HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F
MARTIN ARKOWITZ AND JEFFREY STROM
1
1. Introduction
A map f : X -! Y is said to be detected by a collection F of topological spa*
*ces if
there is a space A 2 F such that the induced map f* : [A; X] -! [A; Y ] of homo*
*topy
sets is nontrivial. It is a standard technique in homotopy theory to use certai*
*n simple
collections F to detect essential homotopy classes. In studying the entire homo*
*topy
set [X; Y ] using this approach, one is led naturally to consider the set of ho*
*motopy
classes which are not detected by F, called the F-trivial homotopy classes. For
example, if S is the collection of spheres, then f : X -! Y is detected by S p*
*recisely
when some induced homomorphism of homotopy groups ssk(f) : ssk(X) -! ssk(Y ) is
nonzero. The S-trivial homotopy classes are those that induce zero on all homot*
*opy
groups. It is also important to determine induced maps on homotopy sets. For th*
*is,
one needs to understand composition of F-trivial homotopy classes. With this in
mind, we study two basic questions in this paper for a fixed collection F: (1) *
*What is
the set of all F-trivial homotopy classes in [X; Y ]? and (2) In the special ca*
*se X = Y ,
how do F-trivial homotopy classes behave under composition? We are particularly
interested in the collections S of spheres, M of Moore spaces and of suspensio*
*ns.
We next briefly summarize the contents of this paper. We write ZF (X; Y ) for
the F-trivial homotopy classes in [X; Y ] and set ZF (X) = ZF (X; X). After so*
*me
generalities on ZF (X; Y ), we observe in Section 2 that ZF (X) is a semigroup *
*under
composition. Its nilpotency, denoted tF (X), is a new numerical invariant of ho*
*motopy
type. For the collection of suspensions, we prove that t (X) dlog2(cat(X))e. *
* In
Section 3 we relate tF (X) to other numerical invariants for arbitrary collecti*
*ons F.
The F-killing length of X, denoted klF(X) (resp., the F-cone length of X, denot*
*ed
clF(X)), is the least number of steps needed to go from X to a contractible spa*
*ce
(resp., from a contractible space to X) by successively attaching cones on wedg*
*es of
spaces in F. We prove that tF (X) klF(X), and, if F is closed under suspension,
that klF(X) clF(X). We also show that klF(X) behaves subadditively with respect
to cofibrations. It is clear that for any X and Y , Z (X; Y ) ZM (X; Y ) ZS(X*
*; Y ),
and we ask in Section 4 if these containments can be strict. It is easy to find*
* infinite
complexes with strict containment. However, in Section 4 we solve the more diff*
*icult
problem of finding finite complexes with this property. From this, we deduce t*
*hat
containment can be strict for finite complexes when X = Y . The next two sectio*
*ns
are devoted to determining ZF (X) and tF (X) for certain classes of spaces. In *
*Section
___________
1Math. Subject Classifications: Primary: 55Q05 Secondary: 55P65, 55P45, 55M30
1
2 MARTIN ARKOWITZ AND JEFFREY STROM
5 we calculate ZF (X) and tF (X) for F = S; M and when X is any real or complex
projective space, or is the quaternionic projective space HPn with n 4. In Sec*
*tion 6
we consider t (Y ) for certain Lie groups Y . We show that 2 t (Y ) when Y = S*
*U(n)
or Sp(n) by proving that the groups [Y; Y ] are not abelian. In addition, we co*
*mpute
t (SO(n)) for n = 3 and 4. The paper concludes in Section 7 with a list of open
problems.
For the remainder of this section, we give our notation and terminology. All *
*topo-
logical spaces are based and connected, and have the based homotopy type of CW
complexes. All maps and homotopies preserve base points. We do not usually di*
*s-
tinguish notationally between a map and its homotopy class. We let * denote the
base point of a space or a space consisting of a single point. In addition to s*
*tandard
notation, we use for same homotopy type, 0 2 [X; Y ] for the constant homotopy
class and id2 [X; X] for the identity homotopy class.
For an abelian group G and an integer n 2, we let M(G; n) denote the Moore
space of type (G; n), that is, the space with a single non-vanishing reduced ho*
*mology
group G in dimension n. If G is finitely generated, we also define M(G; 1) as a*
* wedge
of circles S1 and spaces obtained by attaching a 2-cell to S1 by a map of degre*
*e m.
The nthhomotopy group of X with coefficients in G is ssn(X; G) = [M(G; n); X]. A
map f : X -! Y induces a homomorphism ssn(f; G) : ssn(X; G) -! ssn(Y ; G), and
ss*(f; G) denotes the set of all such homomorphisms. If G = Z, we write ssn(X) *
*and
ssn(f) for the nthhomotopy group and induced map, respectively.
We use unreduced Lusternik-Schnirelmann category of a space X, denoted cat(X).
Thus cat(X) 2 if and only if X is a co-H-space. By an H-space, we mean a space
with a homotopy-associative multiplication and homotopy inverse, i.e., a group-*
*like
space.
For a positive integer n, the cyclic group of order n is denoted Z=n. If X is*
* a space
or an abelian group, we use the notation X(p)for the localization of X at the p*
*rime
p [12]. We let : X -! X(p)denote the natural map from X to its localization.
A semigroup is a set S with an associative binary operation, denoted by juxta-
position. We call S a pointed semigroup if there is an element 0 2 S such that
x0 = 0x = 0 for each x 2 S. A pointed semigroup is nilpotent if there is an int*
*eger n
such that the product x1. .x.nis 0 whenever x1; : :;:xn 2 S. The least such int*
*eger n
is thenilpotency of S. If S is not nilpotent, then we say its nilpotency is 1. *
*Finally,
if x is a real number, then dxe denotes the least integer n x.
2. F-Trivial Homotopy Classes
Let F be any collection of spaces.
Definition 2.1. A homotopy class f : X -! Y is F-trivial if the induced map
f* : [A; X] -! [A; Y ] is trivial for each A 2 F. We denote by ZF (X; Y ) the s*
*ubset of
[X; Y ] consisting of all F-trivial homotopy classes. We denote ZF (X; X) by ZF*
* (X).
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 3
We study ZF (X; Y ) and ZF (X) for certain collections F. The following are s*
*ome
interesting examples.
Examples. fi
(a)S = {Sn fin 1}, the collection of spheres. In this case f 2 ZS(X; Y ) if a*
*nd
only if ss*(f) =f0.i
(b)M = {M(Z=m; n) fim 0; n 1}, the collection of Moore spaces M(Z=m; n).
Here f 2 ZM (X; Y ) if and only if ss*(f; G) = 0 for any finitely generated*
* abelian
group G.
(c) = {A}, the collection of all suspensions. In this case f 2 Z (X; Y ) if and
only if f* = 0 : [A; X] -! [A; Y ] for every space A.
(d)P is the collection of all finite dimensional complexes. Then f 2 ZP (X; Y*
* ) if
and only if f : X -! Y is a phantom map [18].
In this paper our main interest is in the collections S, M and . In Section 7*
* we
will mention a few other collections. However, we begin with some simple, gene*
*ral
facts about arbitrary collections.
Lemma 2.2.
(a)If F F0, then ZF0(X; Y ) ZF (X; Y ) for any X andWY .
(b)If Xff2 F for each ff in some index set, then ZF ( Xff; Y ) = 0 for every *
*Y .
(c)For any X, ZF (X) is a pointed semigroup under the binary operation of comp*
*o-
sition of homotopy classes, and with zero the constant homotopy class.
