, WIRTINGER INEQUALITIES, CAYLEY 4-FORM, AND HOMOTOPY arXiv:math/0608006v1 [math… · 2019-03-03 · arXiv:math/0608006v1 [math.DG] 1 Aug 2006 E7, WIRTINGER INEQUALITIES, CAYLEY

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E7, WIRTINGER INEQUALITIES, CAYLEY 4-FORM,

AND HOMOTOPY

VICTOR BANGERT∗, MIKHAIL G. KATZ∗∗, STEVEN SHNIDER,AND SHMUEL WEINBERGER∗∗∗

Abstract. We study optimal curvature-free inequalities of thetype discovered by C. Loewner and M. Gromov, using a generali-sation of the Wirtinger inequality for the comass. Using a modelfor the classifying space BSU(2) built inductively out of BS1, weprove that the symmetric metrics of certain 2-point homogeneousmanifolds turn out not to be the systolically optimal metrics onthose manifolds. We point out the unexpected role played by theexceptional Lie algebra E7 in systolic geometry, via the calculationof Wirtinger constants.

Contents

1. Inequalities of Pu and Gromov 22. Federer’s proof of Wirtinger inequality 43. Gromov’s inequality for complex projective space 54. Symmetric metric of HP

2 and Kraines 4-form 85. Generalized Wirtinger inequalities 106. BG spaces and a homotopy equivalence 127. Lower bound for quaternionic projective plane 148. E7, Hunt’s trick, and Wirtinger constant of R8 16Acknowledgements 19References 19

Date: 30 july 2006.2000 Mathematics Subject Classification. Primary 53C23; Secondary 55R37,

17B25 .Key words and phrases. BG space, calibration, Cartan subalgebra, Cayley form,

comass norm, Exceptional Lie algebra, Gromov’s inequality, Pu’s inequality, stablenorm, systole, Wirtinger inequality.

∗Partially Supported by DFG-Forschergruppe ‘Nonlinear Partial DifferentialEquations: Theoretical and Numerical Analysis’.

∗∗Supported by the Israel Science Foundation (grants no. 84/03 and 1294/06).∗∗∗Partially supported by NSF grant DMS 0504721.

1

http://arxiv.org/abs/math/0608006v1

2 V. BANGERT, M. KATZ, S. SHNIDER, AND S. WEINBERGER

1. Inequalities of Pu and Gromov

The present text deals with systolic inequalities for the projectiveplanes over the division algebras R, C, and H.In 1952, P.M. Pu [Pu52] proved that the least length, denoted sysπ1,

of a noncontractible loop of a Riemannian metric G on the real projec-tive plane RP2, satisfies the optimal bound

sysπ1(RP2,G)2 ≤ π

2area(RP2,G).

Pu’s inequality is saturated by a round metric, i.e. one of constantGaussian curvature. This inequality extends the ideas of C. Loewner,who proved an analogous inequality for the torus in a graduate courseat Syracuse University in 1949, thereby obtaining the first result insystolic geometry, cf. [Ka06].Defining the optimal systolic ratio SR(M) of a surface M as the

supremum

SR(M) = supG

sysπ1(G)2

area(G)(1.1)

over all Riemannian metrics G on M , we can restate Pu’s inequality asthe calculation of the value

SR(RP2) = π2,

the supremum being attained by a round metric.For a higher dimensional manifold M2k, the appropriate middle-

dimensional invariant is the stable k-systole, defined by setting

stsysk(G) = λ1

(

Hk(M,Z)R, ‖ ‖

)

, (1.2)

where ‖ ‖ is the stable norm in homology, while λ1 is the first successiveminimum of the normed lattice. In other words, the stable k-systole isthe least stable norm of a nonzero integral k-homology class. A detaileddefinition of the stable norm appears in Section 3.By analogy with (1.1), one defines the optimal middle-dimensional

stable systolic ratio, SRk(M2k), by setting

SRk(M) = supG

stsysk(G)2

vol2k(G)

where the supremum is over all Riemannian metrics G on M .In 1981, M. Gromov [Gr81] proved an inequality analogous to Pu’s,

for the complex projective plane CP2. Namely, he evaluated the opti-

mal stable systolic ratio of CP2, which turns out to be

SR2(CP2) = 2,

E7, WIRTINGER INEQUALITIES, CAYLEY 4-FORM, AND HOMOTOPY 3

where, similarly to the real case, the implied inequality is saturated bythe symmetric metric, i.e. the Fubini-Study metric. In fact, Gromovproved a more general optimal inequality

stsys2(CPn,G)n ≤ n! vol2n(CP

n,G), (1.3)

where stsys2 is still defined by (1.2) with k = 2 and M = CPn.

A quaternionic analogue of the inequalities of Pu and Gromov waswidely expected to hold. Namely, the symmetric metric on the quater-nionic projective plane HP

2 gives a ratio equal to 103, calculated by a

calibration argument in Section 4 following [Be72]. It was widely be-lieved that the optimal systolic ratio SR4(HP

2) equals 103, as well. See

also [Gr96, Section 4] and [Gr99, Remark 4.37, p. 262]. Contrary toexpectation, we prove the following theorem.

Theorem 1.1. The quaternionic projective plane HP2 and the complex

projective 4-space CP4 have a common optimal middle dimensional sta-ble systolic ratio, contained in the following interval:

SR4(HP2) = SR4(CP

4) ∈ [6, 14].

The constant 14 which is the upper bound for the optimal ratio, istwice the dimension of the Cartan subalgebra of the exceptional Liealgebra E7, reflected in our title, cf. proof of Proposition 8.1. Theo-rem 1.1 follows by combining Proposition 8.1 and Corollary 7.2. Since

103< 6,

we obtain the following corollary.

Corollary 1.2. The symmetric metric on HP2 is not systolically opti-

mal.

