Contemporary Mathematics
Subgroups of the Group of Self-Homotopy Equivalences
Martin Arkowitz, Gregory Lupton, and Aniceto Murillo
Abstract.Denote by E(Y ) the group of homotopy classes of self-homotopy
equivalences of a finite-dimensional complex Y . We give a selection of *
*results
about certain subgroups of E(Y ). We establish a connection between the *
*Got-
tlieb groups of Y and the subgroup of E(Y ) consisting of homotopy class*
*es of
self-homotopy equivalences that fix homotopy groups through the dimension
of Y , denoted by E# (Y ). We give an upper bound for the solvability cl*
*ass of
E# (Y ) in terms of a cone decomposition of Y . We dualize the latter re*
*sult to
obtain an upper bound for the solvability class of the subgroup of E(Y )*
* con-
sisting of homotopy classes of self-homotopy equivalences that fix cohom*
*ology
groups with various coefficients. We also show that with integer coeffic*
*ients,
the latter group is nilpotent.
1. Introduction and Preliminaries
Let Y be a CW-complex of dimension N and E(Y ) the group of homotopy
classes of self-homotopy equivalences of Y . In this paper we present a sample *
*of
results about a number of subgroups of E(Y ). We denote by E# (Y ) the following
proto-typical such subgroup:
E# (Y ) = {f 2 E(Y ) | f# = 1: ssi(Y ) ! ssi(Y ); for all i N}:
In Section 2, we give a way to construct elements in E# (Y ). This is of intere*
*st since
it provides a connection between the Gottlieb group of Y and certain subgroups *
*of
self-homotopy equivalences (Theorem 2.3). Next, in Section 3, we consider ques-
tions about the solvability and nilpotency of E# (Y ). For example, we show tha*
*t if
Y is the cofibre of a map between two wedges of spheres, then E# (Y ) is an abe*
*lian
group (Corollary 3.5). This result generalizes into a simple upper bound on the
solvability of E# (Y ) in terms of a cone-length invariant of Y (Theorem 3.3). *
* In
Section 4 we dualize these results to obtain upper bounds for the solvability o*
*f the
group of equivalences that fix cohomology with different coefficients (Theorem *
*4.2).
We also show that the subgroup of self-homotopy equivalences which fix the inte*
*gral
cohomology of a finite complex is a nilpotent group (Proposition 4.9).
____________
1991 Mathematics Subject Classification. Primary 55P10; Secondary 55P62, 55*
*Q05.
Key words and phrases. Homotopy equivalences, Gottlieb group, cone-length, *
*nilpotent
group, solvable group.
Oc0000 (copyright holder)
1
2 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
We now review some standard material that we will use. A cofibration sequence
fl j q
Z _____//Y____//X____//_Z;
where X is the mapping cone of fl, gives a homotopy coaction c: X ! X _ Z,
obtained by pinching the `equator' of the cone of Z to a point. The coaction
induces an action of [Z; W ] on [X; W ] for any space W (cf. [Hil65, Chap. 15] *
*for
details). This is defined as follows: If ff 2 [Z; W ] and f 2 [X; W ], then fff*
*is the
composition
f_ff r
X __c_//_X _ Z_____//W _ W_____//W:
The following properties of this action are well-known, and follow easily from *
*the
definitions:
(1) If h: W ! W 0, then h(fff) = (hf)hff.
(2) If ff; fi 2 [Z; W ], then (fff)fi= f(ff+fi).
Next, consider the following portion of the Puppe sequence associated to the ab*
*ove
cofibration sequence:
* j*
[Z; W ]_q___//[X; W_]__//[Y; W ]:
As is also well-known, the orbits of the action are precisely the pre-images of*
* j*.
That is, for f; g 2 [X; W ], we have fff= g for some ff 2 [Z; W ] if and only if
fj = gj.
Next, we review some notation and terminology for groups. Suppose that G is
a group and H and K are subgroups. Then H / G denotes that H is normal in G
and [H; K] denotes the subgroup generated by commutators of elements of H with
elements of K. A normal chain for G is a sequence of subgroups
G = G1 G2 . . .Gk+1 . . .
with Gi+1/ Gi for i 1. If [Gi; Gi] Gi+1 for each i, then the sequence is call*
*ed
a solvability series. If, further, Gk+1 = {1}, then we say that G is solvable o*
*f class
k and write solvG k. Analogously, given a normal chain as above with each
Gi/ G and [G; Gi] Gi+1, then it is called a nilpotency series. In this case, *
*if
Gk+1 = {1}, then we say that G is nilpotent of class k and write nilG k.
