.From Lemma 5.1, we can check that we can always sup-
pose that Y is one-connected.
Point 1). We consider the diagram:
[X, P nY_]______////Hn" ` Ph1F(X, Y )_____////_lim1Hnn
MMM | t~NNNN
MMM | NNN |
MMM | NNNN |
M&&Mfflffl| N'' fflffl|
[X, Y (n)] Ph FD(X, Y )
whose right part is obtained from the left one, by taking lim1.n As the map
Ph 1F(X, Y )! PhFD (X, Y ) is merely an inclusion between two subsets of [X, Y *
*],
the assertion follows.
Point 2). Let An be the image of [X, bY] ! [X, (Yæ)(n-1)] (notice that in
general An as no reason to be a group). We deduce from Lemma 5.4 that the
image of the composition
limAnn,! lim[Xn, (Yæ)(n)] ~=[X, Yæ] -! [X, Y ]
is exactly Ph1F(X, Y ). This set then appears as an orbit set,
Ph 1F(X, Y ) ~=limAnn= [X, bY],
using the action of [X, bY] on limAnninduced by its action on [X, Yæ].
On another hand, a carefull reading of the proof of [14, Theorem 2] reveals *
*the
following purely algebraic fact: the group [X, bY] acts naturally on lim[Xn, *
*bY]=Hbn
in such a way that
i j
lim1bnHn~=lim [nX, bY]=Hbn = [X, bY].
To conclude, it suffices to produce natural bijections [X, bY]=Hbn ! An, wh*
*ich
are moreover maps of sets on which [X, bY] acts, that is to say maps commuting
with the action. This can be done thanks to the exact sequence 1 ! Hbn !
[X, bY] i An induced by the fibration sequence P nY ! bY ! (Y (n))æ '
(Yæ)(n-1). |___|
18 PIERRE GHIENNE
Theorem 2.8 as well as Corollary 5.11 give lim1dnescriptions of the set Ph1F*
*(X, Y ).
To each of these descriptions is associated an algebraic Gray filtration (Defin*
*i-
tion 2.13). The following result interprets geometrically these filtrations.
Proposition 5.13. Let Y be a nilpotent space of finite type, and X be any space.
1) The algebraic Gray filtration on Ph1F(X, Y ) ~=lim1[nX, P nY ] is trivial.
2) Let Ph1,kF(X, Y ) be the k-th term of the algebraic Gray filtration on Ph1F(*
*X, Y ) ~=
lim1Hnn. Then Ph1,kF(X, Y ) is the image of the map Ph1F(X, Y ) ! Ph1F(X, Y )
induced by the k-connected cover Y ! Y .
Proof. Point 1). We shall prove that the algebraic Gray filtration on the lim1o*
*nf
the tower
. .!.[X, P nY ] ! [X, P n-1Y ] ! . .!.[X, P 1Y ] ! [X, bY] ! *
is trivial. The first term L1 of that filtration is by definition the kernel of*
* the nat-
ural map Ph1F(X, Y ) i lim1bnHn. Then L1 is trivial by Point 2) of Corollary 5.*
*11.
Point 2). Let Hn,kbe the image of the map [X, P nY ] ! [X, Y (n)]. By
naturality, we see that the map [X, Y (n)] ! [X, Y (n)] induced by Y !*
* Y
takes Hn,kinto Hn. We claim that the sequence of groups Hn,k! Hn ! Hk is
exact, where Hn ! Hk is induced by [X, Y (n)] ! [X, Y (k)].
Assuming that claim for the moment, consider the diagram:
Hn,k _____________//Hn____________//HkOO
| || |
fflfflfflffl||" |||| ?Ø||
Kn := Im(Hn,k! Hn) Ø_____//Hn____////Hnk:= Im(Hn ! Hk)
Taking lim1,nwe obtain
lim1Hnn,k~=Ph1F(X, Y )
RR
| RRRRRRR
fflfflfflffl||RRRR((RR
lim1Knn____________//_Ph1F(X, Y_)pk////_lim1Hnnk
where the row is an exact sequence of pointed sets. The k-th term of the filtra*
*tion
is by definition the kernel of pk, which then clearly coincide with the image of
Ph 1F(X, Y ) ! Ph1F(X, Y ).
