VOLUMES, MIDDLE-DIMENSIONAL SYSTOLES, AND
WHITEHEAD PRODUCTS
IVAN K. BABENKO, MIKHAIL G. KATZ, AND ALEXANDER I. SUCIU
Abstract. We prove that closed manifolds of dimension 2m 6 with torsion-*
*free
middle-dimensional homology are systolically free.
1. Introduction
Let (X; g) be a closed, orientable Riemannian manifold of even dimension 2m. *
*The
following notion was introduced by Marcel Berger in [5], [6].
Definition 1.1. The k-systole of (X; g), sysk(g), is the infimum of areas of no*
*n-
bounding cycles represented by maps of k-dimensional manifolds into X.
In this note, we will be interested in the following question: Does there ex*
*ist a
constant, C, such that every metric g on X satisfies
(1.1) sys2m(g) C . vol2m(g)?
If there is no such C, we say that X is systolically free.
In the case of surfaces of positive genus the answer to question (1.1)is affi*
*rmative.
For a history of the problem (dating to C. Loewner's solution in the case X = S*
*1xS1),
see M. Berger [6] and P. Sarnak [16]. In the case m 2, this question has been *
*referred
to by M. Gromov as the "basic systolic problem" ([9], p. 357); see also the sub*
*section
"Systolic reminiscences" of [10], p. 267. Progress on the problem became possi*
*ble
once Gromov described a special family of metrics on S1xS3 and surgical procedu*
*res
suitable for generalizations (see [10]).
The purpose of this note is to prove the following result (see Theorem 2.3 fo*
*r a
statement in the more general context of CW-complexes).
Theorem 1.2. If m 3 and Hm (X) is torsion-free, then X is systolically free.
___________
1991 Mathematics Subject Classification. Primary 53C23; Secondary 55Q15.
Key words and phrases. volume, systole, systolic freeedom, coarea inequality,*
* isoperimetric in-
equality, Whitehead product, Hilton-Milnor theorem.
The second author is grateful to the Geometry & Dynamics Program at Tel Aviv *
*University for
hospitality during part of the preparation of this work.
The third author was partially supported by N.S.F. grant DMS-9504833. He is g*
*rateful to the
Institut Elie Cartan at Henri Poincare University - Nancy 1 for hospitality dur*
*ing the completion of
this work.
1
2 IVAN K. BABENKO, MIKHAIL G. KATZ, AND ALEXANDER I. SUCIU
An underlying theme of this paper is the influence of homotopy theory on the
geometric inequality (1.1). Indeed, our basic topological tools here are, on on*
*e hand,
the Hilton-Milnor theorem calculating homotopy groups of a wedge of spheres, an*
*d,
on the other hand, theorems of B. Eckmann and G. Whitehead on composition maps
in homotopy groups of spheres. Our geometric tools are the coarea inequality of*
* [8]
and pullback arguments for simplicial metrics as described by the first author *
*in [1].
The starting point are the metrics on Sm x Sm constructed by the second author
in [12]. An announcement of this paper appeared in [13].
The case m = 2 of (1.1)remains open, but it has been reduced to either CP2 or
S2 x S2 by the first two authors in [2]. Even if we restrict the class of comp*
*eting
metrics to homogeneous ones, the inequality (1.1)is violated in certain cases s*
*uch as
S3 x S3, see [13] and [4].
The structure of the paper is as follows:
oIn section 2, we distill a notion called "meromorphic map" between regular
CW-complexes, which allows us to correlate their systolic freedom. We use t*
*his
technique to give a short proof of the systolic freedom of Sm x Sm .
oIn section 3, we find maps from the (2m - 1)-skeleton of X to a wedge of m-
spheres that induce monomorphisms in Hm (-; Q), and self-maps of _Sm that
send ss2m-1(_Sm ) to the subgroup generated by Whitehead products.
oIn section 4, we prove our theorem in the case where bm (X) = 1, by mapping*
* X
meromorphically to Sm x Sm .
oIn section 5, we present the proof in the general case. This is achieved by
mapping X meromorphically to the 2m-skeleton of a product of sufficiently
many m-spheres.
2.Systolic freedom of CW-complexes and meromorphic maps
In order to prove Theorem 1.2, we will enlarge the class of manifolds to that*
* of
piecewise smooth, simplicial complexes, for which one can still define metrics,*
* vol-
umes, and systoles. We will actually prove our theorem in the context of finit*
*e,
regular CW-complexes. Such a complex K can be triangulated so that the resulting
simplicial complex is a subdivision of K (see [14], p. 80).