Clearly, we have nilG solvG. In addition, we write the identity homomorphism
of a group and the trivial homomorphism between two groups as 1: G ! G and
0: G ! H, respectively. This notation is also used for sets with a distinguish*
*ed
element.
Finally, we fix our topological conventions and notation. By a space, we mean
a connected CW-complex of finite type. Usually, we will be interested in finit*
*e-
dimensional CW-complexes. When we discuss rational spaces, we will specialize to
1-connected CW-complexes. As is well known, such a space X admits a rational-
ization, which is denoted by XQ. Similarly, a map of 1-connected finite complex*
*es
f :X ! Y admits a rationalization map fQ :XQ ! YQ. A general reference for
rationalization is [HMR75 ]. Furthermore, we do not distinguish notationally *
*be-
tween a map and its homotopy class. We write X Y to denote that the spaces
X and Y have the same homotopy type. The identity map of a space X is denoted
: X ! X and the trivial map between two spaces *: X ! Y .
SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES 3
2. A Connection with the Gottlieb Group
We consider the situation as in Section 1 of a mapping cone sequence
fl j q
Z _____//Y____//X____//_Z;
and the induced action of [Z; W ] on [X; W ] which yields fff2 [X; W ] for ff 2
[Z; W ] and f 2 [X; W ]. We are interested in the effect that fffhas on homology
and homotopy groups. This is described in the following result.
Proposition 2.1. For the above cofibration sequence, suppose f 2 [X; W ] and
ff 2 [Z; W ].
(1) The induced homology homomorphism (fff)*: Hi(X) ! Hi(W ) is given
by (fff)*(x) = f*(x) + ff*q*(x), for each x 2 Hi(X).
(2) Suppose that (f; ff): X _ Z ! W factors through the product X x Z.
Then the induced homotopy homomorphism (fff)# :ssi(X) ! ssi(W ) is
given by (fff)# (x) = f# (x) + ff# q# (x), for each x 2 ssi(X).
Proof. (1) This follows directly from the commutative diagram
(f_ff)* r*
Hi(X) ___c*__//Hi(X _ Z)_________//Hi(W _ W )______//Hi(W )
OOO ooo77
OOOO |(p1*;p2*)| (p1*;p2*)|| oooo
(1;q*)OO''OOfflffl| fflffl|o+ooo
Hi(X) Hi(Z) f*ff*//_Hi(W ) Hi(W )
in which the vertical maps are isomorphisms induced by the two projections p1 a*
*nd
p2 and the top row is the homomorphism induced by fff.
(2) Let oe :Si ! Si_ Si denote the standard comultiplication. Write f# (x) +
ff# q# (x) as the composition
(x_qx)oe (f;ff)
Si _____//X _ Z_____//W
and (fff)# (x) as the composition
_cx_//_ _(f;ff)//_
Si X _ Z W :
By hypothesis, we can factor (f; ff) through the product as (f; ff) = aOj :X _Z*
* !
X x Z ! W , for some a: X x Z ! W . It is straightforward to prove that
j(x _ qx)oe = jcx: Si ! X x Z, by checking that their projections onto each
summand are homotopic.
We now specialize to a mapping cone sequence of the form
fl j q n-1 n
Sn-1 _____//Y____//_X____//S S ;
i.e., X = Y [flen. Then we have an action of ssn(X) on [X; X]. We consider elem*
*ents
of the form ff2 [X; X], where is the identity map of X and ff 2 ssn(X). In
general, these maps are not self-homotopy equivalences. However, by adding cert*
*ain
hypotheses, we obtain maps in E# (X), or some other subgroup of E(X). This
approach is similar to that taken in [AL91 ], but instead of assuming j# :ss# (*
*Y ) !
ss# (X) is onto, we shall consider restrictions on the homotopy element ff 2 ss*
*n(X).
Recall that the n'th Gottlieb group of X, denoted Gn(X), consists of those ff 2
4 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
ssn(X) for which there is an associated map a: X xSn ! X such that the following
diagram commutes:
(;ff)
X _ Sn _____//X;;v:
vv
j|| vvavv
|fflfflvv
X x Sn
See [Got69 ] for various results on the groups Gn(X).