__
Any map f : X -! Y gives rise to functions ef: X x Y -! X x Y and f : X _
Y -! X _ Y defined by the diagrams
ef _f
X x Y ____//_XOxOY and X _ Y ____//_XO_OY
p1|| |i2| q1|| j2||
fflffl|f | fflffl|f |
X _________//Y X ________//_Y:
__
Clearly, if f 2 ZF (X; Y ), then ef2 ZF (X x Y ) and f 2 ZF (X _ Y ).
The following lemma, whose proof is obvious, will be used frequently.
Lemma 2.3. The functions
: ZF (X; Y ) -! ZF (X x Y ) and OE : ZF (X; Y ) -! ZF (X _ Y );
__
defined by (f) = efand OE(f) = f are injective. Thus, ZF (X; Y ) 6= 0 implies *
*that
ZF (X x Y ) 6= 0 and ZF (X _ Y ) 6= 0
We conclude this section with some basic results about Z (X; Y ) and Z (X).
Recall that a map f : X -! Y has essential category weight at least n, written
E(f) n, if for every space A with cat(A) n, we have f* = 0 : [A; X] -! [A; Y ]
([29] and [24]).
4 MARTIN ARKOWITZ AND JEFFREY STROM
Lemma 2.4. For any two spaces X and Y ,
Z (X; Y ) = {f | f 2 [X; Y ]; E(f) 2}
= {f | f 2 [X; Y ]; f = 0}:
Proof
Let f 2 Z (X; Y ). If cat(A) 2 then the canonical map : A -! A has a section
s. Thus the diagram
f*
[A; X]_______//_[A; Y ]
* || *||
fflffl|f** fflffl|
[A; X] ____//_[A; Y ]
commutes, and so f* = s*f*** = 0. Since the reverse implication is trivial, th*
*is
establishes the first equality.
Now assume that f* = 0 : [B; X] -! [B; Y ] for every space B. Taking B = X,
we find that f O = 0 : X -! Y . Since this map is adjoint to f, we conclude
that f = 0. Conversely, if f = 0, then f* = 0 : [B; X] -! [B; Y ], which
means that f* = 0 : [B; X] -! [B; Y ]. This completes the proof. *
*||
Remarks.
(a)Since cat(A) 2 if and only if A is a co-H-space, a map f : X -! Y has
E(f) 2 if and only if f* = 0 : [A; X] -! [A; Y ] for every co-H-space A.
(b)By Lemma 2.4, we can regard the set Z (X; Y ) as the kernel of the looping
function : [X; Y ] -! [X; Y ]. We see from (a) that ker = 0 if X is a co-H-
space. The function has been extensively studied in special cases, e.g., w*
*hen
Y is an Eilenberg-MacLane space, then is just the cohomology suspension [3*
*0,
Chap.VII].
Proposition 2.5. Let X be a space of finite category, and let n log2(cat(X)).*
* If
f1; : :;:fn 2 Z (X), then f1O. .O.fn = 0. Thus the nilpotency of the semigroup *
*Z (X)
is at most dlog2(cat(X))e, the least integer greater than or equal to log2(cat(*
*X)).
Proof
Since fi 2 Z (X), Lemma 2.4 shows that E(fi) 2. By the product formula for
essential category weight [28,Thm.9], E(f1O. .O.fn) E(f1) . .E.(fn) 2n cat(X*
*).
From the definition of essential category weight, f1 O . .O.fn = 0. *
* ||
Remark. We shall see later that the semigroup ZS(X) is nilpotent if X is a fin*
*ite
dimensional complex. It follows that this is true for ZM (X) and Z (X) (Remark *
*(b)
following Theorem 3.3).
Definition 2.6. For any collection F of spaces and any space X, we define tF (X*
*),
the nilpotency of X mod F as follows: If X is contractible, set tF (X) = 0;
Otherwise, tF (X) is the nilpotency of the semigroup ZF (X).
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 5
Thus, tF (X) = 1 if and only if X is not contractible and ZF (X) = 0.
The set ZS(X) and the integer tS(X) were considered in [2], where they were
written Z1 (X) and t1 (X). Since S M , we have
0 t (X) tM (X) tS(X) 1
for any space X.
Since cat(A1 x . .x.Ar) r + 1 [15, Prop.2.3], we have the following result.
Corollary 2.7. For any r spaces A1; : :;:Ar,
t (A1 x . .x.Ar) log2(r + 1) :
This paper is devoted to a study of the sets ZF (X; Y ), with emphasis on the
nilpotency of spaces mod F for F = S; M and .
3. F-Killing Length and F-Cone Length
Proposition 2.5 shows that dlog2(cat(X))e is an upper bound for t (X). In th*
*is
section, we obtain upper bounds on tF (X) for arbitrary collections F. We begin*
* with
the main definitions of this section.
Definition 3.1. Let F be a collection of spaces and X a space. Suppose there is*
* a
sequence of cofibrations
Li- ! Xi- ! Xi+1
for 0 i < m such that each Liis a wedge of spaces which belong to F. If X0 X *
*and
Xm *, then this is called an F-killing length decomposition of X with length m.
If X0 * and Xm X, then this is an F-cone length decomposition with length
m. Define the F-killing length and the F-cone length of X, denoted by klF(X)
and clF(X), respectively, as follows. If X *, then klF(X) = 0; otherwise, klF(*
*X)
is the smallest integer m such that there exists an F-killing length decomposit*
*ion of
X with length m. The F-cone length of X is defined analogously.
The main result of this section is that klF(X) is an upper bound for tF (X). *
*We
need a lemma.
f g
Lemma 3.2. If X -! Y -! Z is a sequence of spaces and maps, then there is a
cofiber sequence of mapping cones Cf -! Cgf- ! Cg, where the maps are induced by
f and g.
The proof is elementary, and hence omitted.
Theorem 3.3. If F is any collection of spaces and X is any space, then
tF (X) klF(X):
If F is closed under suspensions, then klF(X) clF(X).
6 MARTIN ARKOWITZ AND JEFFREY STROM
Proof
Assume that klF(X) = m > 0 with F-killing length decomposition
fi pi
Li- ! Xi- ! Xi+1
for 0 i < m. Let g0; : :;:gm-1 2 ZF (X) and consider the following diagram, wi*
*th
dashed arrows to be inductively defined below:
f0 g0 g1 g2 gm-2 gm-1
L0 _____//_X0 X ____//_X____//_X___//_._._.//_X____//X;;v<<
z >>_ ">>
p0|| vv z z __ ""
f1 fflffl|g00vvzz _ _ ""
L1 ________//X1 zzg0 _ "
z 1 _ "
p1|| zz __ ""
f2 fflffl|z _ _ "
L2 ________//X2 _ ""
_ _ g0m-2"
p2|| _ ""
fflffl| _ " g0m-1
.. _ _ ""
. _ "
_ "
pm-2 || __ "
fm-1 fflffl|_ " "
Lm-1 ______//Xm-1 ""
"
pm-1 || ""
fflffl|"
Xm :
Since L0 is a wedge of members of F and g0 2 ZF (X), we have g0 O f0 = 0 by
Lemma 2.2(b). Thus there is a map g00: X1- ! X extending g0. The same argument
inductively defines g0ifor each i, and shows gm-1 . .g.1g0 = g0m-1O(pm-1 . .p.1*
*p0). Now
gm-1 . .g.1g0 = 0 since Xm *. This proves the first assertion.