Definition 2.1. A finite, regular CW-complex K of dimension 2m is called systol*
*i-
cally free if
vol2m(g)
(2.1) inf________= 0;
g sys2m(g)
where the infimum is taken over all metrics g on K. This amounts to the existen*
*ce
of a sequence of metrics {gj} such that
(2.2) sys2m(gj) j vol2m(gj):
VOLUMES, SYSTOLES, AND WHITEHEAD PRODUCTS 3
Remark 2.2. The systolic freedom of K (or the absence thereof) is independent*
* of
the piecewise smooth simplicial structure that one chooses in its homotopy type*
*. This
independence is verified by means of the simplicial approximation theorem and by
the pullback arguments for metrics from [1], [2].
Theorem 2.3. Let K be a finite, regular CW-complex of dimension 2m 6. If
Hm (K) is torsion-free, then K is systolically free.
This theorem (which slightly generalizes Theorem 1.2), will be proved at the *
*end of
section 5. The key to the proof is the following notion, inspired by complex an*
*alysis
and surgery theory.
Definition 2.4. Let X and Y be 2m-dimensional CW-complexes. A "meromorphic
map" from X to Y is a continuous map f : X ! W such that
(i)W is a CW-complex obtained from Y by attaching cells of dimension at most
2m - 1;
(ii)f* : Hm (X) ! Hm (W ) is a monomorphism.
We shall denote such "maps" by f : X --! Y , and drop the quotation marks.
Example 2.5. Let X be a complex surface and bX! X its blow-up at a point p 2 *
*X.
Then the classical meromorphic map X ! bX can be modified in a neighborhood of
p and extended to a continuous map from X to bX[fB3 where the 3-ball is attached
along the exceptional curve.
Proposition 2.6 ([2]).Let X and Y be finite, regular CW-complexes. Suppose X
admits a meromorphic map to Y . If Y is systolically free, then X is also systo*
*lically
free.
S k
Proof (sketch).Let f : X ! W = Y [ iBii be the given meromorphic map. The
attached cells (of dimension ki< 2m) do not affect the 2m-dimensional volume, a*
*nd
thus W is still systolically free, by the cylinder construction of [2], Lemma 6*
*.1. We __
now pull back to X the systolically free metrics on W . Thus X is systolically *
*free. |__|
Proposition 2.7. Let X be an orientable, smooth manifold of dimension n = 2m.
Suppose Y is obtained from X by performing surgery on embedded, framed k-sphere*
*s,
with 1 k < m. Then there exists a meromorphic map from X to Y .
Proof.Let W n+1be the cobordism between X and Y defined by the surgeries. We
claim that the inclusion X ,! W is the desired meromorphic map. Indeed, W is
obtained by attaching handles Dk+1x Dn-k to X x I, or dually, by attaching hand*
*les
Dn-k xDk+1 to Y xI. Hence, W has the homotopy type of Y , with cells of dimensi*
*on
n-k n-1 attached to it, and so condition (i) is satisfied. Since k < m, condit*
*ion_(ii)
is satisfied also. *
* |__|
The main geometric ingredient in the proof of Theorem 2.3 is the following sp*
*ecial
case, first proved in [12] by a longer argument, that did not use CW-complexes.
4 IVAN K. BABENKO, MIKHAIL G. KATZ, AND ALEXANDER I. SUCIU
Proposition 2.8 ([12]).For m 3, the product Sm x Sm is systolically free.