Next, we introduce another subgroup of E(X). Define
E*(X) = {f 2 E(X) | f* = 1: Hi(X) ! Hi(X); for all}i:
We apply Proposition 2.1 and obtain the following consequence.
Corollary 2.2. Let X = Y [flen be a 1-connected CW-complex and ff 2
ssn(X).
(1) ff2 E*(X) if and only if ff*q* = 0: Hn(X) ! Hn(X).
(2) Suppose ff 2 Gn(X), and X is of dimension N. Then ff2 E# (X) if and
only if ff# q# = 0: ssi(X) ! ssi(X) for i N.
Proof. It is immediate from Proposition 2.1 that ffinduces the identity on
homology groups. Since X is a 1-connected CW-complex, ffis a homotopy equiv-
alence. Hence ff2 E*(X). This establishes (1), and (2) follows similarly.
Hence, we are interested in finding situations in which ffq :X ! X induces
the trivial homomorphism, either in homology or homotopy. Our first result is an
integral result. Following this, we shall focus on the rational setting, where *
*more
information can be obtained.
Theorem 2.3. Let X = Y [flen be a 1-connected n-dimensional complex.
Suppose that q# = 0: ssn(X) ! ssn(Sn). Then there is a homomorphism
: Gn(X) ! E# (X);
defined by (ff) = fffor ff 2 Gn(X). This homomorphism restricts to
0:Gn(X) \ kerhn ! E*(X) \ E# (X);
where hn :ssn(X) ! Hn(X) denotes the Hurewicz homomorphism.
Proof. Let ff 2 Gn(X). Since ssi(Sn) = 0 for i < n, the hypothesis gives th*
*at
q# = 0: ssi(X) ! ssi(Sn) for i n. Hence, by Corollary 2.2, ff2 E# (X). Now
suppose fi is any element in ssn(X). Since ff2 E# (X) and fi 2 ssn(X), we have
that ff(fi) = fi. Thus, by the properties of the action listed in the introduct*
*ion, we
have ff ff
fffi= (ff) (fi)= ff+ (fi)= ff+fi:
Therefore is a homomorphism.
Now suppose ff is any element in kerhn. Then ff*: Hn(Sn) ! Hn(X) is zero.
Since Hi(Sn) = 0 for positive i 6= n, Corollary 2.2 implies that ff2 E*(X). Thus
restricts to 0as claimed.
Remark 2.4. It is known that Gn(X) kerhn under certain hypotheses (cf.
[Got69 , Th.4.1]), so the homomorphism and its restriction 0may agree.
We illustrate Theorem 2.3 with an example.
SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES 5
Example 2.5. Take X = S2 x S3 = S2 _ S3 [[i1;i2]e5. This kind of example
has been considered previously (cf. [AM98 , Saw75 ]), but here we put it into *
*the
context discussed above.
As is well-known, S3 is an H-space and therefore satisfies Gi(S3) = ssi(S3) *
*for
all i. Further, the Gottlieb group preserves products so G5(X) = G5(S2) G5(S3).
Since ss5(S3) = Z2, there is at least a non-trivial element of order 2 in G5(X).
Next, consider q# :ss5(X) ! ss5(S5). Since ss5(X) = ss5(S3) ss5(S2) is a fini*
*te
group and ss5(S5) is infinite cyclic, it follows that q# is zero in this dimens*
*ion.
From Theorem 2.3, defines a homomorphism from G5(X) to E# (X). Notice that
h5: ss5(X) ! H5(X) is zero, since H5(X) is infinite cyclic. Therefore, G5(X)
kerh5 and = 0:G5(X) ! E*(X) \ E# (X).
Note that the homomorphism in Theorem 2.3 may have trivial image in
E# (X). For Example 2.5, it follows from the computations in [AM98 , x6] that
is actually injective. However, it seems to be difficult to give general condit*
*ions to
guarantee that is injective. Rather than doing this by placing strong hypothes*
*es
on our spaces, we turn now to the rational setting. For a 1-connected CW-complex
X, the rational Gottlieb group of X is the Gottlieb group of the rationalizatio*
*n of X,
that is, Gn(XQ). Notice that by [Lan75 ], we have Gn(XQ) ~=Gn(X) Q for each
n. In contrast to the ordinary Gottlieb groups, much is known about the rational
Gottlieb groups by results of Felix-Halperin [FH82 ]. For instance, a 1-connec*
*ted,
finite complex has no non-trivial rational Gottlieb groups of even degree, and *
*has
only finitely many non-trivial rational Gottlieb groups of odd degree (see [Fel*
*89]
for details). We only touch on these ideas here and avoid heavy use of rational
techniques.