The Fubini-Study metric gives a ratio equal to 6 for the complexprojective 4-space. We don’t know of any techniques for constructingmetrics on CP

4 with ratio greater than 6. Meanwhile, an analogue ofGromov’s proof for CP2 only gives an upper bound of 14. This is dueto the fact that the Cayley 4-form, cf. [HL82], has a higher Wirtingerconstant than does the Kahler 4-form (i.e. the square of the standardsymplectic 2-form). The resulting stable systolic inequality is thereforenot optimal.In Section 3, we present Gromov’s proof of the optimal stable 2-

systolic inequality (1.3) for the complex projective space CPn, cf. [Gr99,Theorem 4.36], based on the cup product decomposition of its funda-mental class. The proof relies upon the Wirtinger inequality, proved inSection 2. The symmetric metric on the quaternionic projective plane


is analyzed from the systolic viewpoint in Section 4, and the relationto E7 is explored in Section 5.A homotopy equivalence between HP

2 and a suitable CW complexbuilt out of CP

4 is constructed in Section 6 using a map BS1 →BSU(2). Section 7 exploits such a homotopy equivance to build sys-tolically interesting metrics. Section 8 presents a detailed Lie-theoreticanalysis of 4-forms on R8, using an idea of G. Hunt.

2. Federer’s proof of Wirtinger inequality

Following H. Federer [Fe69, p. 40], we prove an optimal upper boundfor the comass norm ‖ ‖ of the exterior powers of a 2-form.Let V be a vector space over C. Let H = H(v, w) be a Hermitian

product on V , with real part v · w, and imaginary part A = A(v, w),

where A ∈∧2 V , the second exterior power of V . Here H is complex

linear in the second variable.

Example 2.1. Let Z1, . . . , Zν ∈∧1(Cν ,C) be the coordinate func-

tions in Cν . We then have the standard (symplectic) 2-form A ∈∧2(Cν ,C), given by

A = i2

ν∑

j=1

Zj ∧ Zj.

Recall that an exterior form is called simple if it can be expressed asa wedge product of 1-forms.

Lemma 2.2. The comass of the standard symplectic form A satis-fies ‖A‖ = 1.

Proof. We can set ξ = v ∧ w, where v and w are orthonormal. Wehave H(v, w) = iA(v, w), hence

〈ξ, A〉 = A(v, w) = H(iv, w) = (iv) · w ≤ 1;

equality holds if and only if iv = w.

Proposition 2.3 (Wirtinger inequality). Let µ ≥ 1. If ξ ∈∧

2µ Vand ξ is simple, then

〈ξ, Aµ〉 ≤ µ! |ξ|;

equality holds if and only if there exist elements v1, . . . , vµ ∈ V suchthat

ξ = v1 ∧ (iv1) ∧ · · · ∧ vµ ∧ (ivµ).

Consequently, ‖Aµ‖ = µ!


Proof. We assume that |ξ| = 1, where | | is the natural Euclidean normin

∧

2µ V . The case µ = 1 was treated in Lemma 2.2.In the general case µ ≥ 1, we consider the 2µ dimensional subspace T

associated with ξ. Let f : T → V be the inclusion map, and considerthe pullback 2-form (∧2f)A ∈

∧2 T . Next, we orthogonally diagonal-ize the skew-symmetric 2-form, i.e. decompose it into 2 × 2 diagonalblocks. Thus, we can choose dual orthonormal bases e1, . . . , e2µ of T

and ω1, . . . , ω2µ of∧1 T , and nonnegative numbers λ1, . . . , λµ, so that

(∧2f)A =

µ∑

j=1

λj (ω2j−1 ∧ ω2j) . (2.1)

By Lemma 2.2, we have

λj = A(e2j−1, e2j) ≤ ‖A‖ = 1 (2.2)

for each j. Noting that ξ = ǫe1 ∧ · · · ∧ e2µ with ǫ = ±1, we compute(

∧2µf)

Aµ = µ!λ1 . . . λµω1 ∧ · · · ∧ ω2µ,

and therefore

〈ξ, Aµ〉 = ǫµ! λ1 . . . λµ ≤ µ! (2.3)

Note that equality occurs in (2.3) if and only if ǫ = 1 and λj = 1.Applying the proof of Lemma 2.2, we conclude that e2j = ie2j−1, foreach j.

Corollary 2.4. Every 2-form A satisfies the comass bound

‖Aµ‖ ≤ µ!‖A‖µ. (2.4)

Proof. An inspection of the proof Proposition 2.3 reveals that the or-thogonal diagonalisation argument, cf. (2.2), applies to an arbitrary2-form A with comass ‖A‖ = 1.

3. Gromov’s inequality for complex projective space

First we recall the definition of the stable norm in the real homol-ogy of a polyhedron X with a piecewise Riemannian metric, following[BaK03, BaK04].

Definition 3.1. The stable norm ‖h‖ of h ∈ Hk(X,R) is the infimumof the volumes

volk(c) = Σi|ri| volk(σi) (3.1)

over all real Lipschitz cycles c = Σiriσi representing h.


Note that ‖ ‖ is indeed a norm, cf. [Fed74] and [Gr99, 4.C].We denote by Hk(X,Z)R the image of Hk(X,Z) in Hk(X,R) and

by hR the image of h ∈ Hk(X,Z) in Hk(X,R). Recall that Hk(X,Z)Ris a lattice in Hk(X,R). Obviously

‖hR‖ ≤ volk(h) (3.2)

for all h ∈ Hk(X,Z), where volk(h) is the infimum of volumes of allintegral k-cycles representing h. Moreover, one has ‖hR‖ = voln(h)if h ∈ Hn(X,Z). H. Federer [Fed74, 4.10, 5.8, 5.10] (see also [Gr99,4.18 and 4.35]) investigated the relations between ‖hR‖ and volk(h) andproved the following.