Proof.Perform surgery on a standard Sk Sm , k 1, to obtain Sk x Sm-k . Let
f : Sm --! Sk x Sm-k be the associated meromorphic map, and take the map
(2.3) idxf : Sm x Sm --! Sm x Sk x Sm-k :
By Proposition 2.6, it suffices to prove that the manifold X = Sm x Sk x Sm-k *
*is
systolically free. Let {gj} be a sequence of metrics on Sm x Sk such that
volm+k(gj)
(2.4) lim ________________= 0
j!1 sysk(gj) sysm(gj)
(cf. [15], [2]). Let hj be a metric on Sm-k of volume volm-k(hj) = sysm(gj)_sy*
*sk(gj). Consider
the metric gj hj on X, and let z be a cycle representing a non-zero multiple of
[Sk x Sm-k ]. Let p : X ! Sm-k be the projection to the last factor. By the co*
*area
inequality, we obtain the following lower bound for the volume of z in (X; gj *
*hj):
Z
(2.5) volm(z) volk(z \ p-1(x)) dx volm-k(hj) sysk(gj) = sysm(gj);
(Sm-k;hj)
where the middle inequality uses intersection numbers for cycles and transversa*
*lity
arguments in the context of maps of manifolds into X (cf. Definition 1.1 and [2*
*],
Lemma 6.1.). Hence
vol2m(gj__hj)_ volm-k(hj) volm+k(gj) volm-k(gj)
(2.6) = ____________________ = ________________---! 0;
sys2m(gj hj) sys2m(gj) sysk(gj) sysm(gj)j!1
*
* __
proving the systolic freedom of X and therefore that of Sm x Sm . *
* |__|
3. Whitehead products and maps to wedges of spheres
In this section, we establish some lemmas that will be needed in the proof of
Theorem 2.3. The main idea is to use high-degree self-maps of the m-sphere Sm *
*as
in R. Thom [18] to handle torsion in homotopy. In what follows, we will denote*
* a
wedge of b copies of Sm by _bSmr, or simply _Sm .
A basic tool is the Hilton-Milnor theorem, which computes the homotopy groups
of a wedge of spheres in terms of the homotopy groups of the factors. In partic*
*ular,
it gives the following splittings (see e.g. [20]):
M
(3.1a) ssk(_Smr)= ssk(Smr) fork 2m - 2;
M r M
(3.1b) ssk(_Smr)= ssk(Smr) Z[er; es]; fork = 2m - 1;
r r~~ 1, we pick OE = _bOEq, with q as in (3.6). The splitting from (3.1b)a*
*nd an__
argument as above insure that OE satisfies (i) and (ii). *
* |__|
Remark 3.3. For m odd, m 3, we can actually choose q so that OE] = 0, since,*
* in
that case, ss2m-1(Sm ) is a finite group, and all its elements have Hopf invari*
*ant 0.
Remark 3.4. Let F be C, H or Ca , and let K = FP2 be the corresponding projec*
*tive
plane. With the usual decomposition into 3 cells for K, we have K2m-1 = FP1 = S*
*m ,
where m = dim(F). The smallest positive integer q for which OEq : Sm ! Sm sati*
*sfies
conditions (i) and (ii) from Lemma 3.2 can be computed explicitly in these exam*
*ples.
Recall that ss2m-1(Sm ) = Z Tm , where Tm is a finite cyclic group, of order e*
*qual to
1, 12, or 120 respectively when F is C, H, or Ca . Let a be the infinite order *
*generator
defined by the Hopf map, and let s be a generator of the torsion part (taken to*
* be 0
when F = C). A result of H. Toda (see [11]) states that [e; e] = 2a s. From *
*this
formula and (3.5)we obtain for q even:
q q q q2
(3.7) OEq](a) = __2a + [e; e] = __s + __[e; e]:
2 2 2 2
Thus, the necessary and sufficient condition for (i) and (ii) to hold is that q*
* be a
non-zero multiple of 2 |Tm.|
VOLUMES, SYSTOLES, AND WHITEHEAD PRODUCTS 7
4.Meromorphic maps to Sm x Sm
In this section, we prove Theorem 2.3 in the particular case where b = 1, by
constructing a meromorphic map to the product of two of m-spheres. The essen-
tial ingredients of the general case are already present here, but the proof is*
* more
transparent in this simpler situation.
Proof of Theorem 2.3 when b = 1.Let K be a finite, regular CW-complex of dimen-
sion 2m 6. Assume Hm (K=Km-1 ) = Z.
By Lemma 3.1, there is a map f : K2m-1 ! Sm such that f* : Hm (K2m-1 =Km-1 ) !
Hm (Sm ) is injective. By Lemma 3.2, there is a map OE = OEmq: Sm ! Sm of de*
*gree
q 6= 0 that maps ss2m-1(Sm ) to the subgroup generated by the Whitehead product
[e; e]. The map OE O f : K2m-1 ! Sm also maps ss2m-1(K2m-1 ) to this subgroup, *
*while
inducing a monomorphism on Hm .