Lemma 2.6. Suppose we have a mapping cone sequence
fl j
Sn-1 ____//_Y____//X
in which Y and X are 1-connected. If flQ 6= *: (Sn-1)Q ! YQ, then (jQ)# :ssn(YQ)
! ssn(XQ) is surjective.
Proof. This can be argued using the long exact homotopy and homology
sequences, together with the relative Hurewicz theorem, for the pair (X; Y ). A*
*lter-
natively, Quillen minimal models can be used. We omit the details.
Remark 2.7. Notice we assert that (jQ)# is onto in degree n only, and not in
all degrees. In the latter case, the cell attachment is called an inert cell at*
*tachment
[HL87 ]. This is one of the hypotheses used in [AL91 ], but it is not satisfie*
*d by
some of the examples we have in mind.
We will see that under our hypotheses, rational equivalences of the form ffa*
*re
contained in a smaller subgroup of E(X) than E# (X). We introduce the following
notation: For r 1, define
E#r(X) = {f 2 E(X) | f# = 1: ssi(X) ! ssi(X); for all i }r:
Note that f 2 E#1 (X) if and only if f induces the identity homomorphism of all
homotopy groups.
The following is our basic rational result.
6 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
Theorem 2.8. Let X = Y [flen be a 1-connected CW complex with n odd and
flQ 6= *: (Sn-1)Q ! YQ. Then there is a homomorphism
: Gn(XQ) ! E#1 (XQ);
defined by (ff) = fffor ff 2 Gn(XQ). This homomorphism restricts to
0:Gn(XQ) \ kerhn ! E*(XQ) \ E#1 (XQ);
where hn denotes the rational Hurewicz homomorphism hn :ssn(XQ) ! Hn(XQ).
Proof. We proceed as in the proof of Theorem 2.3. First, we claim that
(qQ)# = 0: ssi(XQ) ! ssi(SnQ), for all i. Since n is odd, we have ssi(SnQ) = 0*
* for
i 6= n. Hence we must only check that (qQ)# = 0 in degree n. By Lemma 2.6,
(jQ)# :ssn(YQ) ! ssn(XQ) is surjective. Given x 2 ssn(XQ), write x = (jQ)# (y),
for some y 2 ssn(YQ). Then (qQ)# (x) = (qQ)# (jQ)# (y) = (qj)Q # (y) = 0 since
qj = *, and the claim follows.
Now a simple modification of the proof of Lemma 2.2 yields that ff2 E#1 (XQ),
for each ff 2 Gn(XQ). The remainder of the argument follows exactly as in the p*
*roof
of Theorem 2.3.
Remark 2.9. Notice that, unlike Theorem 2.3, there is no restriction on the
dimension of X in Theorem 2.8, and the attached cell need not be top-dimensiona*
*l.
If X is a 1-connected, finite complex, then there is no generality lost in assu*
*ming n
odd since a 1-connected, finite complex has no non-trivial rational Gottlieb gr*
*oups
of even degree.
Although Theorem 2.8 is a rational result, we are able to `de-rationalize' i*
*t to
obtain the following integral consequence.
Theorem 2.10. Let X = Y [flen be a 1-connected finite complex with n odd
and fl 2 ssn-1(Y ) not of finite order. If the homomorphism from Theorem 2.8 is
non-zero, then for each r with dimX r < 1, there are elements of infinite order
in E#r(X).
Proof. Suppose (ff) = ffis not the identity element in E#1 (XQ). Since
this latter is a Q-local group, it contains no non-trivial elements of finite o*
*rder, and
hence (ff)k 6= : XQ ! XQ, for all k. By [Mar89 ], we have E#r(XQ) ~= E#r(X) Q,
for each r with dimX r < 1. From this it follows that for each r there is some
positive integer p and some element f 2 E#r(X) such that fQ = (ff)p. Since fQ is
of infinite order in E#r(XQ), the same must be true of f in E#r(X).