Next we let m = clF(X), and show that klF(X) m. Let
fi pi
Li- ! Xi- ! Xi+1
for 0 i < m be an F-cone length decomposition of X. Set
hi= (pm-1 pm-2 . .p.i+1) O pi: Xi_______//_Xm X
for i < m and hm = id. Since hi= hi+1O pi, Lemma 3.2 yields cofiber sequences
Cpi-! Chi-! Chi+1;
for 0 i < m. This is a killing length decomposition of X. To see this, observe*
* that
Cpi Li, which is a wedge of spaces in F because F is closed under suspension.
Furthermore, h0 : X0 * -! X, so Ch0 Xm X. Finally, Chm 0 because
hm = id: X -! X. ||
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 7
Remarks.
(a)The notion of cone length has been extensively studied. The version in Defi*
*nition
3.1 is similar to the one given by Cornea in [6] (see (c) below). It is pre*
*cisely the
same as the definition of F-Cat given by Sheerer and Tanre [25]. The F-cone
length clF(X) can be regarded as the minimum number of steps needed to build
the space X up from a contractible space by attaching cones on wedges of sp*
*aces
in F. The notion of F-killing length is new and also appears in [2] for the*
* case
F = S. It can be regarded as the minimum number of steps needed to destroy
X (i.e. go from X to a contractible space) by attaching cones on wedges of
spaces in F. We note that Theorem 3.3 appears in [2, Thm. 3.4] for the case
F = S. For the collection S, it was shown in [2, Ex.6.8] that the inequalit*
*ies in
Theorem 3.3 can be strict.
(b)A space need not have a finite F killing length or F-cone length decomposit*
*ion.
For example, kl (CP1 ) = 1 because all 2n-fold cup products vanish in a spa*
*ce
X with kl (X) n. However, if X is a finite dimensional complex, then the
process of attaching i-cells to the (i-1)-skeleton provides X with a S-cone*
* length
decomposition. Thus in this case, klS(X) clS(X) dim(X). Since S M
, it follows that kl (X) klM(X) klS(X) and cl (X) clM(X) clS(X),
and so dim(X) is an upper bound for all of these integers. If X is a 1-conn*
*ected
finite dimensional complex, then a better upper bound for clM(X) is the num*
*ber
of nontrivial positive-dimensional integral homology groups of X. This can *
*be
seen by taking a homology decomposition of X [11, Chap.8].
(c)It follows from work of Cornea [6] that the cone length of a space X, denot*
*ed
cl(X), can be defined exactly like the -cone length cl (X) above, except th*
*at
one does not require L0 2 . It follows immediately that cl(X) cl (X).
(d)The inequality klF(X) clF(X) also follows from work of Sheerer and Tanre
since the function klF satisfies the axioms for F-Cat [25, Thm.2].
We conclude this section by giving a few properties of killing length.
j q
Theorem 3.4. If F is any collection of spaces and X -! Y -! Z is a cofiber *
*se-
quence, then
klF(Y ) klF(X) + klF(Z):
Proof
Write klF(X) = m and klF(Z) = n. Let
fi
Li- ! Xi- ! Xi+1
for 0 i < m be a F-killing length decompositiongoffX. Set g0 = j : X0 X -! Y
and define Y1 by the cofibration L0- ! Y -! Y1. 0By0Lemma 3.2, there is an aux*
*illiary
cofibration
Cf0_____//Cg0f0___//Cg0
|| || ||
|| || ||
|| g1 || ||
X1 ______//Y1_____//_Z
8 MARTIN ARKOWITZ AND JEFFREY STROM
which defines g1. We proceed by induction: given gi: Xi- ! Yi, let Yi+1be the c*
*ofiber
of the map gifi: L0- ! Yi and use Lemma 3.2 to construct an auxilliary cofibrat*
*ion
Cfi _____//Cgifi__//_Cgi
|| || ||
|| || ||
|| gi+1 || ||
Xi+1 _____//Yi+1____//_Z
which defines gi+1. This defines cofiber sequences of the form Lj- ! Yj- ! Yj+1*
* with
0 j < m. Since Xm *, the (m + 1)stcofiber sequence, Xm -! Ym -! Z, shows
that Ym Z. Now adjoin the n cofiber sequences of a minimal F-killing length
decomposition of Z to the first m cofiber sequences to obtain an F-killing leng*
*th
decomposition for Y with length n + m. ||
Finally, we obtain an upper bound for kl (X) and hence an upper bound for t (*
*X).
This provides a useful complement to Proposition 2.5 when cat(X) is not known.
Proposition 3.5. Let X be an N-dimensional complex which is (n - 1)-connected
for some n 1. Then ss
N + 1
kl (X) log2 ______ :
n
Proof
We argue by induction on log2 N+1_n . If log2 N+1_n = 1, then N 2n - 1. It is
well known that this implies that X is a suspension, which means that kl (X) = *
*1.
Now suppose log2 N+1_n = r and the result is known for all smaller values. Let*
* Xk
denote the k-skeleton of X, and consider the cofiber sequence
X2n-1- ! X -! X=X2n-1:
By Theorem 3.4, kl (X) kl (X2n-1) + kl (X=X2n-1). The inductive hypothesis
applies to X2n-1 and to X=X2n-1, so kl (X) 1 + (r - 1) = r. ||
4. Distinguishing ZF for Different F
We have a chain of pointed sets
Z (X; Y ) ZM (X; Y ) ZS(X; Y ):
Simple examples show that each of these containments can be strict. There are n*
*on-
trivial phantom maps CP1 - ! S4 [18]. Each of these lies in ZM (CP1 ; S4) becau*
*se
M P (see Examples in Section 2), but not in Z (CP1 ; S4), by Lemma 2.2(b). For
the other containment, the Bockstein applied to the fundamental cohomology clas*
*s of
M(Z=p; n) [3] corresponds to a map f : M(Z=p; n) -! K(Z=p; n + 1). If p is an o*
*dd
prime, then ssn+1(M(Z=p; n)) = 0 [3, pp.268-69], so f is in ZS(M(Z=p; n); K(Z=p*
*; n+
1)). Since it is essential, f cannot lie in ZM (M(Z=p; n); K(Z=p; n + 1)).
In these examples either the domain or the target is an infinite CW complex.
Thus they leave open the possibility that if X and Y are finite complexes, all *
*of the
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 9
pointed sets above are the same. We will give examples which show that, even f*
*or
finite complexes, these inclusions can be strict. These examples are more diffi*
*cult to
find and verify. They are inspired by an example (due to Fred Cohen) from [8].
Recall that if p is an odd prime, then S2n+1(p)is an H-space [1]. Moreover, i*
*f f is in
the abelian group f 2 [2X; S2n+1] then the order of O f 2 [2X; S2n+1(p)] is ei*
*ther
infinite or a power of p.
Lemma 4.1. Let X be a finite complex and let h : X -! S2n+1 be a map such th*
*at
for some odd prime p, O 2h is nonzero and has finite order divisible p. Then t*
*here
is an s > 0 such that the composite
X -h! S2n+1- i!M(Z=ps; 2n + 1)
is essential.
Proof
Consider the diagram
______________//_2n+1
S2n+1 S(p)
| |
ps|| ps|
h fflffl| fflffl|2n+1
X _________//_MMMS2n+1_________//_S(p)
MMM
MMM i| j|
iOhMMM&&Mfflffl|| fflffl||
M(Z=ps; 2n + 1) __=_//_M(Z=ps; 2n + 1)
in which the vertical sequences are cofibrations and ps denotes the map with de*
*gree
ps. If i O h = 0, then j O O h = 0. It can be shown that ( O h) lifts through *
*the map
ps : S2n+2(p)-!S2n+2(p). Suspending once more, we obtain a lift given by the da*
*shed line
in the diagram
lS2n+3(p)55
ll
l ll l ps||
l l l fflffl|
2n+3
2X _2h__//S2n+3___//_S(p) :
Since X is a finite complex, the torsion in [2X; S2n+3(p)] is p-torsion and has*
* an ex-
ponent e. Since S2n+3(p)is an H-space, the map ps induces multiplication by ps*
* on
[2X; S2n+3(p)]. If s e, then the image of ps : [2X; S2n+3(p)] -! [2X; S2n+3(p*
*)] cannot
contain any nontrivial torsion. But O 2h is nonzero and has finite order. Ther*
*efore
the lift cannot exist, and so i O h 6= 0. *
* ||
10 MARTIN ARKOWITZ AND JEFFREY STROM
Theorem 4.2. Let X be a finite complex, let p be an odd prime and let g : X -*
*! S2n+1
be an essential map.