We show now our claim. From the following diagram,
Hn,k"_`_________//_Hn"_`_______//Hk" `
| | |
| | |
fflffl| fflffl| fflffl|
[X, Y (n)]___//[X, Y (n)]__//[X, Y (k)]
where the bottom row is exact, we see we have only to prove that a map f 2 Hn
projecting to * in Hk belongs to the image of Hn,k.
For such an f, choose g a lift of f through P nY (a lift exists by definiti*
*on of
Hn), and consider the following diagram, where we construct the maps u1, u2 and
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 19
u3 successively:
P nY ________________///bY/____________________
__99__| MMM________ KK
____ | ____MMM__ | KKKK
u __ |_____ MM&& | K%%
3___ __|__ 11 (n)______|______//_b (n)
___u2____|_________Y__ Y
__ ______|__u1_ | | |
_________|_ | | |
______ fflffl| | fflffl| |
X __g__// P nYO________|________//_ bYM |
______________________|OOO MMM |
_____________________|______________________|______OOOOMMM
___________________fflffl|________________fflffl|_______________*
*________________________________________________________________@
f _________//______________________________________________*
*________________________________________________________________@
As f projects to * in Hk, there exists a lift u1 of f through Y (n). We th*
*en
choose u2 thanks to the pullback property of the right vertical face of the cub*
*e.
Again, we choose u3 thanks to the pullback property of the top horizontal face *
*of
the cube. In particular, u3 is a lift of u1 through P nY (we don't claim h*
*owever
that u3 is a lift of g). That means by definition that u1 2 Hn.k, and then f_be*
*longs_
to the image of Hn,k, as required. |__|
Heading now towards the proof of Theorem 2.9, we shall before make a brief
review of variousSexisting universal phantom maps.
Let X = n 1 Xn be a CW-complex, where Xn denotes the n-skeleton of
X. The universal FD-phantom map out of X [6] is the right map in the cofiber
sequence:
` `
Xn -! X -! Xn
n 1 n 1
It's a FD-phantom map and, for any space Y , a map f : XW! Y is FD-phantom
if and only if f factorizes as some composition f : X ! n 1 Xn ! Y .
Dually, for any space Y , the universal FD-phantom map into Y [11] is the le*
*ft
map in the fiber sequence:
n 1 Y (n)-! Y -! n 1 Y (n)
(by fiber sequence it is meaned here that the homotopy fiber of the right map h*
*as
the weak homotopy type of n 1 Y (n)). It's a FD-phantom map and, for any
space X, a map f : X ! Y is FD-phantom if and only if f factorizes as some
composition f : X ! n 1 Y (n)! Y .
Recall now from Section 2 the canonical map X : X ! W (X) of McGibbon-
Strom [18], associated to any space X. It's the universal FD-phantom map with
infinite Gray index out of X. That is that X 2 Ph1FD(X, W (X)) and, for any
space Y , a map f : X ! Y belongs to Ph1FD(X, Y ) if and only if f factorizes as
some composition f : X ! W (X) ! Y .
We now complete this list, defining below two other universal phantom maps.
For any space Y and k 1, set Zk(Y ) := n 1 Y (n), and let `k(Y ) be the
composition Zk(Y ) ! Y ! Y , where the first map is the universal FD-phantom
map into Y .
Proposition 5.14. and Definition.
1) Let Y be any space. The universal FD-phantom map with infinite Gray index
20 PIERRE GHIENNE
into Y is the top map `Y in the following pullback square:
Z(Y )________`Y_______//Y
y
| |
| |
fflffl| fflffl|
k 1 Zk(Y )___________//_ k 1 Y
k 1`k(Y )
Then `Y 2 Ph1FD(Z(Y ), Y ) and, for any space X, a map f : X ! Y belongs to
Ph 1FD(X, Y ) if and only if f factorizes as some composition f : X ! Z(Y ) ! Y*
* .
2) Let Y be a nilpotent space of finite type. The universal F-phantom map with
infinite Gray index into Y is the left map in the fiber sequence
n 1 P nY -! Y -! n 1 P nY,
where the right map is defined by the collection {pn : Y ! P nY }n. This map is*
* a
F-phantom map with infinite Gray index and, for any space X, a map f : X ! Y
belongs to Ph 1F(X, Y ) if and only if f factorizes as some composition f : X !
n 1 P nY ! Y .
Proof. We don't give the details for Point 1): this is a straightforward dualiz*
*ation
of McGibbon-Strom construction [18].