3. Solvability of E# (Y )
A result of Dror-Zabrodsky asserts that if Y is a finite complex, then E# (Y*
* )
is a nilpotent group [DZ79 ]. One can ask, therefore, whether there are reason*
*able
estimates for the nilpotency, or perhaps the solvability, of E# (Y ) in terms o*
*f the
usual algebraic topological invariants of Y . Several results have been establi*
*shed
that relate the nilpotency or solvability of E# (Y ), or some similar group, to*
* the
Lusternik-Schnirelmann category of Y , or related invariants (cf. [AL96 , FM97 *
* ,
FM98 , ST99 ]). Some of these apply in a rational setting, and others in an in*
*tegral
setting. Typically, these results give an upper bound on the nilpotency or solv*
*ability
of the group.
We begin by discussing a topological invariant which appears in our results.
SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES 7
Definition 3.1. For any space X, a spherical cone decomposition of X of
length n, is a sequence of cofibrations
fli ji
Li ____//_Xi___//_Xi+1;
for 0 i < n, such that each Liis a finite wedge of spheres, X0 is contractible*
* and
Xn X. We define the spherical cone-length of X, denoted by scl(X), as follows:
If X is contractible, then set scl(X) = 0. Otherwise, scl(X) is the smallest po*
*sitive
integer n such that there exists a spherical cone decomposition of X of length *
*n.
If no such integer exists, set scl(X) = 1. If X is a finite-dimensional complex*
* and
we have a spherical cone decomposition of X of length n in which, in addition,
dimLi < dimX for i = 0; : :;:n - 1, then this is called a restricted spherical *
*cone
decomposition of X of length n. We then define the restricted spherical cone-le*
*ngth
of X, denoted rscl(X), using only restricted spherical cone decompositions in p*
*lace
of ordinary spherical cone decompositions.
Remark 3.2. Spherical cone-length has been considered in [Cor94 , ST99 ]. If
we denote the Lusternik-Schnirelmann category of X by cat(X), then it is known
that cat(X) scl(X). Note that a space X with scl(X) = 1 is homotopy equivalent
to a wedge of spheres and that a space X with scl(X) 2 is homotopy equivalent *
*to
the cofibre of a map between wedges of spheres. Furthermore, the cell-structure*
* of
a finite-dimensional complex X provides a restricted spherical cone decompositi*
*on
of length the number of dimensions in which there are positive-dimensional cel*
*ls.
We introduce a bit more notation before proving the main result of this sect*
*ion.
Once again, Y is a complex of dimension N. We define
Ek(Y ) = {f 2 E(Y ) | f* = 1: [X;[YX];!Y ]; for every complex X
with dim X N and rscl(X) k}:
In particular, we have E1(Y ) = E# (Y ). Also, there is a chain of subgroups
(1) E# (Y ) = E1(Y ) E2(Y ) . . .Ek(Y ) . .:.
Clearly we have Ek(Y ) / Ek-1(Y ): For if f 2 Ek(Y ), g 2 Ek-1(Y ), dimX N and
rscl(X) k, then f*g-1*= g-1*:[X; Y ] ! [X; Y ]. Hence
(gfg-1)* = g*f*g-1*= g*g-1*= 1;
and so gfg-1 2 Ek(Y ). Therefore, the series (1) is a normal chain. Furthermore*
*, if
rscl(Y ) k, then Ek(Y ) = 1. Then we have a normal chain
E# (Y ) E2(Y ) . . .Ek(Y ) = {1}:
The following is the main result of this section.
Theorem 3.3. The series (1) is a solvability series, i.e., [Ei(Y ); Ei(Y )]
Ei+1(Y ). Consequently, we have
solvE# (Y ) rscl(Y ) - 1:
Proof. Let f; g 2 Ei(Y ) and X be a complex with rscl(X) = i+1 and dim X
N. It suffices to show that f*g*(h) = g*f*(h) for every h 2 [X; Y ]. Consider t*
*he
last cofibre sequence in a length-(i + 1) restricted spherical cone decompositi*
*on of
X,
fl j
Li ____//_Xi___//_Xi+1;
8 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
where Li is a wedge of spheres and Xi+1 X. Now, since f 2 Ei(Y ), it follows
that j*(fh) = f*(hj) = j*(h). Thus, from the properties of the coaction reviewed
in Section 1, there is some ff 2 [Li; Y ] such that fh = hff. Similarly, there *
*is some
fi 2 [Li; Y ] such that g*(h) = hfi. Note also that ffi = fi since f 2 Ei(Y ) *
*E1(Y ),
and Li is a wedge of spheres of dimension N. Now we have
f*g*(h) = f(hfi) = (fh)ffi= (hff)fi= hff+fi:
A similar computation yields g*f*(h) = hfi+ff. Since [Li; Y ] is abelian, the p*
*roof
is complete.