(a)Assume that ss2n+1(X) is a finite group, and that O 2g is nonzero with fin*
*ite
order divisible by pn+1. Then there is an s > 0 such that the composite
________l___________________________________________*
*______________________________________________________________________@
_______________________________**___________________________*
*______________________________________________________________________@
X __g_//_S2n+1pn_//_S2n+1i__//M(Z=ps; 2n + 1)
is essential and ss*(l) = 0.
(b)Assume that ssk(X) = 0 for k = 2n and 2n + 1, and that O 2g is nonzero with
finite order divisible by p2n+1. Then there is an s > 0 such that the compo*
*site
f
____________________________________________________*
*______________________________________________________________________@
_______________________________**___________________________*
*______________________________________________________________________@
X __g_//_S2n+1p2n//_S2n+1i__//M(Z=ps; 2n + 1)
is essential, and ss*(f; G) = 0 for any finitely generated abelian group G.
Proof
In part (a), the composition O 2(pn O g) has finite order divisible by p. Ther*
*efore
Lemma 4.1 shows that l = i O pn O g is essential if s is large enough. Similarl*
*y, if s is
large enough, the map f in part (b) is essential. From now on, we assume that s*
* has
been so chosen. We use the commutative diagram
g pk i s
X ____//_CS2n+1__//_S2n+1___//M(Z=p ; 2n + 1)
CC | |
CCC | | |=
Og C!!Cfflffl| fflffl| fflffl||
pk 2n+1j
S2n+1(p)__//_S(p)___//_M(Z=ps; 2n + 1):
We take k = n in part (a) and k = 2n in part (b).
Proof of (a) Since M(Z=ps; 2n + 1) is p-local, there is only p-torsion to cons*
*ider.
By results of Cohen, Moore and Neisendorfer [5, Cor. 3.1], the p-torsion in ss**
*(S2n+1(p))
has exponent n. Since S2n+1(p)is an H-space, pn : S2n+1(p)-!S2n+1(p)annihilate*
*s all p-
torsion in homotopy groups. Thus ss*(l) can be nonzero only in dimension 2n + *
*1.
But ss2n+1(g) is a homomorphism from a finite group to Z, so ss*(l) = 0.
Proof of (b) It suffices to show that ssm (f; G) = 0 for any cyclic group G; b*
*y part
(a) we need only consider G = Z=pr. For each r 1 and each m 0, there is the
exact coefficient sequence [11, Chap.5]
0 -! Ext (Z=pr; ssm+1 (Y )) -! ssm (Y ; Z=pr) -! Hom (Z=pr; ssm (Y )) -!*
* 0:
Let Y = S2n+1(p). Since the p-torsion in ss*(S2n+1(p)) has exponent n [5], the *
*exact sequence
shows that the p-torsion in ssm (S2n+1(p); Z=pr) has exponent at most 2n if m 6*
*= 2n. Thus
the map p2n : S2n+1(p)-!S2n+1(p)induces 0 on the mthhomotopy groups with coeffi*
*cients
in any finite abelian group if m 6= 2n. Taking Y = X in the coefficient seque*
*nce,
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 11
we have ss2n(X; Z=pr) = 0. Therefore ss*(f; G) = 0 for any finitely generated a*
*belian
group G. ||
We apply this theorem to construct examples of finite complexes which disting*
*uish
the various ZF .
Our first example shows that ZM (X) can be different from ZS(X) even when X is
a finite complex. Using the coefficient exact sequence for homotopy groups, we *
*find
that
[M(Z=pr; 2n); S2n+1] = ss2n(S2n+1; Z=pr) ~=Z=pr
for each r; this is a stable group. Therefore, if r > n, there are essential m*
*aps
g : M(Z=pr; 2n) -! S2n+1 with finite order divisible by pn+1. Applying part (a*
*) of
Theorem 4.2, we have the following example.
Example. Let r > n > 1. For p an odd prime and s large enough, there are
essential maps
l : M(Z=pr; 2n) -! M(Z=ps; 2n + 1)
such that ss*(l) = 0. Therefore by Lemma 2.3,
r s
ZS M(Z=p ; 2n) _ M(Z=p ; 2n + 1) 6= 0
while, of course,
r s
ZM M(Z=p ; 2n) _ M(Z=p ; 2n + 1) = 0
by Lemma 2.2 (b). It can be shown that any s r will suffice in this example.
Freyd's generating hypothesis [9] is the conjecture that no stably nontrivial*
* map
between finite complexes can induce zero on stable homotopy groups. The map
l in this example is stably nontrivial, but our argument does not show that lar*
*ge
suspensions of l induce zero on homotopy groups; the difficulty is that after t*
*wo
suspensions, l factors through pn : S2n+3(p)-!S2n+3(p), which need not annihila*
*te all
p-torsion.
i 2n+1 j
Our second example is a map f : 2n-2 CPp =S2 -! M(Z=ps; 2n + 1) which
we use to show that ZM (X) can be different from Z (X) when X is a finite compl*
*ex.
We need some preliminary results to show that Theorem 4.2 applies to this situa*
*tion.
Lemma 4.3. Let f : n+1CPm -! Sn+3. The degree of f|n+1S2 is divisible by
lcm(1; : :;:m), the least common multiple of 1; : :;:m.
Proof
We may assume that f is in the stable range. If f|n+1S2 has degree d, then
f n+3 n+1 m
n+1CPm -! S ,! CP :
has degree d in Hn+3(n+1CPm ) and is trivial in all other dimensions. According*
* to
McGibbon [18, Thm.3.4], d is divisible by lcm(1; : :;:m). *
* ||
12 MARTIN ARKOWITZ AND JEFFREY STROM
Proposition 4.4. The image of the n-fold suspension map
t 2 3 n pt 2 n+3
n : [CPp =S ; S ] -! [ (CP =S ); S ]
contains elements of order pt for every n 1 and t 1.
Proof
Write m = pt and examine the commutative diagram
[CPm ; S3] _________//_[S2; S3]_________//[CPm =S2; S3]
|n| n|| |n|
fflffl| fflffl| fflffl|
[n+1CPm ; Sn+3] _____//[n+1S2; Sn+3]____//[n(CPm =S2); Sn+3]
|*| *|| |*|
fflffl| fflffl| fflffl|
[n+1CPm ; Sn+3(p)]___//[n+1S2; Sn+3(p)]//_[n(CPm =S2); Sn+3(p)]:
To show that the image of * O n : CPm =S2; S3] -! [n(CPm =S2); Sn+3(p)] contains
elements of order pt, we modify the above diagram as follows: the image and cok*
*er-
nel of [CPm ; S3] -! [S2; S3] ~= Z are kZ and Z=k, respectively, for some integ*
*er
k; similarly for [n+1CPm ; Sn+3] -! [n+1S2; Sn+3] and [CPm ; S3(p)] -! [S2; S3(*
*p)].