Proposition 3.2. If h ∈ Hk(X,Z), 1 ≤ k < n, then

‖hR‖ = limi→∞

1

ivolk(ih). (3.3)

Equation (3.3) is the origin of the term stable norm for ‖ ‖. Recallthat the stable k-systole of a metric (X,G) is defined by setting

stsysk(G) = λ1

(

Hk(X,Z)R, ‖ ‖

)

, (3.4)

cf. (1.2). Let us now return to systolic inequalities on projective spaces.

Theorem 3.3 (M. Gromov). Let G be a Riemannian metric on complexprojective space CP

n. Then

stsys2(G)n ≤ n! vol2n(G);

equality holds for the Fubini-Study metric on CPn.

Proof. Following Gromov’s notation in [Gr99, Theorem 4.36], we let

α ∈ H2(CPn;Z) = Z (3.5)

be the positive generator in homology, and let

ω ∈ H2(CPn;Z) = Z

be the dual generator in cohomology. Then the cup power ωn is agenerator of H2n(CPn;Z) = Z. Let η ∈ ω be a closed differential2-form. Since wedge product ∧ in Ω∗(X) descends to cup productin H∗(X), we have

1 =

∫

CPn

η∧n. (3.6)

Now let G be a metric on CPn. Recall that the pointwise comass

norm for a simple k-form coincides with the natural Euclidean normon k-forms associated with G. In general, the comass is defined by eval-uating on k-tuples of unit vectors, and taking the maximal value. Thecomass norm of a differential k-form is, by definition, the supremum


of the pointwise comass norms. Then by the Wirtinger inequality andCorollary 2.4, we obtain

1 ≤

∫

CPn

‖η∧n‖ dvol

≤ n! (‖η‖∞)n vol2n(CPn,G)

(3.7)

where ‖ ‖∞ is the comass norm on forms (see [Gr99, Remark 4.37] fora discussion of the constant in the context of the Wirtinger inequality).The infimum of (3.7) over all η ∈ ω gives

1 ≤ n! (‖ω‖∗)n vol2n (CPn,G) , (3.8)

where ‖ ‖∗ is the comass norm in cohomology. Denote by ‖ ‖ the stablenorm in homology. Recall that the normed lattices (H2(M ;Z), ‖ ‖)and (H2(M ;Z), ‖ ‖∗) are dual to each other [Fe69]. Therefore theclass α of (3.5) satisfies

‖α‖ =1

‖ω‖∗,

and hencestsys2(G)

n = ‖α‖n ≤ n! vol2n(G). (3.9)

Equality is attained by the two-point homogeneous Fubini-Study met-ric, since the standard CP

1 ⊂ CP n is calibrated by the Fubini-StudyKahler 2-form, which satisfies equality in the Wirtinger inequality atevery point.

Example 3.4. Every metric G on the complex projective plane satisfiesthe optimal inequality

stsys2(CP2,G)2 ≤ 2 vol4(CP

2,G).

This example generalizes to the manifold obtained as the connectedsum of a finite number of copies of CP2 as follows.

Proposition 3.5. Every Riemannian nCP2 satisfies the inequality

stsys2(

nCP2)2

≤ 2 vol4(

nCP2)

. (3.10)

Proof. We define two varieties of conformal 2-systole as follows. We let

Confsys2 = λ1(H2(X ;Z), ‖ ‖2)

andconfsys2 = λ1(H2(X ;Z), | |2).

Since in dimension 4, every 2-form is either simple, or decomposes asthe sum of two orthogonal simple pieces, cf. (2.1), we have an inequality

|x|2 ≤ 2‖x‖2


between the pointwise Euclidean norm and the pointwise comass, forall x ∈

∧2(nCP2). It follows that, dually, we have

Confsys22 ≤ 2 confsys22 .

For a metric of unit volume we have

stsysk ≤ Confsysk .

Thus,stsys22 ≤ 2 confsys22 .

Recall that the intersection form of nCP2 is given by the identitymatrix. Every metric on a connected sum nCP2 satisfies the iden-tity confsys2 = 1 because of the identification of the L2 norm and theintersection form. We thus reprove Gromov’s optimal inequality

stsys22 ≤ 2 vol4,

but now it is valid for the connected sum of n copies of CP2.

In fact, the inequality can be stated in terms of the last successiveminimum λn of the integer lattice in homology with respect to thestable norm ‖ ‖.

Corollary 3.6. The last successive minimum λn satisfies the inequality

λn

(

H2(nCP2,Z), || ||

)2≤ 2 vol4(nCP

2)

The proof is the same as before. This inequality is in fact optimalfor all n, though equality may not be attained.

Question 3.7. What is the asymptotic behavior for the stable systoleof nCP2 when n → ∞? Can the constant in (3.10) be replaced by afunction which tends to zero as n → ∞?

4. Symmetric metric of HP2 and Kraines 4-form

The quaternionic projective plane HP2 has volume vol8(HP

2) = π4

5!for the symmetric metric with sectional curvature 1 ≤ K ≤ 4, whilefor the projective line with K ≡ 4 we have vol4(HP1) = π2

3!, cf. [Be72,

formula (3.10)]. This gives a systolic ratio

stsys4(HP2)2

vol8(HP2)

= 103

(4.1)

for the symmetric metric.In more detail, we endow HP

n with the natural metric as the basespace of the Riemannian submersion from the unit sphere S4n+3 ⊂Hn+1. A projective line HP

1 ⊂ HPn is a round 4-sphere of (Riemann-

ian) diameter π2and sectional curvature +4, attaining the maximum


of sectional curvatures of HPn. The extension of scalars from R to H

gives rise to an inclusion R3 → H3, and thus an inclusion RP2 → HP2.