Remark 3.4. A result analogous to Theorem 3.3 has been proved by Scheerer
and Tanre in [ST99 , Th.6]. We note the differences and similarities between th*
*ese
results. Theorem 6 in [ST99 ] is proved for the group of equivalences of a spac*
*e Y
relative to certain fixed classes of spaces (though our proof could be easily m*
*odified
to hold for these classes). When the class consists of wedges of spheres, the c*
*orre-
sponding group of equivalences is E#1 (Y ). The upper bound for the solvability*
* of
the group of equivalences E#1 (Y ) relative to the class of wedges of spheres g*
*iven in
[ST99 ] is then the so-called spherical category of Y , which is less than or e*
*qual to
the spherical cone-length of Y minus one. On the other hand, the group of equiv*
*a-
lences E# (Y ) that we consider in Theorem 3.3 is larger than E#1 (Y ). Further*
*more,
the two proofs are similar, but the solvability series in Theorem 3.3 appears t*
*o be
different from the one in [ST99 , Th.6].
Theorem 3.3 easily gives the next two corollaries.
Corollary 3.5. If rscl(Y ) 2, that is, Y is the cofibre of a map between
wedges of spheres, then E# (Y ) is abelian.
Corollary 3.6. For Y any finite-dimensional complex, E# (Y ) is solvable,
with solvE# (Y ) k - 1, where k is the number of dimensions in which there are
positive-dimensional cells.
Proof. This follows by Remark 3.2.
We can modify much of the previous material to deal with equivalences that
fix all homotopy groups. This also allows us to deal with the case in which Y i*
*s an
arbitrary space which is not necessarily a finite-dimensional complex. Define
E0k(Y ) = {f 2 E(Y ) | f* = 1: [X; Y ] ! [X; Y ]; for all X withscl(X) k}:
Then there is a normal chain
(2) E#1 (Y ) = E01(Y ) E02(Y ) . . .E0k(Y ) . .:.
Now the proof of Theorem 3.3 yields the following analogous results.
Theorem 3.7. [ST99 , Th. 6] The series (2) is a solvability series. Therefor*
*e,
solvE#1 (Y ) scl(Y ) - 1:
We note that from a spectral sequence of Didierjean [Did85 ], one can also
obtain a different upper bound on the solvability of E#1 (Y ) in terms of the c*
*oho-
mology of Y with coefficients in the homotopy groups of Y .
Corollary 3.8. If scl(Y ) 2, then E#1 (Y ) is abelian.
SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES 9
In view of the results in this section, together with the bounds found in [A*
*L96 ,
FM97 , FM98 , ST99 ], it is natural to believe that, for a finite-dimensional *
*complex
Y , the nilpotency class of E# (Y ) is bounded above by scl(Y )-1. We have not *
*been
able to prove this, and so we leave it as a conjecture.
Conjecture 3.9. For a finite-dimensional complex Y ,
nilE# (Y ) rscl(Y ) - 1 and nilE#1 (Y ) scl(Y ) - 1:
We note that Scheerer and Tanre have conjectured that nilE#1 (Y ) is bounded
above by the spherical category of Y [ST99 , x7, (6)].
Conjecture 3.9 would be established by showing that each of the series (1) a*
*nd
(2) is a nilpotency series. A direct proof of this would also give an independ*
*ent
proof of the Dror-Zabrodsky result on the nilpotency of E# (Y ).
4. Equivalences that Fix Cohomology Groups
In this section, we dualize some of the ideas of the previous section. Altho*
*ugh
we did not use homotopy groups with coefficients there, we do use coefficients *
*here,
since this is more common with cohomology.
Let G be a collection of abelian groups and X be a space. Define
E*G(X) = {f 2 E(X) | f* = 1: Hi(X; G) ! Hi(X; G); for all i and all G}2:G
The following cases are of special interest:
(1) G = {Z}. We write E*G(X) as E*(X) in this case.
(2) G = all cyclic groups. Then f 2 E*G(X) if and only if f 2 E(X) and
f* = 1: Hi(X; G) ! Hi(X; G) for every finitely-generated abelian group
G. We write E*G(X) as E*fg(X) in this case. Note that E*fg(X) E*(X).
Next we define a topological invariant that plays a role dual to that of sph*
*erical
cone-length in the previous section.
