[PPT] - On the explicit systolic inequality from the cup-product Hoil Ryu PowerPoint Presentation

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On the explicit systolic inequality from the cup-product

Hoil Ryu

Graduate School of Mathematics, Kyushu University

EACAT4 Dec 7, 2011

On the explicit systolic inequality from the cup-product Hoil Ryu

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Systoles

... The medical term systole comes from the Greek word for contraction ... by Marcel Berger, Notice of the AMS, 2008 Mn : connected closed Riemannian manifold sysπ1(M) : fundamental 1–systole syshq(M) : homology q–systole

On the explicit systolic inequality from the cup-product Hoil Ryu

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Question

What is the relationship between systoles and volume of M? Loewner showed sysπ1(T2)2 ≤ 2 √ 3 · vol(T2) for an arbitrary metric on T2. This inequality seems to be related with T2 = S1 × S1. Gromov showed a theorem and generalized to ...

On the explicit systolic inequality from the cup-product Hoil Ryu

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Theorem (Gromov ’83, Bangert-Katz ’02)

Let Mn be a connected closed orientable Riemannian manifold. If there exist α1 ∪ · · · ∪ αk = 0 with αi ∈ Hdi(M; R), then there exist ηi ∈ Hdi(M; Z), ξ ∈ Hd1+···+dk(M; Z) and 0 < C < ∞ satisfying

k

∏

i=1

stm(ηi) ≤ C · stm(ξ) where C is only determined by H∗(M; R) and stm is a norm on free part

f H∗(M; Z) which is called the stable mass.

Problem : Orientable manifolds only. What are ηi (and C)? ⇒ We will argue in this talk.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Systolic categories [Katz & Rudyak ’06]

If ” some kind of ” systole satisfies

k

∏

i=1

sysdi ≤ C · vol(M) then the ” some kind of ” systolic category of M is defined by the maximum number of k among inequalities.

The homology systolic category catsysh is invariant under

homotopy equivalences [Katz & Rudyak ’08].

If M is orientable, then the stable systolic category catstsys is

lower bounded by the real cup-length.

A corollary of [Gromov ’83].

If M is 0–universal, then catstsys(M) is invariant under rational

equivalences [Ryu ’11].

On the explicit systolic inequality from the cup-product Hoil Ryu

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Systolic freedom

Orientable cases are stabilized by [Gromov, Katz] : e.g. catsysh(S1) = catsysh(S3) = 1 but catsysh(S1 × S3) = 1. catstsys(S1) = catstsys(S3) = 1 and catstsys(S1 × S3) = 2. A non-stabilizable case [Iwase] : catsysh(S1 ∨ S2) = catstsys(S1 ∨ S2) = 1. A non-orientable case [Ryu] : catsysh(RP2 × S3) = catstsys(RP2 × S3) = 0 and catsysh(RP2 × S3; Z/2Z) = 1. Compare [Pu] : catsysh(RP2) = catstsys(RP2) = 0 but catsysh(RP2; Z/2Z) = 2.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Mass and comass on cubes

M : connected closed smooth n–manifold with a Riemannian metric G. For a differential q–form ω and a normal q–current T, m∗(ω) := sup |ωx(τ)| : x in M, orthonormal q–frame τ

m(T) := sup
T(ω) : m∗(ω) ≤ 1
where m∗(ω) is called the comass of ω and m(K) is called the mass of K.

For a smooth singular cube κ : Iq → M, there is a current K K(ω) :=

Iq κ#ω.

Define the mass of κ by m(κ) := m(K)

On the explicit systolic inequality from the cup-product Hoil Ryu

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Stabilization and systole with local coefficients

We can consider local coefficients ← → locally constants sheaf A A locally constants sheaf is a covering space, if we consider the product metric on A, we can extend the mass. If A0 is a Z–lattice in some real vector space R0 and ι : A ⊂ R is a sheaf homomorphism, then stm(η) := m(ι∗η) is said to be the stable mass of η ∈ H∗(M; A) where the mass is a norm on H∗(M; R). The stable q–systole is stm[Free Hq(M; A) \ {0}].

On the explicit systolic inequality from the cup-product Hoil Ryu

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Twisted cohomology and lattice

Let Z (and R) be the twisted integers (and twisted real numbers) on M. Free Hq(M; Z) is a lattice in Hq

dR(M; R).

An ordered basis Eq = (e∗

1, · · · , e∗ k) for Free Hq(M; Z) is said to be

reduced for stable comass, if stm*(e∗

i ) = stm*

0 + Free Hq(M; Z) \ spanZ{e∗

1, · · · , e∗ i−1}

is satisfied for each 1 ≤ i ≤ k.