Then RP2 ⊂ HP2 is a totally geodesic submanifold of diameter π

2and curvature +1, attaining the minimum of the sectional curvaturesof HP

2, cf. [CE75, p. 73].The following proposition was essentially proved by V. Kraines [Kr66]

and M. Berger [Be72]. The invariant 4-form was briefly discussed in[HL82, p. 152].

Proposition 4.1. There is a 4-form κHP

representing a generator ofH4(HP

2,Z) = Z, with

|κ2HP| =

103‖κHP‖

2 (4.2)

and|κHP|

2 = 103‖κHP‖

2, (4.3)

where | | and ‖ ‖ are, respectively, the Euclidean norm and the comassof the unit volume symmetric metric on HP

2.

Proof. Consider the quaternionic vector space Hn = R

4n. Each of thethree quaternions i, j, and k defines a complex structure on Hn, i.e.an identification Hn = C2n. The imaginary part of the associatedHermitian inner product on C2n is the standard symplectic exterior2-form, cf. Example 2.1. Let ωi, ωj, and ωk be the triple of 2-formson Hn defined by the three complex structures. We take their wedgesquares ω2

i , ω2j , and ω2

k. Note that ωi is the sum of 4 terms, while ω2i is

twice the sum of 6 terms. We define an exterior 4-form κn, first writtendown explicitly by V. Kraines [Kr66], by setting

κn = 16

(

ω2i + ω2

j + ω2k

)

. (4.4)

The coefficient 16normalizes the form to unit comass, cf. Lemma 2.2.

The form κn is invariant under transformations in Sp(n)×Sp(1) [Kr66,Theorem 1.9]. The corresponding parallel differential 4-form κ

HP∈

Ω4HPn is closed. The form κ

HPrepresents a nonzero element of the

line H4(HPn,R) = R.

The form 3κ2 on H2 decomposes into a sum of 18 simple 4-forms,i.e. “monomials” in the 8 coordinates. The 18 monomials are not alldistinct. Two of them, denoted m0 and its Hodge star ∗m0, occur withmultiplicity 3. Thus, we obtain a decomposition as a linear combinationof seven pairs

3κ2 = 3(m0 + ∗m0) +6

∑

ℓ=1

(mℓ + ∗mℓ), (4.5)

where ∗ is the Hodge star operator.


Lemma 4.2. The Kraines form has unit comass: ‖κ2‖ = 1.

This was proved in [Be72, DHM88]. Meanwhile, from (4.5) we have

(3κ2)2 = 2 (9 vol+6 vol) ,

where vol = e1 ∧ e2 ∧ · · · ∧ e8 is the volume form of H2 = R8. Hence∣

∣(3κ2)2∣

∣ = 2 · 15 = 30,

proving identity (4.2). Meanwhile, |3κ2|2 = 9 + 9 + 12 = 30, proving

identity (4.3).

Remark 4.3. There is a misprint in the calculation of the systolic con-stants in [Be72, Theorem 6.3], as is evident from [Be72, formula (6.14)].Namely, in the last line on page [Be72, p. 12], the formula for the co-efficient s4,b lacks the exponent b over the constant 6 appearing in thenumerator. The formula should be

s4,b =6b

(2b+ 1)!.

5. Generalized Wirtinger inequalities

Definition 5.1. The Wirtinger constant Wirtn of R2n is the maxi-

mal ratio |ω2|‖ω‖2

over all n-forms ω ∈ ΛnR2n. The modified Wirtinger

constant Wirt′n is the maximal ratio |ω|2

‖ω‖2over all n-forms ω on R2n.

When n is even, let Wirtsd be the Wirtinger constant maximized overselfdual n-forms on R2n.

The calculation of Wirtn can thus be thought of as a generalisationof the Wirtinger inequality of Section 2.

Lemma 5.2. One has Wirtn = Wirtsd ≤ Wirt′n if n is even.

Proof. If ω is a middle-dimensional form, then

‖ω2‖ = |ω2| = 〈ω, ∗ω〉 ≤ |ω| |∗ ω| = |ω|2, (5.1)

proving that Wirtn ≤ Wirt′n.Let η be a form with nonnegative wedge-square. If n is even, the

Hodge star is an involution. Let η = η+ + η− be the decompositioninto selfdual and anti-selfdual parts under Hodge ∗. Then

η2 = (η+ + η−)2

= η2+ + η2−(5.2)


Thus|η2| = |η2+| − |η2−|

≤ |η2+|

≤ Wirtsd‖η+‖2.

(5.3)

Meanwhile,

‖η+‖ = 12(‖η + ∗η‖) ≤ 1

2(‖η‖+ ‖ ∗ η‖) = ‖η‖

by the triangle inequality. Thus, ‖η+‖ ≤ ‖η‖ and we therefore concludethat |η2| ≤ Wirtsd‖η‖

2.

Proposition 5.3. Let X be an orientable, closed manifold of dimen-sion 2n, with bn(X) = 1. Then

SRn(X) ≤ Wirtn.

Proof. By Poincare duality, the fundamental cohomology class in thegroup H2n(X ;Z) = Z is the cup square of a generator of Hn(X ;Z)R =Z. The inequality is now immediate by applying the method of proofof (3.7).

Recall that the cohomology ring for CPn is polynomial on a single

2-dimensional generator, truncated at the fundamental class. The co-homology ring for HP

n is the polynomial ring on a single 4-dimensionalgenerator, similarly truncated. Thus the middle dimensional Bettinumber is 1 if n is even and 0 if n is odd.

Corollary 5.4. Let n ∈ N. We have the following bounds for themiddle-dimensional stable systolic ratio:

SR4n(HP2n) ≤ Wirt4n

SR2n(CP2n) ≤ Wirt2n

SR8(M16) ≤ Wirt8

where M16 is the Cayley projective plane.