Definition 4.1. For G a collection of abelian groups, call an Eilenberg-Mac-
Lane space K(G; m), with G 2 G, a G-Eilenberg-MacLane space. For any space X,
a G-fibre decomposition of X of length n, is a sequence of fibrations
Xi+1 _ji__//Xi_pi_//Ki;
for 0 i < n, such that each Kiis a finite product of G-Eilenberg-MacLane space*
*s,
X0 is contractible and Xn X. We define the G-fibre-length of X, denoted by
G-fl(X), by dualizing Definition 3.1 in a straightforward way.
Note that a space X with G-fl(X) = 1 is homotopy equivalent to a product of
G-Eilenberg-MacLane spaces. A space X with G-fl(X) 2 is homotopy equivalent
to the fibre of a map between products of G-Eilenberg-MacLane spaces. Note also
that when we mention a product of G-Eilenberg-MacLane spaces, we allow factors
with homotopy groups in different dimensions, so that a product of G-Eilenberg-
MacLane spaces is not itself a G-Eilenberg-MacLane space in general.
Now define subgroups of E(X) as follows:
E*G;s(X) = {f 2 E(X) | f* = 1: [X; Y ] ! [X; Y ]; for all Y with G-fl(Y )}:s
Then there is a normal chain of subgroups
(3) E*G(X) = E*G;1(X) E*G;2(X) . . .E*G;s(X) . .:.
The proof of normality for (3) is similar to the proof of normality for (1) abo*
*ve.
10 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
A straightforward dualization of the proof of Theorem 3.3 yields the followi*
*ng
result.
Theorem 4.2. The series (3) is a solvability series. Thus
solvE*G(X) G-fl(X) - 1:
In Corollary 3.6, we showed that the number of dimensions in which Y has cel*
*ls
may be used to estimate the spherical cone length. We now indicate briefly how
the preceding notions can be modified for the dual result.
Definition 4.3. A space X is called a Postnikov piece if there is some N such
that ssi(X) = 0 for all i > N. The smallest such N is called the homotopical
dimension of X, and is denoted h-dimX.
For a Postnikov piece X, we define another subgroup of E(X) as
E*0G(X) = {f 2 E(X) | f* = 1: Hi(X; G) ! Hi(X;fG);or all i h-dimX
and all G 2}G:
We then define a restricted G-fibre decomposition of a Postnikov piece X, of le*
*ngth
n, as above but with the additional condition that h-dimKi h-dimX + 1, for all
i. This yields the restricted G-fibre-length of X, denoted by r-G-fl(X).
Remark 4.4. Let X be a 1-connected space and X(N) be the N'th Postnikov
section of X. Set G = all cyclic groups and s = the number of non-trivial homot*
*opy
groups of X(N). Then by taking the Postnikov decomposition of X(N), we see that
r-G-fl(X(N)) s.
Now if X is a Postnikov piece, define
E*0G;s(X) = {f 2 E(X)f|*= 1: [X; Y ] ! [X; Y ]; for all Postnikov pieces Y
such that h-dimY h-dimX and r-G-fl(Y ) s}:
Then once again we have a normal chain of subgroups
(4) E*0G(X) = E*0G;1(X) E*0G;2(X) . . .E*0G;s(X) . .:.
Here also, the proof of normality is similar to the proof of normality for (1) *
*above.
A further adaptation of the proof of Theorem 3.3 gives the following result.
Theorem 4.5. The series (4) is a solvability series. Thus ,
solvE*0G(X) r-G-fl(X) - 1;
for a Postnikov piece X.
This leads to the dual of Corollary 3.6.
Corollary 4.6. If X is a 1-connected finite complex, then E*fg(X) is solvabl*
*e.
In particular, if X has dimension N and there are s non-trivial homotopy groups
in degrees N, then solvE*fg(X) s - 1.
Proof. Any map f :X ! X induces a corresponding map (f): X(N) !
X(N) of N'th Postnikov sections. This gives us a homomorphism :E*G(X) !
E*0G(X(N)) which is one-one. Therefore, solvE*G(X) solvE*0G(X(N)). But if G
is the collection of all cyclic groups, then it is a consequence of Remark 4.4 *
*and
Theorem 4.5 that solvE*0G(X(N)) s - 1.