To prove that the homotopy fiber of Y ! n 1 P nY is weakly equivalent to
n 1 P nY , we follow the same line of arguments than in [11, Theorem 3]. The *
* __
universal property is easily checked with Point 1) of Theorem 2.8. *
*|__|
Proof of Theorem 2.9.Consider the following diagram, where the bottom fibration
sequence is obtained by CW-approximation of the upper fibration sequence:
{pn}n n n {pn}n n
Y _______//_ n 1 P Y______//_ n 1 P Y______//_Y______//_ n 1 P Y
|| OO OO ||
|| h|' h|' ||
|| ______ | | ||
|| {pn}n ___________| ___________| ||
Y _______//_ n 1 P nY______//_ n 1 P nY______//_Y
From universality (Proposition_5.14),_we see that Ph1F(X, Y ) = * for any space*
* X
if and only if the map n 1 P nY ! Y , which is itself a F-phantom map with
infinite_Gray_index, is trivial. By classical argument, this holds if and only *
*if the
map__{pn}n_admits_a retraction._We then only have to show that if Y is a retra*
*ct
of n 1 P nY, then {pn}n itself admits a retraction.
Suppose then there exists maps ~iand ~rsuch that ~rO ~iis identity on Y :
pn
Y _________________________// P_nY
___
~i|| _in_____
_____fflffl|_h fflffl_____
n 1 P nY _'__//_ n 1 P nYproj.//_ P nY
~r||
fflffl|
Y
It is not to hard to see that P nY ' P n-1( Y ), and that, through this equiva*
*lence,
the map pn is identified as pn-1. Using Point 3) of Lemma 5.9, we deduce the
existence of the dotted map in making the diagram commutative. The map h O~iis
then the composition
n}n in
Y {p-! n 1 P nY - ! n 1 P nY,
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 21
and we deduce that the map ~iis the composition
_______{pn}_________n___________
Y - ! n 1 n P nY- i! n 1 P nY ,
____ ____ _______
where inis a CW-approximation of___in._ Then ~rO ~i' ~rO in O {pn}n is
identity on Y , which means that {pn}n itself admits a retraction. *
*|___|
Proof of Proposition 2.10.For any finite type rationally elliptic space L, the *
*map
ßn(L) ! ßn(bL) induced in homotopy by the completion is an isomorphism for n
large enough, and then pn : Y ! P nY is an equivalence by Point 1) of Lemma 5.9.
If Y splits as stated, set Ln as the product of all those Lffsuch that Lff'
P nLff, whereas Lffis not equivalent to P n-1Lff. If no such Lffexists, set Ln *
*= *.
We then clearly have a weak equivalence Y -'! n 0 Ln. We deduce a weak
equivalence P k Y -'! n 0 P kLn for any k, and finally a weak equivalence
___________ _____________ '
k 1 P kY ' k 1 P k-1 Y -! k 1 n 0 P k-1Ln = k 0 n 0 P kLn.
As Lk ' P kLk, then Lk is a retract of n 0 P kLn. We deduce that k 0 Lk is
a retract of k 0_n_0_P_kLn._Using the weak equivalences above, we see that Y
is a retract of k 1 P kY. |___|
6.The algebraic Gray filtration on snt sets
We give below the proofs of main theorems of Section 3. Notice that the base
point of SNT (X), described as lim1Anut(X(n)), is the homotopy type of the space
X itself. Therefore, when we say that some subset A of SNT (X) is trivial, we m*
*ean
in fact that A = {X}.
In what follows we suppose that F is a functor, from the category of pointed
homotopy types of connected CW-complexes to itself, and that there exists, for *
*any
space X and any n, a natural equivalence F (X(n)) ' (F X)(n). We then simply
write F X(n).
In the following diagram between exact sequences, commutativity of the right
square follows by naturality:
1_____//AutF(X(n+1))____//Aut(X(n+1))____//Aut(F X(n+1))
__
____ | |
____ | |
fflffl____ fflffl| fflffl|
1______//AutF(X(n))______//Aut(X(n))______//Aut(F X(n))
We then deduce the existence of the dotted arrow. This gives a map of tower
{Aut F(X(n))}n ! {Aut (X(n))}n, whose lim1wne denote by
jF : lim1AnutF(X(n)) ! lim1Anut(X(n)) ~=SNT (X).