The systolic ratio of the symmetric metric of CP2n is (2n)!/(n!)2.Since by Proposition 8.1 we have Wirt4 = 14 > 6, it is in principleimpossible to calculate the optimal systolic ratio for either HP

2 or CP4

by any direct generalisation of the method of proof of (3.10) or (3.7).

Meanwhile, HPn has volume π2n

(2n+1)!. Hence the middle dimensional

systolic ratio of the symmetric metric g0 of HP2n is

SR(HP2n, g0) =

(4n+ 1)!

((2n+ 1)!)2.


ThusSR(CP4n, g0)

SR(HP2n, g0)

=(2n+ 1)2

4n+ 1> 1,

pointing to the same phenomenon as in real dimension 8.The detailed calculation of the Wirtinger constant Wirt4 appears in

Section 8.

6. BG spaces and a homotopy equivalence

Systolically interesting metrics can be constructed as pullbacks byhomotopy equivalences. A particularly useful one is described below.

Proposition 6.1. The complex projective 4-space CP4 admits a de-

gree 1 map to the quaternionic projective plane HP2.

Proof. We consider CP4 as the 8-skeleton of CP∞. The latter is thoughtof as the classifying space BS1 of the circle. We consider HP

2 asthe 8-skeleton of HP

∞. The latter is thought of as the classifyingspace BSU(2), using the identification of SU(2) with unit quaternions.Namely, BG can be characterized as the quotient of a contractible

space S by a free G action. But HP∞ is such a quotient for S = S∞

and G = S3.The map CP

4 → HP2 is then associated with the inclusion of S1 as a

subgroup of SU(2). The map CP4 → CP

∞ → HP∞ is then compressed,

using the cellular approximation theorem, to the 8-skeleton. In matrixterms, the element u ∈ S1 goes to

[

u 00 u−1

]

∈ SU(2). (6.1)

The induced map on cohomology is computed for the infinite dimen-sional spaces, and then restricted to the 8-skeleta. The cohomology ofBSU(2) is Z[c2], i.e. a polynomial algebra on a 4-dimensional genera-tor, given by the second Chern class (thinking of an SU(2) bundle asa particular kind of complex bundle). Thus, to compute the inducedhomomorphism on H4, we need to compute c2 of the sum of the tau-tological line bundle L on CP

∞ and its inverse, cf. (6.1). By the sumformula, it is

−c1(L)2,

but this is a generator of H4(CP∞). In other words, the map

H4(BSU(2)) → H4(BS1)

is an isomorphism. From the structure of the cohomology algebra, wesee that the same is true for the induced homomorphism in H8.


H6(CP4 ∪ e3)

α

nicemap // π6(CP4 ∪ e3)

H7(HP2,CP4 ∪ e3)

β−1

// π7(HP2,CP4 ∪ e3)

γ

OO

Figure 6.1. Commutation of boundary and Hurewicz homomorphisms

The inclusions of the 8-skeleta of these BG spaces are isomorphismson cohomology H8, as well, in view of the absence of odd dimensionalcells. Hence the conclusion follows for these finite-dimensional projec-tive spaces.

A similar argument argument goes through for any CP2n → HP

n.The cell attaching argument below works as well, inductively.The lower bound of Theorem 1.1 for the optimal systolic ratio of HP

2

follows from the two propositions below.

Proposition 6.2. We have H∗(BS3) = Z[v], where v is 4 dimensional.Meanwhile, H∗(BS1) = Z[c] where c is 2-dimensional. Here i∗(v) =−c2 (with usual choices for basis), S3 = SU(2), and v is the secondChern class.

Now restrict attention to the 8 skeleta of these spaces. We obtain amap

CP4 → HP

2 (6.2)

which is degree one (from the cohomology algebra).

Proposition 6.3. There exists a map HP2 → CP

4 ∪ e3 ∪ e7 defining ahomotopy equivalence.

Proof. Coning off a copy of CP1 ⊂ CP4, we note that the map (6.2)

factors through the CW complex CP4 ∪ e3.

The map CP4 ∪ e3 → HP

2 is an isomorphism on homology thoughdimension 5, and a surjection in dimension 6. We consider the pair

(HP2,CP4 ∪ e3).

Its homology vanishes through dimension 6 by the exact sequence of apair. The relative group H7(HP

2,CP4∪e3) is mapped by the boundarymap to H6(CP

4 ∪ e3) = Z, generated by an element h ∈ H6(CP4 ∪ e3).

We therefore obtain an isomorphism

α : H6(CP4 ∪ e3) → H7(HP

2,CP4 ∪ e3),

cf. Figure 6.1.


Both spaces are simply connected and the pair is 6-connected as apair. Applying the relative Hurewicz theorem, we obtain an isomor-phism

β : π7(HP2,CP4 ∪ e3) → H7(HP

2,CP4 ∪ e3).

Applying the boundary homomorphism

γ : π7(HP2,CP4 ∪ e3) → π6(CP

4 ∪ e3),

we obtain an element

h′ = γ β−1 α(h) ∈ π6(CP4 ∪ e3) (6.3)

which generates H6 and is mapped to 0 ∈ π6(HP2).

We now attach a 7-cell to the complex CP4∪ e3 using the element h′

of (6.3). We obtain a new CW complex

X =(

CP4 ∪ e3

)

∪h′ e7,

and a map X → HP2, by choosing a nullhomotopy of the composite

map to HP2. The new map is an isomorphism on all homology. Since

both spaces are simply connected, the map is a homotopy equivalence.Reversing the arrow, we obtain a homotopy equivalence from HP

2 tothe union of CP4 with cells of dimension 3 and 7.

7. Lower bound for quaternionic projective plane

In this section, we show how to apply the homotopy equivalenceobtained in Section 6, so as to obtain systolically interesting metrics.