Thus we have a commutative diagram with exact rows
kZ _______//Z_____//Z=k
| |
| |~ |
| |= |
fflffl| fflffl| fflffl|
lZ _______//Z_____//Z=l
|| || ||
fflffl| fflffl| fflffl|
lpZ(p)____//Z(p)___//Z=lp
for some integers k, l and lp, where lp is the largest power of p which divides*
* l. Lemma
4.3 shows that lp is divisible by pt. The composite Z=k -! Z=l -! Z=lp is surje*
*ctive,
and this completes the proof. *
*||
It follows from Proposition 4.4 that part (b) of Theorem 4.2 applies to the s*
*pace
2n+1 2
2n-2(CPp =S ) for each n > 1, and so we obtain our second example.
Example. For each odd prime p and each n 1, there is an s > 0 such that there
are essential maps
i 2n+1 j
f : 2n-2 CPp =S2 -! M(Z=ps; 2n + 1)
which induce zero on homotopy groups with coefficients. Therefore,
i i 2n+1 j j
ZM 2n-2 CPp =S2 _ M (Z=ps; 2n + 1) 6= 0
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 13
while, of course,
i i 2n+1 j j
Z 2n-2 CPp =S2 _ M (Z=ps; 2n + 1) = 0:
The map f can be chosen to be stably nontrivial. As in the previous example, the
suspensions of f might not be trivial on homotopy groups with coefficients.
2n+1 2 r s
Finally, let A = 2n-2(CPp =S ), B = M(Z=p ; 2n) and C = M(Z=p ; 2n + 1)
for s large. Then
Z (A _ B _ C) < ZM (A _ B _ C) < ZS(A _ B _ C);
so both of these inequalities can be strict for a single finite complex.
5. Projective Spaces
We show that for projective spaces FPn with F = R; C or H,
Z (FPn) = ZM (FPn) = ZS(FPn);
and we completely determine these sets for F = R and C and all n. We also deter*
*mine
tS(HPn), for n 4.
5.1. General Facts. We first prove some general results that will be applied la*
*ter.
W
Proposition 5.1. If X Snff, then ZS(X; Y ) = ZM (X; Y ) = Z (X; Y ) for
any space Y .
Proof f
Let f 2 ZS(X;WY ). The map f is adjoint to the composition X -! X -! Y .
Since X Snff, f O = 0, and so f = 0. Thus f 2 Z (X; Y ). ||
By Lemma 2.4, the condition ZS(X; Y ) = Z (X; Y ) is equivalent to the condit*
*ion
that if f : X -! Y induces zero on homotopy groups, then f = 0.
W Proposition 5.1 applies to X = Sn+1 because, according to James [13], Sn+1
1 nk+1 n+1
k=1S . Of course ZS(S ; Y ) = 0. Since (A x B) A _ B _ (A ^ B)
for any A and B [11, 11.10], James's result allows us to apply Proposition 5.1 *
*to any
space whose loop space splits as a finite type product of spheres and loop spac*
*es on
spheres. Moreover, if X and X0 both satisfy the hypotheses of Proposition 5.1, *
*then
so does X x X0.
For F = R; C or H, let d = 1; 2 or 4, respectively. For each n 1 there is a
homotopy equivalence FPn Sd-1 x S(n+1)d-1[10].
Corollary 5.2. For F = R; C or H and each n 1, ZS(FPn) = ZM (FPn) =
Z (FPn).
14 MARTIN ARKOWITZ AND JEFFREY STROM
Another corollary of Proposition 5.1 applies to intermediate wedges of sphere*
*s. For
spaces X1; X2; : :;:Xn, the elements (x1; : :;:xk) 2 X1 x . .x.Xk with at least*
* j
coordinates equal to the base point form a subspace Tj(X1; : :;:Xk) X1x . .x.X*
*k.
Porter has shown [23, Thm. 2] that Tj(Sn1; : :;:Snk) has the homotopy type of
a product of loop spaces of spheres for each 0 j k. Our previous discussion
establishes the following.
Corollary 5.3. For any n1; : :;:nk 1 and any 0 j k,
ZS(Tj(Sn1; : :;:Snk)) = ZM (Tj(Sn1; : :;:Snk)) = Z (Tj(Sn1; : :;:Snk)):
Remarks.
(a)Taking j = 0 in Corollary 5.3, we deduce from Corollary 2.7 that
tS(Sn1 x . .x.Snk) log2(k + 1) :
This reproves [2, Prop.6.2] by a different method.
(b)It is proved in [2, Prop. 6.5] that for any positive integer n, there is a *
*finite
product of spheres X with tS(X) = n. By Corollary 5.3, the same is true for
t (X) and tM (X). Thus the integers tF (X) for F = S; M or and any X take
on all positive integer values.
Finally, we observe that the splitting of FPn gives a useful criterion for de*
*ciding
when a map f : FPn -! Y lies in ZS(FPn; Y ).
Proposition 5.4. Let i be the inclusion Sd = FP1 ,! FPn, and let p : S(n+1)d-1*
*-! FPn
be the Hopf fiber map. Then the map (i; p) : Sd _ S(n+1)d-1-! FPn induces a sur*
*jec-
tion on homotopy groups. Therefore, a map f : FPn -! Y satisfies ss*(f) = 0 if *
*and
only if f O i = 0 and f O p = 0.
5.2. Complex Projective Spaces. Next we show that certain skeleta X of Eilenber*
*g-
MacLane spaces have the property that ZS(X) = 0. We apply this to CPn and nCP2
for each n.
Let G be a finitely generated abelian group. Give the Eilenberg-MacLane space
K(G; n) with n 2 a homology decomposition [11, Chap. 8] and denote the mth
section by K(G; n)m . Thus K(G; n) is filtered
* K(G; n)n . . .K(G; n)m . . .K(G; n)
and there are cofiber sequences M(Hm+1 (K(G; n)); m) -! K(G; n)m -! K(G; n)m+1*
* .
Theorem 5.5. If the group Hm (K(G; n)) is torsion free and Hm+1 (K(G; n)) = 0,
then ZS(K(G; n)m ) = 0.
Proof
We write X = K(G; n)m . Then Hk(K(G; n); X) = 0 for k m + 1. By Whitehead's
theorem [30, Thm. 7.13], the induced map ssk(X) -! ssk(K(G; n)) is an isomorphi*
*sm
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 15
for k m. Since Hm (K(G; n)) is torsion free, X has dimension at most m, and so
X has a CW decomposition
_
Sn = Xn Xn+1 . . .Xm X:
For f 2 ZS(X) we prove by induction on k that f factors through X=Xk for each
k m. The first step is trivial since ssn(f) = 0 implies f|Xn = 0. Inductively,*
* assume
that f factors through X=Xk with n k < m. There is a cofibration
_
Sk+1 Xk+1=Xk -! X=Xk -! X=Xk+1:
Since n < k + 1 m, it follows that ssk+1(X) ~=ssk+1(K(G; n)) = 0, so f extends*
* to
X=Xk+1. Taking k = m, we find f = 0. ||
Remark. Clearly, ssk(K(G; n)m ) = 0 for n < k < m. The hypotheses in Theorem
5.5 are needed to conclude further that ssm (K(G; n)m ) = 0.
As an application of Theorem 5.5, we have the following calculations.
Theorem 5.6.
(a)ZF (CPn) = 0 for each n 1 and each F = ; M or S.
(b)ZF (nCP2) = 0 for each n 1 and each F = ; M or S.