Proof of Theorem 3.2.We owe the idea of the proof to a similar result of F'elix-
Thomas [3, Theorem 1], in a slightly different context. InsteadLof F , they con*
*sider
the functor associating to a space X the öh motopy group" k 1 ßk(X). Their
proof however works as well here, and we content ourselves to describe some ste*
*ps,
without further details.
22 PIERRE GHIENNE
Let PF denotes the set of pairs (Y, (fn)n 1), where fn : Y (n)! X(n) are
homotopy equivalences such that the following diagram commutes:
F(fn+1)
F Y (n+1)_____//F X(n+1)
| |
| |
fflffl|F(fn) fflffl|
F Y (n)_______//F X(n)
We introduce now an equivalence relation on PF . Two pairs (Y, (fn)n 1) and
(Z, (gn)n 1) in PF are said equivalent if there is an homotopy equivalence OE :*
* Z ! Y
such that, for any n, the maps fn O OE O g-1nbelongs to AutF (X(n)). There is t*
*hen
a well-defined map PF = ~ -æ!SNT (X), associating to the class of (Y, (fn)n 1) *
*the
homotopy type of Y .
The key step of the proof is then to show that the bijection ` : SNT (X) !
lim1Anut(X(n)) defined by Wilkerson [24] induces another map `F such that the
following square commutes:
PX = ~ _____æ______//SNT(X)
`F|| |`|
fflffl| fflffl|
lim1AnutF(X(n))__j__//lim1Aut(X(n))
F n
Using Wilkerson techniques, we show also that `F is still a bijection.
As the image of æ is clearly SNT F (X) (as defined in Section 3), the_result_
follows. |__|
Notice that from the exact sequence of pointed sets
lim1AnutF(X(n)) jF-!lim1Anut(X(n)) ! lim1Anut(F X(n)),
we deduce that SNT F(X) belongs to the kernel of the natural map SNT (X) !
SNT (F X), Y 7! F Y . The following example shows that, in general, SNT F(X) is
a proper subset of that kernel.
Example 6.1. Let X be a 1-connected finite type space, and let F = -(0)be
the rationalisation functor. Then Ker SNT (X) ! SNT (X(0)) is SNT (X) itself,
whereas SNT (0)(X) is trivial.
Proof. The groups Aut(0)(X(n)) are finite for all n, by [9, Corollary II.5.4]. *
*There-
fore, lim1Anut(0)(X(n)) ~=*, and SNT (0)(X) is trivial by Theorem 3.2. On anoth*
*er
hand, SNT (X(0)) = {X(0)} by [24, Corollary II.b], and the example follows. *
* |___|
It seems however difficult in general to characterize algebraically the "dif*
*ference"
between SNT F(X) and Ker SNT (X) ! SNT (F X) . One reason is that the image
of Aut(X(n)) ! Aut(F X(n)) need not to be normal.
Proof of Theorem 3.1.Let F be the functor -(k), the k-th Postnikov section. Ap-
plying lim1tno the exact sequence (n k)
1 ! AutF(X(n)) = Autk(X(n)) ! Aut(X(n)) ! Gnk! 1,
where Gnkstands for Im Aut(X(n)) ! Aut(X(k)) , we obtain the following exact
sequence of pointed sets:
pk 1 n
lim1AnutF(X(n)) jF-!lim1Anut(X(n)) i lim Gnk.
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 23
This identifies SNT k(X) := Kerpk with the image of jF . The result follows_fro*
*m_
Theorem 3.2 and the definition of SNT F(X). |__|
Proof of Theorem 3.3.The finite type hypothesis on X implies that the groups
Aut (X(n)) are countable. Then Point 1) follows directly from Theorem 2.15.
Let Gn := Aut(X(n)). Then the group Autk(X(k+1)) is Ker(Gk+1 ! Gk), and
the first statement of Point 2) follows from Point 2) of Theorem 4.4.