Proposition 7.1. One can approximate the map of Proposition 6.3 bya simplicial map, and choose a point in a cell of maximal dimensionin CP

4 ⊂ CP4 ∪ e3 ∪ e7 with a unique inverse image.

Proof. The inverse image of a little ball around such a point is a unionof balls mapping the obvious way to the ball CP4∪ e3 ∪ e7. We need tocancel balls occurring with opposite signs. Take an arc connecting theboundaries of two such balls where the end points are the same pointof the sphere. Apply homotopy extension to make the map constanton a neighborhood of this arc (π1 of the target is 0). Then the unionof these balls and fat arc is a bigger ball and we have a nullhomotopicmap to the sphere on the boundary. We can homotope the map to thedisc relative to the boundary to now lie in the sphere.

Corollary 7.2. The optimal middle dimensional stable systolic ratioof HP

2 equals that of CP4.


Proof. We have SR4(CP4) ≥ SR4(HP

2) since metrics on HP2 can be

pulled back to CP4 by the degree one map (6.2) which can be made

monotone by the work of A. Wright [Wr74].Let us prove that SR4(CP

4) ≤ SR4(HP2). Once the map

f : HP2 → CP

4 ∪ e3 ∪ e7 (7.1)

is one-to-one on an 8-simplex

∆ ⊂ CP4 ∪ e3 ∪ e7

of the target (by Proposition 7.1), we argue as follows. The imagesof the attaching maps of e3 and e7 may be assumed to lie in a hyper-plane CP

3 ⊂ CP4. Take a self-diffeomorphism

φ : CP4 → CP4 (7.2)

preserving the hyperplane, and sending the 8-simplex ∆ to the com-plement of a thin neighborhood of the hyperplane, so that most of thevolume of the symmetric metric of CP4 is contained in the image of ∆.Now pull back the metric of the target by the composition φ f of

the maps (7.1) and (7.2). The resulting “metric” on HP2 is degenerate

on certain simplices. The metric can be inflated slightly to make thequadratic form nondegenerate everywhere, without affecting the totalvolume significantly, proving the corollary.The proof is completed by the following proposition.

Proposition 7.3. The symmetic metric of CP4 can be extended to

the 3-cell and the 7-cell in such a way as to decrease the stable systoleby an arbitrarily small amount.

Proof. We work with piecewise smooth metrics on polyhedra, cf. [Ba02].Here volumes and systoles are defined, as usual, cell by cell. When at-taching a cell along its boundary, the attaching map is always assumedto be simplicial, so that all systolic notions are defined on the newspace, as well.The metric on the attached cells needs to be chosen in such a way

as to contain a long cylinder capped off by a hemisphere.To justify the attachment of a cell ep, we argue as follows. Normal-

ize X to unit 4-systole. Let W = X ∪ ep, and consider a metric on ep

which includes a cylinder of length L >> 0, based on a sphere Sp−1,of radius R chosen in such a way that the attaching map ∂ep → X isdistance-decreasing. Here R is fixed throughout the argument (and inparticular is independent of L).


Now consider an n-fold multiple of the generator g ∈ H4(W ), wellapproximating the stable norm in the sense of (3.3). Consider a sim-plicial 4-cycle M with integral coefficients, in the class ng ∈ H4(W ).We are looking for a lower bound for the stable norm ‖g‖ in W . Herewe have to deal with the possibility that the 4-cycle M might “spill”into the cell ep. Applying the coarea formula along the cylinder, weobtain a 3-dimensional section S of M of 3-volume at most

vol3(S) =n‖g‖

L, (7.3)

i.e. as small as one wishes compared to the 4-volume of M itself.Here M decomposes along S as the union

M = M+ ∪M−

where M+ admits a distance decreasing projection to the manifold X ,whileM− is entirely contained in ep. For any 4-chain C ⊂ Sp−1 filling S,the new 4-cycle

M ′ = M+ ∪ C

represents the same homology class ng ∈ H4(W ), since the difference 4-cycle M −M ′ is contained in a p-ball whose homology is trivial. Nowwe apply the linear (without the exponent n+1

n) isoperimetric inequality

in Sp−1. This allows us to fill the section S = ∂M+ by a 4-chain C ⊂Sp−1 of volume at most f(R)n‖g‖/L by (7.3), where f(R) is a suitablefunction of R. The corresponding cycle M ′ has volume at most

(

n+n

L

)

‖g‖ = n||g||(1 + f(R)/L).

Since M ′ admits a short map to X , its volume is bounded below by n,which yields a lower bound for ‖g‖ which is arbitrarily close to 1. Notethat similar arguments have appeared in the work of I. Babenko andhis students [Ba93, Ba02, Ba04, BB05].

8. E7, Hunt’s trick, and Wirtinger constant of R8

Proposition 8.1. We have Wirt2 = 2, while Wirt4 = 14.

Proof. By the Wirtinger inequality and Corollary 2.4, we obtain thevalue Wirt2 = 2. The square η = τ 2 of the Kahler form τ on C4

satisfies |η|2

‖η‖2= 6, while the Cayley form ω1 specified in [DHM88, p. 14]

has unit comass, satisfies |ω1|2 = 14, and is shown there to have the

maximal ratio among all selfdual forms on R8. The E7 approach wasnot clarified in [DHM88]. Thus, the “very nice seven-dimensional cross-section” referred to in [DHM88, p. 3, line 8] and [DHM88, p. 12, line 5],is in fact the Cartan subalgebra of E7.