Next, we compare the subgroups E*(X) and E*fg(X). Below, we give a sim-
ple example to illustrate that these two subgroups are distinct in general. Fi*
*rst,
however, we obtain conditions under which they agree.
SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES 11
Proposition 4.7. For any space X, if
i+1 i
Hom Tor(H (X); G); H (X) G = 0
for all i and all finitely-generated groups G, then E*(X) = E*fg(X):
Proof. If f 2 E*(X), then the universal coefficient theorem (cf. [Spa66 ,
Thm. 5.10]) gives a commutative diagram with exact rows
0 ____//_Hi(X) G____//_Hi(X; G)_ss//_Tor(Hi+1(X); G)___//0
|1| |OE| |1|
|fflffl fflffl|ss fflffl|
0 ____//_Hi(X) G____//_Hi(X; G)___//_Tor(Hi+1(X); G)___//0
where the middle homomorphism OE can be either f* or the identity 1. Thus there
is a homomorphism ae: Tor(Hi+1(X); G) ! Hi(X) G such that f* - 1 = aess. By
hypothesis, ae = 0 and so f* = 1. Therefore, f 2 E*fg(X).
The following example illustrates that E*(X) and E*fg(X) may differ.
Example 4.8. Let X be a Moore space M(G; n), for n 2 and G any infi-
nite, finitely-generated abelian group with torsion. Then it follows from resul*
*ts of
[AM98 ] that E*(X) 6= E*fg(X)
In Corollary 4.6, we showed that for a 1-connected, finite-dimensional compl*
*ex
X, E*fg(X) is solvable. This raises the question of whether or not the group is
nilpotent. We conclude the paper by showing that E*(X), and therefore E*fg(X),
is a nilpotent group. We will use the following notation: Suppose that a group
G acts on an abelian group A. We define inductively a decreasing sequence of
subgroups of A by setting G1(A) = A and Gi(A) is the subgroup generated by
{ga - a | g 2 G; a 2 Gi-1(A)}. We say that the action is nilpotent if, for some*
* i,
Gi(A) = {0}.
Proposition 4.9. For any nilpotent finite complex X, E*(X) is a nilpotent
group.
Proof. We shall prove that E*(X) acts nilpotently on H*(X). Then it follows
from [DZ79 ] that E*(X) is a nilpotent group.
Let f 2 E*(X) and consider the diagram with exact rows obtained from the
universal coefficient theorem
0_____//Ext(H*-1(X); Z)___//_H*(X)____//Hom(H*(X); Z)____//0
Ext(f*;Z)|| f*=1|| |Hom(f*;Z)|
fflffl| fflffl| fflffl|
0_____//Ext(H*-1(X); Z)___//_H*(X)____//Hom(H*(X); Z)____//0:
Then both Ext(f*; Z) and Hom (f*; Z) are identity maps. Write H*(X) = F T as
the sum of its free and torsion parts and let pT :F T ! T and pF :F T ! F
be the projections. Since Ext(Z=m; Z) = Z=m, it follows that Ext(H*(X); Z) =
Ext(T; Z) = T . Thus Ext(f*; Z) = 1 implies that pT O f*|T :T ! T is the identi*
*ty.
In the same way, since Hom (f*; Z) = 1, we have that pF O f*|F :F ! F is also
the identity. Therefore, f*: F T ! F T can be written as f*(x;*y) = (x; y+OE(x)*
*),
with OE: F ! T a homomorphism that depends on f. Hence E1(X)(H*(X)) is
generated by elements of the form f*(x; y) - (x; y) = (0; OE(x)). On these elem*
*ents,
12 MARTIN ARKOWITZ, GREGORY LUPTON, AND ANICETO MURILLO
*(X)
any g 2 E*(X) satisfies g*(0; OE(x)) = (0; OE(x)). Hence E2 (H*(X)) = {0}, and
the result follows.
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Department of Mathematics, Dartmouth College, Hanover NH 03755 U. S. A.
E-mail address: Martin.Arkowitz@Dartmouth.edu
Department of Mathematics, Cleveland State University, Cleveland OH 44115
U. S. A.
E-mail address: Lupton@math.csuohio.edu
Departmento de Algebra, Geometria y Topologia, Universidad de Malaga, Ap. 59,
29080 Malaga, Spain
E-mail address: Aniceto@agt.cie.uma.es