(k)
Look at now the fibration X(k+1)-p! X(k)! K(ßk+1(X), k + 2) where the
right map is the k-th Postnikov invariant of X. Consider its associated exact
(k))*
sequence [X(k+1), K(ßk+1(X), k + 1)] ! [X(k+1), X(k+1)] (p-! [X(k+1), X(k)]. If
f 2 Aut k(X(k+1)), then (p(k))* . f = p(k)= (p(k))* . Id, which means that f
and Id are in the same orbit through the action of [X(k+1), K(ßk+1(X), k + 1)] =
Hk+1(X; ßk+1(X)) on [X(k+1), X(k+1)]. If this last cohomology group is finite,
the same must be true for Aut k(X(k+1)) (for a systematic study of the group
Aut k(X(k+1)), see [19]). __
Point 3) follows directly from Points 1) and 2). |_*
*_|
The following example might seem a little bit artificial, but it has the adv*
*antage
to illustrate drastically Remark 2.14. It is easily proved with Theorem 3.1.
Example 6.2. Let X1 and X2 be non equivalent spaces of the same n-type for
all n. Then the set L := SNT (X1) = SNT (X2) inherits in particular two differe*
*nt
algebraic Gray filtrations: the first, whose subsets are denoted Lk1, is inher*
*ited
from the bijection L ~=lim1Anut(X(n)1), whereas the second one, whose subsets a*
*re
denoted Lk2, is inherited from the bijection L ~=lim1Anut(X(n)2).
This two filtrations are indeed in general distincts: suppose X2 =2Lk1for so*
*me
k. This automatically holds for example if X1 is rationally elliptic, by Point *
*3) of
Theorem 3.3. Then X1 2 L11 (as X1 is the base point!), but X1 =2Lk2.
We now decompose the proof of Theorem 3.4 in a sequence of three lemmas,
each of which being a generalization, as well as a consequence, of similar lemm*
*as
of [14, 15]. In what follows, when we say that a group homomorphism G ! H has
a finite cokernel, it is meaned in fact that the image of G has finite index in*
* H.
Lemma 6.3. If X is as in Point 1) of Theorem 3.4, then, for all n k 1, the
map Autk(X(n)) ! Autk(H n (X; ZP)) has finite kernel and finite cokernel. If X
is as in Point 2) of Theorem 3.4, then the map Autk(X(n)) ! Autk(ß n X) has
finite kernel and finite cokernel, for all n k 1.
Proof. For the first claim, consider the following natural commutative diagram:
1 _______//_Autk(X(n))__________//_Aut(X(n))_________//_Aut(X(k))
| | |
| | |
fflffl| fflffl| fflffl|
1 ____//_Autk(H n (X; ZP))___//Aut(H n (X; ZP))___//Aut(H k (X; ZP))
where the rows are exact. By [14, Lemma 3.1.a], the two vertical maps on the ri*
*ght
have finite kernel and cokernel. By chasing in this diagram, we then deduce the
same is true for the left vertical map. __
The second claim is proved similarly, using [15, Lemma 1]. |_*
*_|
The following key algebraic result is not stated this way in [14]. It is how*
*ever
not too hard to see that the following more general formulation stays valid.
24 PIERRE GHIENNE
Lemma 6.4. [14] Let {Gn}n be a tower of countable groups. Suppose there exists
N such that each map Gk+1 ! Gk, k N, has a finite cokernel. Then lim1Gnn~=*
if and only if, for each k N, the canonical map limGnn! Gk has a finite coker*
*nel.
Lemma 6.5. Let X be 1-connected.
1) If the n-th Postnikov invariant X(n) ! K(ßn+1(X), n + 2) is torsion, then
Aut k(X(n+1)) ! Autk(X(n)) has finite cokernel. If moreover ßn+1(X) Q ~=0,
then it has also a finite kernel.
2) If X is a co-H0-space, then, for any n, Autk(X(n+1)) ! Autk(X(n)) has finite
cokernel. Furthermore, if Hn+1(X; Q) ~=0 this map has also a finite kernel.
3) If Hn(X; Q) ~=0 for all n > N, then Autk(X(n+1)) ! Autk(X(n)) has a finite
kernel and a finite cokernel for all n N.
Proof. We first show that for any space X, if, for some n, the map Aut(X(n+1)) !
Aut (X(n)) has finite kernel and/or cokernel, then the same must be true, for a*
*ny
k, for the map Autk(X(n+1)) ! Autk(X(n)). Indeed, if n < k, there is nothing to
be proved. If n k, this follows by chasing in the following natural commutati*
*ve
diagram, with exact rows:
1_____//Autk(X(n+1))____//Aut(X(n+1))____//Aut(X(k))
| | ||
| | ||
fflffl| fflffl| ||
1______//Autk(X(n))______//Aut(X(n))_____//Aut(X(k))
Then Points 1) and 2) follow from their corresponding statement in the case
k = 0, to be found in [14, 15].
