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
Systoles ... The medical term systole comes from the Greek word for contraction ... by Marcel Berger, Notice of the AMS, 2008 M n : connected closed Riemannian manifold sys π 1 ( M ) : fundamental 1 –systole sysh q ( M ) : homology q –systole On the explicit systolic inequality from the cup-product Hoil Ryu
Question What is the relationship between systoles and volume of M ? Loewner showed 2 sys π 1 ( T 2 ) 2 ≤ 3 · vol ( T 2 ) √ for an arbitrary metric on T 2 . This inequality seems to be related with T 2 = S 1 × S 1 . Gromov showed a theorem and generalized to ... On the explicit systolic inequality from the cup-product Hoil Ryu
Theorem (Gromov ’83, Bangert-Katz ’02) Let M n be a connected closed orientable Riemannian manifold. If there exist α 1 ∪ · · · ∪ α k � = 0 with α i ∈ H d i ( M ; R ) , then there exist η i ∈ H d i ( M ; Z ) , ξ ∈ H d 1 + ··· + d k ( M ; Z ) and 0 < C < ∞ satisfying k ∏ stm ( η i ) ≤ C · stm ( ξ ) i = 1 where C is only determined by H ∗ ( M ; R ) and stm is a norm on free part of H ∗ ( M ; Z ) which is called the stable mass . Problem : Orientable manifolds only. What are η i (and C )? ⇒ We will argue in this talk. On the explicit systolic inequality from the cup-product Hoil Ryu
Systolic categories [Katz & Rudyak ’06] If ” some kind of ” systole satisfies k ∏ sys d i ≤ C · vol ( M ) i = 1 then the ” some kind of ” systolic category of M is defined by the maximum number of k among inequalities. • The homology systolic category cat sysh is invariant under homotopy equivalences [Katz & Rudyak ’08] . • If M is orientable, then the stable systolic category cat stsys is lower bounded by the real cup-length. A corollary of [Gromov ’83] . • If M is 0 –universal, then cat stsys ( M ) is invariant under rational equivalences [Ryu ’11] . On the explicit systolic inequality from the cup-product Hoil Ryu
Systolic freedom Orientable cases are stabilized by [Gromov, Katz] : e.g. cat sysh ( S 1 ) = cat sysh ( S 3 ) = 1 but cat sysh ( S 1 × S 3 ) = 1 . cat stsys ( S 1 ) = cat stsys ( S 3 ) = 1 and cat stsys ( S 1 × S 3 ) = 2 . A non-stabilizable case [Iwase] : cat sysh ( S 1 ∨ S 2 ) = cat stsys ( S 1 ∨ S 2 ) = 1 . A non-orientable case [Ryu] : cat sysh ( R P 2 × S 3 ) = cat stsys ( R P 2 × S 3 ) = 0 and cat sysh ( R P 2 × S 3 ; Z /2 Z ) = 1 . Compare [Pu] : cat sysh ( R P 2 ) = cat stsys ( R P 2 ) = 0 but cat sysh ( R P 2 ; Z /2 Z ) = 2 . On the explicit systolic inequality from the cup-product Hoil Ryu
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 , � | ω x ( τ ) | : x in M , orthonormal q –frame τ m ∗ ( ω ) : = sup � T ( ω ) : m ∗ ( ω ) ≤ 1 � � m ( T ) : = sup where m ∗ ( ω ) is called the comass of ω and m ( K ) is called the mass of K . For a smooth singular cube κ : I q → M , there is a current K � I q κ # ω . K ( ω ) : = Define the mass of κ by m ( κ ) : = m ( K ) On the explicit systolic inequality from the cup-product Hoil Ryu
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 A 0 is a Z –lattice in some real vector space R 0 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 H q ( M ; A ) \ { 0 } ] . On the explicit systolic inequality from the cup-product Hoil Ryu
Twisted cohomology and lattice Let Z (and R ) be the twisted integers (and twisted real numbers) on M . Free H q ( M ; Z ) is a lattice in H q dR ( M ; R ) . An ordered basis E q = ( e ∗ 1 , · · · , e ∗ k ) for Free H q ( M ; Z ) is said to be reduced for stable comass, if stm * ( e ∗ i ) = stm * � 0 + Free H q ( M ; Z ) \ span Z { e ∗ 1 , · · · , e ∗ � i − 1 } is satisfied for each 1 ≤ i ≤ k . Free H q ( M ; Z ) ∼ = Free H q ( M ; Z ) and the twisted Poincar´ e duality imply that there is a basis ( e 1 , · · · , e k ) for Free H q ( M ; Z ) with e ∗ i ( e j ) = δ ij . On the explicit systolic inequality from the cup-product Hoil Ryu