More specifically, let dx1, dx2, dx3, dx4 denote the dual basis tothe standard real basis 1, i, j, k for the quaternion algebra H andlet dxi, dxi′, i = 1, . . . , 4, be the dual basis for H2. The three sym-plectic forms on H2 defined by the three complex structures are

ωi = dx1 ∧ dx2 + dx3 ∧ dx4 + dx1′ ∧ dx2′ + dx3′ ∧ dx4′ ,

ωj = dx1 ∧ dx3 − dx2 ∧ dx4 + dx1′ ∧ dx3′ − dx2′ ∧ dx4′ ,

ωk = dx1 ∧ dx4 + dx2 ∧ dx3 + dx1′ ∧ dx4′ + dx2′ ∧ dx3′

Let dxabcd := dxa∧dxb∧dxc∧dxd, where a, b, c, d ⊂ 1, ..., 4, 1′, ..., 4′.The corresponding wedge squares satisfy12ω2i = (dx1234 + dx1′2′3′4′) + (dx121′2′ + dx343′4′) + (dx123′4′ + dx341′2′),

12ω2j = (dx1234 + dx1′2′3′4′) + (dx131′3′ + dx242′4′)− (dx132′4′ + dx241′3′),

12ω2k = (dx1234 + dx1′2′3′4′) + (dx141′4′ + dx232′3′) + (dx142′3′ + dx231′4′)

(8.1)The seven distinct self-dual 4-forms appearing in the decomposition(4.5) of the Kraines form, which are also displayed in parenthesesin (8.1), form the basis of a 7-dimensional Cartan subalgebra h of theexceptional real Lie algebra E7, as we now explain. The Lie algebra E7

can be decomposed as a direct sum sl(8) ⊕ Λ4(8), cf. [Ad96, p. 74],which can be refined into a Cartan decomposition, as follows.Recall that the latter consists of a maximal compact subalgebra, on

which the restriction of the Killing form is negative definite, and anorthogonal positive definite complement. We write

sl(8) = so(8)⊕ sym(8),

where sym(8) is the set of 8 × 8 symmetric matrices. Similarly, wewrite

Λ4(8) = Λ4+(8)⊕ Λ4

−(8),

where the subscripts + and − indicate “selfdual” and “anti-selfdual”forms, respectively. Then the Cartan decomposition is given by

E7 = k⊕ p

k = so(8)⊕ Λ4−(8)

p = sym(8)⊕ Λ4+(8).

The 7 self-dual forms shown in (8.1) span a maximal abelian subalge-bra h in p. The span of these forms is an abelian subalgebra becausethe E7 Lie bracket of two simple 4-forms, described in [Ad96, p. 74],vanishes whenever they have a common dxi ∧ dxj factor, and it is easyto see that this condition is satisfied for all the Lie brackets of pairs ofsimple forms which occur in the Lie brackets of the 7 self-dual forms.


Since E7 is of rank 7, the dimension of a maximal abelian subalgebraof p is also 7.Since every selfdual 4-form is contained in a maximal abelian sub-

algebra of Λ4+(8), any self-dual 4-form is orthogonally equivalent to an

element of h by Lemma 8.2.

Such a representation of a self-dual 4-form was apparently first de-scribed explicitly by L. Antonyan [An81], in the context of the studyof θ-groups by V. Kac and E. Vinberg [GV78] and E. Vinberg andA. Elashvili [VE78]. The Cayley form [Jo00, Definition 10.5.1] is thesigned sum of the 7 basis elements of h, with a coefficient −1 for the lastterm in the second row and the last two terms in the third row of (8.1)and the remaining terms with coefficient +1, see [Jo00, equation 10.19].

Lemma 8.2. Any two maximal abelian subalgebras of Λ4+(8) are con-

jugate under the action of SO(8).

Remark 8.3. The Killing form is positive definite both on the sym-metric matrices in sl(8), and on Λ4

+. The standard (diagonal) Car-tan subalgebra of sl(8) is conjugate to the one in Λ4

+, but by the ac-tion of the full maximal compact subgroup of the adjoint group of E7,cf. [Wa88, §2.1.9], which is isomorphic to SU(8), much bigger thanmerely SO(8,R).

Proof of Lemma 8.2. The lemma can be proved by a slight modificationof a proof, due to G. Hunt [Hu56], of the K-conjugacy of any twomaximal abelian subalgebras of p.Let ai, i = 1, 2 be two maximal abelian subalgebras of Λ4

+(8), andlet Hi ∈ ai be regular elements, such that

ai = X ∈ p | [X,Hi] = 0 .

Consider the following function on SO(8):

f(k) = B(Ad(k)H1, H2),

where B(−,−) is the Killing form on E7. Since SO(8) is compact, thefunction attains a minimum at some point u, and we have

0 = ddt|t=0B(Ad(exp(tY )u)H1, H2)

= B([Y,Ad(u)H1], H2)

= B(Y, [Ad(u)H1, H2])

by the Ad-invariance of the Killing form, for all Y ∈ so(8). Since

[X1, X2] ∈ so(8) ⊂ sl(8)


for any pair X1, X2 ∈ Λ4+(8), and the Killing form on so(8) is negative

definite, it follows that [Ad(u)H1, H2] = 0. Thus Ad(u)H1 ∈ a2, and

a2 = X ∈ p | [X,Ad(u)H1] = 0.

Thus Ad(u)(a1) ⊂ a2 and by equality of dimensions, we finally ob-tain Ad(u)(a1) = a2.

Acknowledgements

We are grateful to D. Alekseevsky, R. Bryant, A. Elashvili, D. Joyce,V. Kac, C. LeBrun, and F. Morgan for helpful discussions.

References

[Ad96] Adams, J.: Lectures on exceptional Lie groups. With a foreword by J. Pe-ter May. Chicago Lectures in Mathematics. University of Chicago Press,Chicago, IL, 1996.

[An81] Antonyan, L.: Classification of four-vectors of an eight-dimensional space.(Russian) Trudy Sem. Vektor. Tenzor. Anal. No. 20, (1981), 144–161.