It remains to show Point 3) in the case k = 0. As far as we know, this it not
clearly stated in the literature, although our present proof uses techniques al*
*ready
present in [15].
We shall find a rational equivalence W -f! X, where W is a N-dimensional
CW-complex (see for example the proof of Theorem 4 in [10]). Fix n N. Let us
define
(f(n)) := {(ff, fi) 2 Aut(W (n)) x Aut(X(n)) | f(n)O ff ' fi O f(n)}.
Consider now the commutative diagram
Aut (W (n+1))oo__ (f(n+1))_____//Aut(X(n+1))
~=|| || ||
fflffl| fflffl| fflffl|
Aut (W (n))oo____ (f(n))_______//Aut(X(n))
where the horizontal maps are given by the obvious projections. It follows from*
* [25,
Theorem 2.3] that these horizontal maps have finite kernel and cokernel. As n is
up the dimension of W , the left vertical map is in fact an isomorphism. By cha*
*sing
in the diagram, we deduce the middle, and then the right vertical map has_finit*
*e_
kernel and cokernel. |__|
Proof of Theorem 3.4.We only give the details for Point 1), the proof of Point *
*2)
being similar.
The equivalence of conditions (ii) and (iii) follows directly from Lemma 6.3.
The finite type hypothesis on X implies that the groups Autk(X(n)) are count-
able, being subgroups of the countable groups Aut (X(n)). Now, the canonical
surjection Aut(X) i limAunt(X(n)) is easily seen to induce, for any k, a surjec*
*tion
Aut k(X) i limAuntk(X(n)). Using Lemma 6.4, whose hypothesis is seen to be
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 25
satisfied (with N = 1) thanks to Point 1) of Lemma 6.5, we deduce that condition
(ii) is equivalent to lim1Anutk(X(n)) ~=*.
To conclude, we use now Point 1) of Theorem 4.4. Indeed, with its notation, *
*we
have Knk:= Autk(X(n)) and Lk := SNT k(X). Therefore SNT k(X) is trivial if and
only if lim1Anutk(X(n)) ~=*, and the result follows. |*
*___|
Proof of Example 3.5.The space BSU(m) is rationally equivalent to the product
K(Z, 2) x K(Z, 4) x . .x.K(Z, 2m). Then ßk(BSU(m)) Q ~=0 for k 2m + 1,
and we deduce from Theorem 3.3 that SNT 2m(BSU(m)) is trivial.
To show that SNT 2m-1 (BSU(m)) is non trivial, we use Point 1) of Theo-
rem 3.4, condition (iii). The image of Aut(BSU(m)) ! Aut (H*(BSU(m); Z))
is finite (see the proof of [14, Theorem 4]). Then, for any n 2m, the im-
age of Aut 2m-1(BSU(m)) ! Aut 2m-1(H n (BSU(m); Z)) is also clearly finite.
It then suffices to show that Aut2m-1 (H n (BSU(m); Z)) is infinite. The coho-
mology algebra H*(BSU(m); Z) is the polynomial algebra Z[e2, e4, . .,.e2m ] on
generators e2i of degree 2i. For u 2 Z, let OEu be the automorphism sending e2i
to itself for 1 1 m - 1, and sending e2m to e2m + u . e2e2m-1 . Now, if we
associate to u the restriction of OEu to H n (BSU(m); Z), we define an inclusion
Z ,! Aut2m-1(H n (BSU(m); Z)), and the result follows. __
The assertions about BSp(m) are showed similarly. |__|
To close this section, we provide the following weaker version of Theorem 3.*
*4,
available for spaces with a finiteness condition on their rational homotopy typ*
*es.
Theorem 6.6. Let X be a 1-connected space, of finite type over some subring of
the rationals. Suppose there exists N such that ßn(X) Q ~=0 for any n > N, or
such that Hn(X; Q) ~=0 for any n > N. Then SNT k(X) = {X} if and only if one,
and hence all, of the following equivalent conditions holds:
(i) The image of Autk(X) ! Autk(X(N)) has finite index.
(ii) The image of Autk(X) ! Autk(X(n)) has finite index, for some n N.