Cup-product and stable mass Theorem Let E be a stable comass reduced basis for the free part of H ∗ ( M ; Z ) . k ) in E × k such that its cup-product If there is a k –tuple ( η ∗ 1 , · · · , η ∗ γ ∗ : = η ∗ 1 ∪ · · · ∪ η ∗ k is non-zero, then k k � � ( d i + ··· + d k ∏ ∏ stm ( η i ) ≤ ) · b i ! · stm ( γ ) d i i = 1 i = 1 where b i is the rank of Free H d i ( M ; Z ) . Remark that the constant is not optimal. On the explicit systolic inequality from the cup-product Hoil Ryu
Corollary The twisted stable systolic category is lower bounded by the twisted real cup-length. Corollary The twisted stable systolic category of R P 2 × S 3 is 2 . On the explicit systolic inequality from the cup-product Hoil Ryu
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 η i ∈ H d i ( M ; Z ) , there is α i ∈ H d i ( M ; R ) such that m ∗ ( α i ) · stm ( η i ) = 1. α i ( η i ) = 1 and In general, α i ( η j ) is not 0 for i � = j and α i is not contained in H d i ( M ; Z ) . (Of course, stm * ( η ∗ i ) · stm ( η i ) ≥ 1 .) On the explicit systolic inequality from the cup-product Hoil Ryu
Lemma (1) There are inequalities m ∗ ( α i ) ≤ stm * ( η ∗ i ) ≤ b i ! · m ∗ ( α i ) where b i is the rank of H d i ( M ; R ) . (Remark that the constant b i ! is not optimal, but metric invariant.) So we can see 1 b i ! stm ( η i ) ≤ stm * ( η ∗ i ) ≤ stm ( η i ) which implies that some suitable inequality for comass gives the result. On the explicit systolic inequality from the cup-product Hoil Ryu
Lemma (2) The inequality � p + q � m ∗ ( ω 1 ∧ ω 2 ) ≤ · m ∗ ( ω 1 ) · m ∗ ( ω 2 ) p is satisfied for all twisted de Rham cohomology classes ω 1 ∈ H p dR ( M ; R ) and ω 2 ∈ H q dR ( M ; R ) . (The comass is determined by local information of the harmonic form.) � d 1 + · · · + d k � stm * ( η ∗ 1 ∪ · · · ∪ η ∗ · stm * ( η ∗ 1 ) · stm * ( η ∗ 2 ∪ · · · ∪ η ∗ k ) ≤ k ) d 1 k � d i + · · · + d k � · stm * ( η ∗ ∏ ≤ i ) . d i i = 1 � On the explicit systolic inequality from the cup-product Hoil Ryu
Proof of lemma (1) Let E d i = ( e ∗ 1 , · · · , e ∗ b i ) = E ∩ Free H d i ( M ; Z ) . Fact stm * ( e ∗ j ) ≤ stm * � Free H d i ( M ; Z ) \ span Z ( E d i \ { e ∗ � j } ) is satisfied for all 1 ≤ j ≤ b i . µ i j : = stm * ( η ∗ i ) / stm * ( e ∗ j ) j · e ∗ � µ i � L i : = span Z j : 1 ≤ j ≤ b i On the explicit systolic inequality from the cup-product Hoil Ryu
We define � � i : = ∑ j a j a ′ ∑ j a j µ i j · e ∗ j , ∑ j a ′ j µ i j · e ∗ j j which is the Euclidean inner product. The volume of a subset in H d i ( M ; R ) is obtained from this inner product. Fact stm * ( η ∗ i ) ≤ stm * � L i \ { 0 } � x ∈ H d i ( M ; R ) : m ∗ ( x ) < stm * ( η ∗ � � For the open ball B i : = i ) , L i ∩ B i is only zero from the fact. On the explicit systolic inequality from the cup-product Hoil Ryu
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 ) ≤ 2 k · det ( L ) where the determinant of L is the volume of the parallelepiped whose edges are a basis for L . det ( L i ) = 1 for the parallelepiped in H d i ( M ; R ) . Therefore vol ( B i ) ≤ 2 b i . On the explicit systolic inequality from the cup-product Hoil Ryu
Next we observe the height h of B i from the vector subspace V : = span R { µ i j · e ∗ j } j \ { η ∗ i } . We define another norm on H d i ( M ; R ) by � 0 � ∑ j a j µ i i : = ∑ j | a j | · | µ i j · e ∗ � � j | . j The � · � 0 i –unit open ball B 0 i is contained in B i . Therefore h ≥ 1 . In addition, B 0 i ∩ V is the convex set (and a polytope) with minimal volume whose boundary contains { µ i j · e ∗ j } \ { η ∗ i } . 1 vol ( B 0 ( b i − 1 ) ! · 2 b i − 1 i ∩ V ) = On the explicit systolic inequality from the cup-product Hoil Ryu
The interior of the cone with the vertex v which gives the height h of B i over B 0 i ∩ V , is contained in B i . This cone is the convex subset with minimal volume whose boundary contains given vertices. Therefore vol ( B 0 i ∩ V ) × h × 1 b i × 2 ≤ vol ( B i ) ≤ 2 b i implies h ≤ b i ! Furthermore v ( η i ) = h and m ∗ ( v ) = stm * ( η ∗ i ) . We define α i : = v / h then stm * ( η ∗ i ) = h · m ∗ ( α i ) and this implies the result m ∗ ( α i ) ≤ stm * ( η ∗ i ) ≤ b i ! · m ∗ ( α i ) . � On the explicit systolic inequality from the cup-product Hoil Ryu
Recommend
More recommend