Free Hq(M; Z) ∼ = Free Hq(M; Z) and the twisted Poincar´ e duality imply that there is a basis (e1, · · · , ek) for Free Hq(M; Z) with e∗

i (ej) = δij.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Cup-product and stable mass

Theorem

Let E be a stable comass reduced basis for the free part of H∗(M; Z). If there is a k–tuple (η∗

1, · · · , η∗ k ) in E×k such that its cup-product

γ∗ := η∗

1 ∪ · · · ∪ η∗ k is non-zero, then k

∏

i=1

stm(ηi) ≤

k

∏

i=1

(di+···+dk

di

) · bi!

· stm(γ)

where bi is the rank of Free Hdi(M; Z). Remark that the constant is not optimal.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Corollary

The twisted stable systolic category is lower bounded by the twisted real cup-length.

Corollary

The twisted stable systolic category of RP2 × S3 is 2.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Proof of theorem

Remark that we must find inequalities and equalities which do not depend on the metric. We observe the mass-comass duality. For each ηi ∈ Hdi(M; Z), there is αi ∈ Hdi(M; R) such that αi(ηi) = 1 and m∗(αi) · stm(ηi) = 1. In general, αi(ηj) is not 0 for i = j and αi is not contained in Hdi(M; Z). (Of course, stm*(η∗

i ) · stm(ηi) ≥ 1.)

On the explicit systolic inequality from the cup-product Hoil Ryu

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Lemma (1)

There are inequalities m∗(αi) ≤ stm*(η∗

i ) ≤ bi! · m∗(αi)

where bi is the rank of Hdi(M; R). (Remark that the constant bi! is not optimal, but metric invariant.) So we can see 1 stm(ηi) ≤ stm*(η∗

i ) ≤

bi! stm(ηi) which implies that some suitable inequality for comass gives the result.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Lemma (2)

The inequality m∗(ω1 ∧ ω2) ≤ p + q p

· m∗(ω1) · m∗(ω2)

is satisfied for all twisted de Rham cohomology classes ω1 ∈ Hp

dR(M; R)

and ω2 ∈ Hq

dR(M; R).

(The comass is determined by local information of the harmonic form.) stm*(η∗

1 ∪ · · · ∪ η∗ k ) ≤

d1 + · · · + dk d1

· stm*(η∗

1) · stm*(η∗ 2 ∪ · · · ∪ η∗ k )

≤

k

∏

i=1

di + · · · + dk di

· stm*(η∗

i ).

On the explicit systolic inequality from the cup-product

Hoil Ryu

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Proof of lemma (1)

Let Edi = (e∗

1, · · · , e∗ bi) = E ∩ Free Hdi(M; Z).

Fact

stm*(e∗

j ) ≤ stm*

Free Hdi(M; Z) \ spanZ(Edi \ {e∗

j })

is satisfied for all 1 ≤ j ≤ bi.

µi

j := stm(η∗ i )/ stm(e∗ j )

Li := spanZ

µi

j · e∗ j : 1 ≤ j ≤ bi

On the explicit systolic inequality from the cup-product

Hoil Ryu

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We define

∑j ajµi

j · e∗ j , ∑j a′ jµi j · e∗ j

i := ∑j aja′

j

which is the Euclidean inner product. The volume of a subset in Hdi(M; R) is obtained from this inner product.

Fact

stm*(η∗

i ) ≤ stm*

Li \ {0}

For the open ball Bi :=
x ∈ Hdi(M; R) : m∗(x) < stm*(η∗

i )

,

Li ∩ Bi is only zero from the fact.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Fact (Minkowski)

Let L be a k–lattice in a Euclidean vector space V and U be an open convex subset U of V. If the intersection U ∩ L is only zero, then vol(U) ≤ 2k · det(L) where the determinant of L is the volume of the parallelepiped whose edges are a basis for L. det(Li) = 1 for the parallelepiped in Hdi(M; R). Therefore vol(Bi) ≤ 2bi.

On the explicit systolic inequality from the cup-product Hoil Ryu

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Next we observe the height h of Bi from the vector subspace V := spanR{µi

j · e∗ j }j \ {η∗ i }.

We define another norm on Hdi(M; R) by

∑j ajµi

j · e∗ j

i := ∑j |aj| · |µi

j|.

The · 0

i –unit open ball B0 i is contained in Bi. Therefore h ≥ 1.

In addition, B0

i ∩ V is the convex set (and a polytope)

with minimal volume whose boundary contains {µi

j · e∗ j } \ {η∗ i }.

vol(B0

i ∩ V) =

1 (bi − 1)! · 2bi−1

On the explicit systolic inequality from the cup-product Hoil Ryu