[Ba93] Babenko, I.: Asymptotic invariants of smooth manifolds. Russian Acad.Sci. Izv. Math. 41 (1993), 1–38.

[Ba02] Babenko, I.: Forte souplesse intersystolique de varietes fermees et depolyedres. Annales de l’Institut Fourier 52, 4 (2002), 1259-1284.

[Ba04] Babenko, I.: Geometrie systolique des varietes de groupe fondamental Z2,Semin. Theor. Spectr. Geom. Grenoble 22 (2004), 25-52.

[BB05] Babenko, I.; Balacheff, F.: Geometrie systolique des sommes connexes etdes revetements cycliques, Math. Annalen 333 (2005), no. 1, 157-180.

[BaK03] Bangert, V.; Katz, M.: Stable systolic inequalities and cohomology prod-ucts, Comm. Pure Appl. Math. 56 (2003), 979–997. Available at the sitearXiv:math.DG/0204181

[BaK04] Bangert, V; Katz, M.: An optimal Loewner-type systolic inequality andharmonic one-forms of constant norm. Comm. Anal. Geom. 12 (2004),no. 3, 703-732. See arXiv:math.DG/0304494

[Be72] Berger, M.: Du cote de chez Pu. Ann. Sci. Ecole Norm. Sup. (4) 5 (1972),1–44.

[CE75] Cheeger, J.; Ebin, D.: Comparison theorems in Riemannian geometry.North-Holland Mathematical Library, Vol. 9. North-Holland PublishingCo., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., NewYork, 1975.

[DHM88] Dadok, J.; Harvey, R.; Morgan, F.: Calibrations on R8. Trans. Amer.Math. Soc. 307 (1988), no. 1, 1–40.

[Ec41] Eckmann, B.: Ueber die Homotopiegruppen von Gruppenraumen. Com-ment. Math. Helv. 14 (1941), 234–256.

[EK62] Eells, J.; Kuiper, N.: Manifolds which are like projective planes. Inst.Hautes Etudes Sci. Publ. Math. 14 (1962), 5–46.

[Ep66] Epstein, D.: The degree of a map. Proc. London Math. Soc. 13 (1966)369-383.


[Fe69] Federer, H.: Geometric Measure Theory. Grundlehren der mathematis-chen Wissenschaften, 153. Springer-Verlag, Berlin, 1969.

[Fed74] Federer, H.: Real flat chains, cochains, and variational problems. IndianaUniv. Math. J. 24 (1974), 351–407.

[GV78] Gatti, V.; Viniberghi, E.: Spinors of 13-dimensional space. Adv. in Math.30 (1978), no. 2, 137–155.

[Gr81] Gromov, M.: Structures metriques pour les varietes riemanniennes.Edited by J. Lafontaine and P. Pansu. Textes Mathematiques, 1. CEDIC,Paris, 1981.

[Gr83] Gromov, M.: Filling Riemannian manifolds. J. Diff. Geom. 18 (1983),1-147.

[Gr96] Gromov, M.: Systoles and intersystolic inequalities. Actes de la TableRonde de Geometrie Differentielle (Luminy, 1992), 291–362, Semin.Congr., 1, Soc. Math. France, Paris, 1996.www.emis.de/journals/SC/1996/1/ps/smf sem-cong 1 291-362.ps.gz

[Gr99] Gromov, M.: Metric structures for Riemannian and non-Riemannianspaces. Progr. Math. 152, Birkhauser, Boston, 1999.

[HL82] Harvey, R.; Lawson, H. B.: Calibrated geometries. Acta Math. 148

(1982), 47–157.[Hu56] Hunt, G.: A theorem of Elie Cartan. Proc. Amer. Math. Soc. 7 (1956),

307–308.[Jo00] Joyce, D.: Compact manifolds with special holonomy. Oxford Mathemat-

ical Monographs. Oxford University Press, Oxford, 2000.[Ka06] Katz, M.: Systolic geometry and topology. Mathematical Surveys and

Monographs, to appear. American Mathematical Society, Providence, R.I.[Kr66] Kraines, V.: Topology of quaternionic manifolds. Trans. Amer. Math.

Soc. 122 (1966), 357–367.[Pu52] Pu, P.M.: Some inequalities in certain nonorientable Riemannian mani-

folds. Pacific J. Math. 2 (1952), 55–71.[VE78] Vinberg, E.; Elashvili, A.: A classification of the three-vectors of nine-

dimensional space. (Russian) Trudy Sem. Vektor. Tenzor. Anal. 18

(1978), 197–233.[Wa88] Wallach, N.: Real reductive groups. I. Pure and Applied Mathematics,

132, Academic Press, Inc., Boston, MA, 1988.[Wh78] Whitehead, G.: Elements of homotopy theory. Grad. Texts in Math., 61,

Springer-Verlag, New York, 1978.[Wr74] Wright, A.: Monotone mappings and degree one mappings between PL

manifolds. Geometric topology (Proc. Conf., Park City, Utah, 1974),pp. 441–459. Lecture Notes in Math. 438, Springer, Berlin, 1975.


Mathematisches Institut, Universitat Freiburg, Eckerstr. 1, 79104

Freiburg, Germany

E-mail address : [email protected]

Department of Mathematics, Bar Ilan University, Ramat Gan 52900

Israel


Department of Mathematics, Bar Ilan University, Ramat Gan 52900

Israel


Department of Mathematics, University of Chicago, Chicago, IL

60637


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, WIRTINGER INEQUALITIES, CAYLEY 4-FORM, AND HOMOTOPY arXiv:math/0608006v1 [math… · 2019-03-03 · arXiv:math/0608006v1 [math.DG] 1 Aug 2006 E7, WIRTINGER INEQUALITIES, CAYLEY