Proof
By Proposition 5.1 it suffices to consider the case F = S. Since CP1 = K(Z; 2) *
*and
the CPn are the sections of a homology decomposition of CP1 , part (a) follows *
*from
Theorem 5.5. Recall from [7] that for n 2
ae
Z if k = n or n + 2
Hk(K(Z; n)) = 0 if k = n + 1 or n + 3.
Since Sq2 is nontrivial on Hn(K(Z; n); Z=2), we have K(Z; n)n+2 n-2CP2. Thus
Theorem 5.5 applies to n-2CP2. ||
This theorem immediately shows that tF (CPn) = tF (nCP2) = 1 for F = ; M
or S and each n 1.
5.3. Real Projective Spaces. In this subsection we completely calculate ZS(RPn).
By the Hopf-Whitney theorem [30, Cor. 6.19], [RP2n; S2n] ~=H2n(RP2n) ~=Z=2. The
unique nontrivial map q : RP2n- ! S2n is theqquotientpmap obtained by factoring*
* out
RP2n-1. Let f2n denote the composite RP2n- ! S2n -! RP2n where p is the univers*
*al
covering map.
Theorem 5.7. For F = ; M or S and each n 1,
(a)ZF (RP2n-1) = 0
(b)ZF (RP2n) = {0; f2n}.
16 MARTIN ARKOWITZ AND JEFFREY STROM
Proof
Let f : RPm -! RPm with ss1(f) = 0. Because ssk(RPm ) = 0 for 1 < k < m,
an argument similar to the proof of Theorem 5.5 shows that f must factor through
q : RPm -! Sm . For m > 1, any map Sm -! RPm lifts through p : Sm -! RPm .
Thus there is a map g : Sm -! Sm of degree d which makes the following diagram
commute
p f
Sm G____//_GRPm___//_RPmOO
GGG q| p|
qOpGG##Gfflffl|| |
g |
Sm _____//_Sm :
First let m = 2n - 1. We may assume n > 1. The composite q O p : S2n-1- ! S2n*
*-1
is known to have degree 2. Since ssi(p) is an isomorphism for i > 1, f O p repr*
*esents
2d 2 Z ~=ss2n-1(RP2n-1). If f 2 ZS(RP2n-1), then d must be 0, and so f = 0. This
proves (a).
Now take m = 2n. The composite q O p : S2n -! S2n is trivial because it is ze*
*ro
on homology. Therefore f O p = 0, and since ss1(f) = 0, Proposition 5.4 shows t*
*hat
f 2 ZS(RP2n). Since RP2n is connected, there is a bijection
~= fi
p* : [RP2n; S2n]________//f fif 2 [RP2n; RP2n]; ss1(f) = 0 = ZS(RP2n):
Since [RP2n; S2n] = {0; q} as noted above, ZS(RP2n) = {0; f2n}, where f2n = p O*
* q. ||
Remark. This argument actually shows that, if ssk(Y ) = 0 for 1 < k < 2n+1, th*
*ere
is a bijection between ZS(RP2n+1; Y ) and the set of elements ff 2 ss2n+1(Y ) s*
*uch that
2ff = 0.
Corollary 5.8. For each n 1,
(a)tF (RP2n-1) = 1 for F = ; M and S.
(b)tF (RP2n) = 2 for F = ; M and S.
Proof
It suffices to prove part (b) for F = S. Since ZS(RP2n) 6= 0, tS(RP2n) 2. The
only possibly nonzero product in this semigroup is f2nO f2n. But this is zero b*
*ecause
ZS(RP2n) is nilpotent by Theorem 3.3. ||
5.4. Quaternionic Projective Spaces. The quaternionic projective spaces are not
skeleta of Eilenberg-MacLane spaces, and it is much more difficult to compute t*
*heir
nilpotency.
Let f 2 [HPn+1; HPn+1], and assume that f is cellular. Then f|HPn : HPn -! HPn
and the homotopy class f|HPn is well defined.
Lemma 5.9. If f 2 ZS(HPn+1), then f|HPn 2 ZS(HPn).
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 17
Proof
Let f 2 ZS(HPn+1) and let g = f|HPn. Consider the diagram
_ __ h___ _//4n+3
S4n+3 S
p || |p|
i fflffl| g fflffl|
S4 _____//_HPn___________//HPnJ
JJJ
j || j|| JlJJJ
fflffl| f fflffl|JJ$$m
HPn+1 __________//HPn+1 ____//_HP1
q || |q|
fflffl|h fflffl|
S4n+4 _ _ _ _ _ //_S4n+4:
p n l 1
where i; j; m and l are inclusions. Since S4n+3- ! HP -! HP can be regarded *
*as
a fibration and l O (g O p) = m O (f O (j O p)) = 0, it follows that g O p lift*
*s to the
map h. Since f 2 ZS(HPn+1), f induces zero on H4(HPn+1), and hence is zero in
cohomology. Therefore h is zero in cohomology and hence is trivial. Thus h = 0,*
* so
g O p = 0. Also, g O i = 0, so g 2 ZS(HPn) by Proposition 5.4. *
* ||
Next we indicate how we will apply Lemma 5.9. If ZS(HPn) = 0 and f 2
ZS(HPn+1), then f|HPn = 0, so f factors through q : HPn+1 -! S4n+4. By Proposi-
tion 5.4, if i : S4 ,! HPn+1, then ss4n+4(i) is surjective, so f factors as in *
*the diagram
f m 1
HPn+1 _____//HPn+1____//_HPOO::t
tt
q || |i| tttt
fflffl|g | ttt l
S4n+4 ______//_S4:
By cellular approximation, f is essential if and only if m O f is essential. T*
*he map
l O g : S4n+4- ! HP1 is adjoint to a map g0 : S4n+3- ! S3, which in turn is ad*
*joint
to l O g0. By cellular approximation again, i O g = i O g0, so we may assume th*
*at g
is in the image of the suspension : ss4n+3(S3) -! ss4n+4(S4).
The proof of our main result about quaternionic projective spaces requires so*
*me
detailed information about homotopy groups of spheres. Since we refer to Toda's*
* book
[30] for this information, we use his notation here. For example, jk : Sk+1 -! *
*Sk and
k : Sk+3 -! Sk are suspensions of the Hopf fiber maps.
Theorem 5.10.
(a)ZF (HPn) = 0 for F = S; M or and n = 1; 2 and 3
(b)ZF (HP4) 6= 0 for F = S; M or .
18 MARTIN ARKOWITZ AND JEFFREY STROM
Proof
First HP1 = S4, so ZS(HP1) = 0. If f 2 ZS(HP2), then there is a commutative
diagram
f m 1
HP2 _____//HP2____//_HPOO;;v
vv
q || |i| vvv
fflffl|g | vvv l
S8 _____//_S4;;w
ww
5 || www
fflffl|j4www
S5
in which the vertical sequence is a cofibration. If g = 0, then f = 0, so we m*
*ay
assume that g 6= 0. We know that ss8(HP1 ) ~=ss7(S3) ~=Z=2, generated by j3O4 [*
*29,
p.43-44]. Thus we can take g = j4 O 5. Since 5 O q = 0, we conclude that g O q *
*= 0,
so f = 0. This shows that ZF (HP2) = 0.
The proof that ZS(HP3) = 0 is similar. Let f 2 ZS(HP3) and apply Lemma 5.9
to get a similar factorization. The resulting map g : S12- ! S4 is either ffl3*
* or 0
[29, Thm. 7.1]. If g = ffl3, then results of [14, (2.20a)] and [29, Thm. 7.4] s*
*how that
f O p 6= 0. Thus g = 0 and so f = 0.