(iii) The image of Autk(X) ! Autk(X(n)) has finite index, for all n N.
Proof. As in the proof of Theorem 3.4, we see that SNT k(X) = {X} if and only if
lim1Anutk(X(n)) ~=*.
By Point 1) or Point 3) of Lemma 6.5 , we see that all the maps Autk(X(n+1))*
* !
Aut k(X(n)), n N, have finite kernel and cokernel. This shows the equivalence
of the three conditions. This shows also that the hypothesis of Lemma 6.4 are
satisfied, and we then see that lim1Anutk(X(n)) ~=* if and only if condition (i*
*ii) is
satisfied. |___|
7. From phantom maps to snt-theory
In that section we emphasize on the connection between phantom maps and
SNT-theory, which is given by the maps
SNT (Y _ X) Cof-PhFD (X, Y ) Fib-!SNT(X x Y )
associating to a FD-phantom map its homotopy cofiber or its homotopy fiber. In
what follows we suppose all our spaces to be 0-connected.
In [20] another connection is defined in an algebraic way. Let us define a g*
*roup
homomorphism _n : [X, Y (n)] ! Aut(X x Y )(n-1), associating to a map f :
X ! Y (n)' ( Y )(n-1)the homotopy equivalence
(n-1)
X(n-1)x ( Y )(n-1)! X(n-1)x ( Y )(n-1), (x, !) 7! x, ! * f (x)
26 PIERRE GHIENNE
where * denotes here loop mutiplication.
The maps _n fit together in a map of towers {[X, Y (n)]}n ! {Aut (X x
Y )(n-1)}n and give rise to a map
: PhFD (X, Y ) ~=lim1[nX, Y (n)] -! lim1Anut(X x Y )(n-1)~=SNT (X x Y )
We don't detail the dual version, which starts from maps _0n: [ X, Y (n)] !
Aut (Y _ X)(n)(there is no shift of degree here) and gives rise to a map
0: PhFD (X, Y ) ~=lim1[n X, Y (n)] -! lim1Anut(Y _ X)(n)~=SNT (Y _ X)
It is said in [20] that it is not known if the maps and 0are naturally eq*
*uiv-
alent, respectively, to Fib and Cof. The key result of that section is that ind*
*eed,
they are.
Theorem 7.1. The map associates to a phantom map its homotopy fiber, whereas
the map 0 associates to a phantom map its homotopy cofiber.
Proof. We only give the proof for the map . It may be dualized naturally to
describe 0.
Let f : X ! Y be a FD-phantom map, and let F be the fiber of f. Working
simplicially, we can construct the following strictly commutative diagram:
qn q1
F _____//._._.__//Fn+1_____//_Fn____//_._._._//_F1
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
X ______. ._.____//X_________X _______. ._.____X
f|| fn+1|| fn|| f1||
fflffl| fflffl| fflffl| fflffl|
Y _____//._._._//Y (n+1)ßn_//Y (n)__//_._._._//Y (1)
where the bottom row is a Postnikov system for Y , and we define Fn as the homo-
topy fiber of the map fn := X ! Y ! Y (n). Then Fn is the set of pairs (x, !n),
where x 2 X and !n is a path in Y (n), starting at the base point, ending at fn*
* (x).
The natural maps qn are given by (x, !n+1) 7! x, ßn O !n+1).
We see that the maps in the upper row induce equivalences F (n)' (Fn+1)(n).
It follows they induce as well an equivalence from F to the homotopy inverse li*
*mit
of the tower {Fn+1 qn!Fn}n.
Suppose now that, through the bijection Ph FD(X, Y ) ~= lim1[nX, Y (n)], the
map f is described by some family {un : X ! Y (n)}n . By construction of that
bijection, that means we can choose homotopies Hn : X x I ! Y (n), between fn
and the trivial map (Hn(x, 0) = fn (x) and Hn(x, 1) = *), such that un is the
"difference" between ßn O Hn+1 and Hn. More precisely, we have the formula:
i j i j
un : X ! Y (n), x 7! ßn O t 7! Hn+1(x, 1 - t) * t 7! Hn(x, t)
Now, homotopies Hn can be used to construct homotopy equivalences:
n
vn : Fn -'! X x Y (n), (x, !n) 7! x, !n * (t 7! H (x, t))
PHANTOM MAPS, SNT-THEORY, AND NATURAL FILTRATIONS ON lim1SETS 27
Let hn : X x Y (n+1)! X x Y (n)by given by the formula (x, !) 7! x, ßn . ! *
un(x) . Checking carefully, we see that we have construct a commutative diagram
vn+1
Fn+1__'__//X x Y (n+1)_____//(X x Y )(n)
| | |
| | |
| | fflffl|
qn || |hn| (X x Y )(n-1)
| |
| | _ (u )|
| | n n |
fflffl|vn |fflffl fflffl|
Fn___'___//X x Y (n)_____//(X x Y )(n-1)
where the unlabelled maps are induced by Postnikov sections, and _n are those
maps involved in the definition of .