Now let a1 and a2 be the generators of ssm (Sm xSm ), corresponding to the in*
*clusions
of the factors. Recall that Sm x Sm = Sm _ Sm [[a1;a2]B2m . Attaching an (m + 1*
*)-cell
along the diagonal map (1; 1) : Sm ! Sm xSm , we obtain the 2m-dimensional regu*
*lar
CW-complex
(4.1) W = Sm x Sm [a1+a2Bm+1 :
Since m 2, the complex W satisfies condition (i) in Definition 2.4. Let ff : S*
*m ! W
(1;0)m m
be the composite Sm - -! S x S ,! W . Then
(4.2) ff([e; e])= [a1; a1] sinceff(e) = a1
= [a1; -a2] sincea1 + a2 = 0 inssm (W )
= 0 since[a1; a2] = 0 inss2m-1(Sm x Sm ):
f m OE m ff
Now let h : K2m-1 ! W be the composite K2m-1 -! S -! S -! W . By the
above, h] : ss2m-1(K2m-1 ) ! ss2m-1(W ) is the 0 map. Thus, h extends over the *
*2m-
cells of K, to a map h : K ! W . Since h is clearly injective on Hm , we have d*
*efined
a meromorphic map from K to Sm x Sm . By Proposition 2.8, Sm x Sm is systolical*
*ly __
free. Hence, by Proposition 2.6, K is also systolically free. *
* |__|
5. Meromorphic maps to skeleta of products of spheres
Before proving the general case of Theorem 2.3, we establish the systolic fre*
*edom
of a model space by a "long cylinder" argument.
Let X be a triangulated manifold of dimension n. Let A be the n-skeleton of X*
* x I
where I is an interval. Then A = X x @I [ Xn-1 x I. Let g+ and g- be two metrics
on X, and g0 another metric dominating both g+ and g- . Let g be the metric on
A obtained by restricting the metric gt dt2 of X x I, where I = [-L; L], with
8 IVAN K. BABENKO, MIKHAIL G. KATZ, AND ALEXANDER I. SUCIU
L = ` + 1 > 1, and
(
g0 if|t| `;
(5.1) gt=
(1 - )g0 + g ift = (` + ); with 0 1:
Lemma 5.1. For k 2 and ` sufficiently large, we have sysk(g) fi, where
1
(5.2) fi = __min (sysk(g+ ); sysk(g-:))
2
Proof.Suppose z is a non-bounding k-cycle in A such that volk(z) < fi. Let p : *
*A !
[0; L] be the restriction of the map X x I ! [0; L] given by (x; t) 7! |t|. The*
* coarea
inequality yields a point t0 2 [0; `] such that the (k - 1)-cycle fl = z \ p-1(*
*t0) satisfies
1
(5.3) volk-1(fl) __volk(z):
`
By the isoperimetric inequality for cycles of small volume ([8], Sublemma 3.4*
*.B0)
applied to g0|Xn-1, there is a constant C = C(g0|Xn-1) with the following prope*
*rty:
Every (k-1)-cycle fl in Xn-1 with vol(fl) < 1_Cbounds a k-chain D in Xn-1, of v*
*olume
_k_
(5.4) volk(D) C volk-1(fl)k-1:
By choosing ` > fiC we insure that the isoperimetric inequality applies to fl. *
*More-
fi_k_
over, we needpto_choose_` so that volk(D) < fi. Thus we also require C _`k-1 <*
* fi,
that is, ` > k fiCk-1.
Write D = D- + D+ where D Xn-1 x {t0}. Consider the decomposition of z
into a sum of cycles, z = z- + z0 + z+ , where z- = p-1([0; t0]) \ (X x [-L; 0]*
*) + D- ,
z0 = p-1([0; t0]) + D+ - D- , and z+ = p-1([0; t0]) \ (X x [0; L]) - D+ . Now l*
*et ffl = 0,
+, or -. We have
(5.5) volk(zffl) volk(z) + volk(D) < fi + fi = min (sysk(g+ ); sysk(g-:))
Hence zfflis a boundary for every ffl and so [z] = 0. The contradiction proves*
*_the
lemma. |__|
Lemma 5.2. Let B = (xcSm )2m be the 2m-dimensional skeleton of a product of c
copies of the m-sphere, c 2, m 3. Then B is systolically free.
Proof.We choose the following representative B0 in the homotopy class of B. Take
the Cartesian product of the wedge _cSm with the wedge of c2intervals Irs= [0;*
* L]
for sufficiently large L = ` + 1. At the end of each interval, attach a 2m-cell*
* along
the Whitehead product, [er; es], of the fundamental classes of the spheres Smrx*
* {L}
and Smsx {L} in _cSm x Irs:
i_ c j i_ j [
(5.6) B0 = Sm x Irs [ D2mrs:
r~~