For part (b), we make use of the diagram preceding Theorem 5.10 and the fact *
*that
g can be taken to be a suspension map. If f 2 ZS(HP4), then we have
p f m 1
S19 ____//_EHP4___//_HP4___//_HPOO;;
EE | | vvv
EEE q| |i vvvv
E""Efflffl|g| vv l
S16 _____//_S4:
According to Toda [30], ss16(S4) = (ss15(S3)) ~=Z=2 Z=2. By [14, (2.20a)], q O*
* p is
4162 ss19(S16). Then g O (416) = 4g O 16 because 16 is a suspension. Since 4g =*
* 0,
any map HP4- ! HP4 which factors through q lies in ZS(HP4). Marcum and Randall
show in [17] that the map
(i O 0) O 7) O q : HP4__________//HP4
is essential, where 0 2 ss6(S3) generates the 2-torsion and 7 2 ss16(S7) genera*
*tes a
Z=2 summand [29, Thm.7.2]. Thus ZS(HP4) 6= 0, and so tS(HP4) 2. ||
As before, we obtain the nilpotency.
Corollary 5.11.
(a)tF (HP1) = tF (HP2) = tF (HP3) = 1 for F = S; M or
(b)tF (HP4) = 2 for F = S; M or .
Proof
It suffices to prove that tS(HP4) 2. Suppose f; g 2 ZS(HPn). The proof of Theo*
*rem
5.10 shows that f factors through S16. Now g O f = 0 because g 2 ZS(HP4). *
*||
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 19
6. H-Spaces
In this section we study the nilpotency of H-spaces Y mod . We make calculati*
*ons
for specific Lie groups such as SU(n) and Sp(n) and show that Z is nontrivial *
*in
these cases. If Y is an H-space, the Samelson product of ff 2 ssm (Y ) and fi 2*
* ssn(Y )
is written 2 ssn+m (Y ) [30, Chap.X].
We first give a few general results which are needed later.
Lemma 6.1. If Y is an H-space and 6= 0 for some ff 2 ssm (Y ) and fi*
* 2 ssn(Y ),
then [Sm x Sn; Y ] is not abelian.
Proof
The quotient map q : Sm x Sn -! Sm ^ Sn Sm+n induces a monomorphism q* :
[Sm ^ Sn; Y ] -! [Sm x Sn; Y ] such that q* = [ffp1; fip2], the commuta*
*tor of ffp1
and ffp2. ||
It is well known that if an H-space Y is a finite complex, then it has the s*
*ame
rational homotopy type as a product of spheres S2n1-1x . .x.S2nr-1with n1 . . .
nr. If p is an odd prime such that
2n -1 2n -1 2n -1 2n -1
Y(p) S 1 x . .x.S r (p) S(p1) x . .x.S(pr) ;
then p is called a regular prime for Y . If Y is a simply-connected compact L*
*ie
group, then p is regular for Y if and only if p nr [16, Sec.9-2].
We need a second product decomposition for p-localized Lie groups. By [20, Se*
*c.2]
there are fibrations S2k+1- ! Bk(p) -! S2k+2p-1for k = 1; 2; : :.:An odd prime *
*p is
called quasi-regular for the H-space Y if
!
Y Y
Y(p) S2ni-1x Bmj(p) :
i j (p)
6.1. The Groups SU(n) and Sp(n). We apply the notions of regular and quasi-
regular primes to the Lie group SU(n), which has the rational homotopy type of
S3 x S5 x . .x.S2n-1, and to the Lie group Sp(n), which has the rational homoto*
*py
type of S3 x S7 x . .x.S4n-1. It is well known [20, Thm.4.2] that if p is an od*
*d prime
then
(a)p is regular for SU(n) if and only if p n; p is quasi-regular for SU(n) if*
* and
only if p > n_2
(b)p is regular for Sp(n) if and only if p 2n; p is quasi-regular for Sp(n) i*
*f and
only if p > n.
It is also known [4, Thm. 1] that if n 3 and r + s + 1 = n, there are generato*
*rs
ff 2 ss2r+1(SU(n)) ~=Z, fi 2 ss2s+1(SU(n)) ~=Z and fl 2 ss2n(SU(n)) ~=Z=n! such*
* that
= r!s!fl. If p is an odd prime and ff02 ss2r+1(SU(n)(p)), fi0 2 ss2s+1*
*(SU(n)(p))
and fl0 2 ss2n(SU(n)(p)) are the images of ff, fi and fl under the localization*
* homo-
morphism * : ss*(SU(n)) -! ss*(SU(n)(p)) ~=ss*(SU(n))(p), then
= r!s!fl02 ss2n(SU(n))(p)~=Z=n! Z(p):
20 MARTIN ARKOWITZ AND JEFFREY STROM
Now we prove the main result of this section.
Theorem 6.2. The groups
(a)[SU(n); SU(n)] for n 5 and
(b)[Sp(n); Sp(n)] for n 2
are not abelian.
Proof
Consider SU(n) for n 5 and let p be the largest prime such that n_2< p < n. If
n 12, then it follows from Bertrand's postulate [26,p.137] that there are two *
*primes
p and q such that n_2< q < p < n. This implies that 2n + 6 < 4p. For 5 n < 12,
and n 6= 5; 7; 11, it is easily verified that 2n + 6 < 4p.
Assume that n 5 and that n 6= 5; 7 or 11. Since p > n_2, it follows that p *
*is
quasi-regular for SU(n). Since 2n + 6 < 4p, the spheres S2n-2p+3 and S2p-3 both
appear in the resulting product decomposition. Thus we have
2n-2p+3 2p-3 2p-1
SU(n)(p) B1(p) x . .x.Bn-p(p) x S x . .x.S x S (p):
Assume [SU(n); SU(n)] is abelian. Then [SU(n)(p); SU(n)(p)] is abelian, and the*
*refore
[S2n-2p+3x S2p-3; SU(n)(p)] is abelian.
There are ff02 ss2n-2p+3(SU(n)(p)), fi02 ss2p-3(SU(n)(p)) and fl02 ss2n(SU(n)*
*(p)) so
that
= (n - p + 1)!(p - 2)!fl0
in ss2n(SU(n)(p)) ~= Z=n! Z(p)~= Z=p. Since fl0 is a generator of Z=p, we have
6= 0. By Lemma 6.1, [S2n-2p+3x S2p-3; SU(n)(p)] is not abelian, and*
* so
[SU(n); SU(n)] is not abelian.
It remains to prove that [SU(n); SU(n)] is not abelian for n = 5; 7; 11. The *
*argu-
ment we now give applies to SU(p) for any prime p 5. Notice that p is regular *
*for
SU(p), so it suffices to show that [S3 x S2p-3; SU(p)(p)] is nonabelian. Since*
* p is a
regular prime for SU(p), we choose generators ff 2 ss3(SU(p)), fi 2 ss2p-3(SU(p*
*)) and
fl 2 ss2p(SU(p)) so that
= (p - 2)!fl06= 0 2 Z=p! Z(p)~=Z=p:
Therefore [S3 x S2p-3; SU(p)(p)] is nonabelian by Lemma 6.1.
The proof that [Sp(n); Sp(n)] is not abelian for n 2 is analogous: one uses *
*Bott's
result for Samelson products in ss*(Sp(n)) [4, Thm. 2] together with a quasi-re*
*gular
decomposition for Sp(n). We omit the details. *
*||
Corollary 6.3.
(a)For n 5, Z (SU(n)) 6= 0, and 2 t (SU(n)) dlog2(n)e.
(b)For n 2, Z (Sp(n)) 6= 0, and 2 t (Sp(n)) d2 log2(n + 1)e.