We deduce first that F is homotopy equivalent to the homotopy inverse limit *
*of
the tower {X x Y (n+1)hn!X x Y (n)}n. Looking at homotopy groups, we then
see that F is also homotopy equivalent to the homotopy inverse limit of the tow*
*er
(n) (n-1)_n(un) (n-1)
(X x Y ) -! (X x Y ) -! (X x Y ) n
By [24], that precisely means that F is described as the family {_n(un)}n, thro*
*ugh
the bijection lim1Anut(X x Y )(n-1)~=SNT (X x Y ).
To summarize, we have shown that if f 2 Ph FD(X, Y ) is described by some
family {un}n, its homotopy fiber F is described by the family {_n(un)}n. _That_
exactly means that (f) = F = Fib(f). |__|
Proof of Theorem 3.6.By naturality of the construction of the algebraic Gray fi*
*l-
tration, we see that a map lim1Gnn! lim1Gn0n, which is the lim1onf a map of tow*
*ers
{Gn ! G0n}n, must respect the filtrations. Then Theorem 3.6 is certainly true f*
*or
(Point 1) and 0 (Point 2). Notice however the shift of degree for , due to
the same shift for the maps _n : [X, Y (n)] ! Aut(X x Y )(n-1)involved in its
definition. __
Therefore Theorem 3.6 follows directly from Theorem 7.1. |__|
The following proposition generalizes Example 3.7, providing generic examples
of spaces Z for which the algebraic Gray filtration on SNT (Z) is highly non tr*
*ivial.
Proposition 7.2. Suppose that X ' K(Z, 2n), n 1, and that Y , finite type with
torsion fundamental group, is equivalent to jW , for some j 0 and some finite
CW-complex W . Suppose furthermore that Hn(X; ßn+1(Y ) Q) is non zero for
infinitely many n. Then there are infinitely many strict inclusions in the alge*
*braic
Gray filtration on SNT (X x Y ).
Proof. Let f : X ! Y be any phantom map. By [7, Lemma 3.3], the homotopy
fiber Fib(f) of f is not homotopy equivalent to X x Y , provided f is essentia*
*l.
Now, by Theorem 2.7, which is seen to apply thanks to Proposition 5.5, we deduce
that there are, for infinitely many n, essential phantom maps fn : X ! Y of
Gray index n + 1. As Fib(fn) 2 SNT n(X x Y ) (Theorem 3.6), we deduce that
SNT n(X x Y ) is non trivial for infinitely many n, and then that SNT n(X x Y*
* )
is non trivial for all n < 1.
Suppose now only finitely many inclusions are strict in the Gray filtration *
*on
SNT (X x Y ). Then it would exist some integer k such that SNT k(X x Y ) =
SNT 1 (X x Y ). By Point 1) of Theorem 3.3, this would imply SNT k(X x Y )_is_
trivial, a contradiction. |__|
28 PIERRE GHIENNE
Proof of Example 3.8.Let W (X)=X be the cofiber of X : X ! W (X). As X 2
Ph 1FD(X, W (X)), we deduce from Point 2) of Theorem 3.6 that W (X)=X belongs
to SNT 1 (W (X) _ X).
Suppose X is finite type, and that its cohomology is not locally finite as a
module over the Steenrod Algebra for some prime p. It is then proved in [18] th*
*at
X is essential. Looking carefully the proof, we see it is in fact shown that *
*X is
not a retract of W (X)=X. In particular, W (X)=X is not homotopy equivalent_to
W (X) _ X. |__|
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Matematisk Institut, Universitetsparken 5, DK-2100 København
E-mail address: ghienne@math.ku.dk
*