Proof
For an H-space Y , a commutator in [X; Y ] is an element of Z (X; Y ) [28, Thm.*
* 7].
Thus, if [Y; Y ] is nonabelian, 2 t (Y ). The upper bound for t (SU(n)) follo*
*ws
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 21
from Proposition 2.5 since Singhof has shown that cat(SU(n)) = n [28]. The upper
bound on t (Sp(n)) follows from Proposition 3.5. *
*||
Schweitzer [25,Ex.4.4] has shown that cat(Sp(2)) = 4, so it follows from Prop*
*osition
2.5 that t (Sp(2)) = 2.
6.2. Some Low Dimensional Lie Groups. Here we consider the Lie groups SU(3),
SU(4), SO(3) and SO(4) and make estimates of t by either quoting known results
or by ad hoc methods. We first deal with SU(3) and SU(4).
Proposition 6.4. The groups [SU(3); SU(3)] and [SU(4); SU(4)] are not abelian.
Proof
For the group SU(3) this follows from results of Ooshima [21, Thm. 1.2] For SU(*
*4),
observe that the prime 5 is regular for both SU(4) and Sp(2), so
SU(4)(5) (S3 x S5 x S7)(5) and Sp(2)(5) (S3 x S7)(5):
If [SU(4); SU(4)] is abelian, then so is [SU(4)(5); SU(4)(5)] ~=[S3 x S5 x S7; *
*SU(4)(5)],
and thus [S3 x S7; SU(4)(5)] is abelian.
If ff02 ss3(Sp(2)(5)) and fi02 ss7(Sp(2)(5)) are the images of generators of *
*ss3(Sp(2)) ~=
Z and ss7(Sp(2)) ~= Z then it follows from [4] that 6= 0 2 ss10(Sp(2*
*)(5)) ~=
Z=5! Z(5)~=Z=5.
Now we relate SU(4) to Sp(2) via the fibration Sp(2) -i!SU(4) -! S5. The exact
homotopy sequence of a fibration shows that ss10(i) is an isomorphism after loc*
*alizing
at any odd prime. Since i is an H-map,
* = i* 6= 0 2 ss10(SU(4)(5)):
Thus [S3 x S7; SU(4)(5)] is not abelian, so [SU(4); SU(4)] cannot be abelian. *
* ||
Corollary 6.5.
(a)Z (SU(3)) 6= 0, and t (SU(3)) = 2
(b)Z (SU(4)) 6= 0, and t (SU(4)) = 2.
Proof
Since the groups [SU(n); SU(n)] are not abelian for n = 3 and 4, t (SU(3)) and
t (SU(4)) are at least 2. But cat(SU(n)) = n by [28], so the reverse inequalit*
*ies
follow from Proposition 2.5. *
*||
Next we investigate the nilpotence of SO(3) and SO(4). This provides us with
examples of non-simply-connected Lie groups.
Proposition 6.6. Z (SO(3)) = 0 and Z (SO(4)) 6= 0.
Proof
Since SO(3) is homeomorphic to RP3, the first assertion follows from Theorem 5.*
*7.
For the second assertion, recall that SO(4) is homeomorphic to S3 x SO(3). For
notational convenience, we write X = SO(3) and Y = S3. We show that Z (XxY ) 6=
22 MARTIN ARKOWITZ AND JEFFREY STROM
0. Let j : X _ Y -! X x Y be the inclusion and q : X x Y -! X ^ Y be the quot*
*ient
map. Consider
q* : [X ^ Y; X x Y ] -! [X x Y; X x Y ]:
Notice that Im(q*) Z (X x Y ) because q 2 Z (X x Y; X ^ Y ), so q induces a
function q** : [X ^ Y; X x Y ] -! Z (X x Y ). Consider the exact sequence of gr*
*oups
j* q*
[(X x Y ); X x Y ] -! [(X _ Y ); X x Y ] -! [X ^ Y; X x Y ] -! [X x Y; X x Y ]:
Since j* has a left inverse, ker(q*) = 0. Thus q** is one-one, so it suffices t*
*o show
that [X ^ Y; X x Y ] ~= [3SO(3); SO(3)] [3SO(3); S3] is nonzero. This follows
from [31, Cor.2.12], where it is shown that [3SO(3); S3] ~=Z=4 Z=12. *
*||
Corollary 6.7. t (SO(3)) = 1 and t (SO(4)) = 2.
Proof
We only have to show that t (SO(4)) 2. The remark following Theorem 5.7 shows
that if f 2 Z (X x Y ) then f|X_Y = 0, so q** is onto. Thus f factors through*
* a
sphere, so we can proceed as in the proof of Corollary 5.11. *
* ||
6.3. The Group E (Y ). We conclude the section by relating Z (Y ) to a certain
group of homotopy equivalences of Y . For any space X, let E (X) [X; X] be the
group of homotopy equivalences f : X -! X such that f = id. This group has been
studied by Felix and Murillo [8] and by Pavesic [22]. We note that if Y is an H*
*-space,
then the function
: Z (Y ) -! E (Y )
defined by (g) = id+ g is a bijection of pointed sets. In general does not pre*
*serve
the binary operation in Z (Y ) and E (Y ). Thus E (Y ) is nontrivial whenever Z*
* (Y )
is nontrivial.
Proposition 6.8. The groups E (Y ) are nontrivial in the following cases: Y =
SU(n), n 3; Y = Sp(n), n 2; and Y = SO(4). The groups E (Y ) are triv-
ial in the following cases: Y = SU(2), Sp(1), SO(2) and SO(3).
7. Problems
In this brief section we list, in no particular order, a number of problems w*
*hich
extend the previous results or which have been suggested by this material.
1.Calculate tF (X) for F = S; M or and various spaces X. In particular, what
is t (HPn) for n > 4, and t (Y ) for compact Lie groups Y not considered in
Section 6?
2.Find general conditions on a space X such that Z (X; Y ) = ZS(X; Y ). One s*
*uch
was given in Section 5. Is Z (Y ) = ZS(Y ) if Y is a compact simply-connect*
*ed
Lie group without homological torsion, such as SU(n) or Sp(n)?
3.Find lower bounds for tF (X) in the cases F = S; M or in terms of other
numerical invariants of homotopy type.
HOMOTOPY CLASSES THAT ARE TRIVIAL MOD F 23
4.With F = S; M or , characterize those spaces X such that ZF (X) = 0.
5.What is the relation between kl (X) and dlog2(cat(X))e? In particular, if
kl (X) < 1, does it follow that cat(X) < 1? Notice that both of these
integers are upper bounds for t (X).
6.Find an example of a finite H-complex Y such that Z (Y ) 6= ZS(Y ) (see Sec*
*tion
4). In the notation of Section 6, this would yield a finite complex Y for w*
*hich
E (Y ) 6= ES(Y ). Such an example which is not a finite complex was given i*
*n [8].
7.Examine tF (X); klF(X) and clF(X) for various collections F such as the col*
*lec-
tion of p-local spheres or the collection of all cell complexes with at mos*
*t two
positive dimensional cells.
8.Investigate the Eckman-Hilton dual of the results of this paper. One define*
*s a
map f : X -! Y to be F-cotrivial if f* = 0 : [Y; A] -! [X; A] for all A 2 *
*F.
One could then study the set ZF (X; Y ) of all F-cotrivial maps X -! Y and*
* ,
in particular, the semigroup ZF (X) = ZF (X; X).
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Martin Arkowitz
Dartmouth College
Hanover NH, 03755
martin.arkowitz@dartmouth.edu
Jeffrey Strom
Dartmouth College
Hanover NH, 03755
jeffrey.strom@dartmouth.edu
*