This is the accepted version of the journal article: Balacheff, Florent Nicolas; Karam, Steve; Parlier, Hugo. «The minimal length product over homology bases of manifolds». Mathematische Annalen, Vol. 380, Issue 1-2 (June 2021), p. 825-854. DOI 10.1007/s00208-021-02150-5 This version is available at https://ddd.uab.cat/record/287702 under the terms of the license THE MINIMAL LENGTH PRODUCT OVER HOMOLOGY BASES OF MANIFOLDS FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER A BSTRACT. Minkowski’s second theorem can be stated as an inequality for n-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsler manifolds. This includes manifolds for which first Betti number and dimension do no necessarily coincide, a prime example being the case of surfaces. This class of manifolds is described by a non-vanishing condition for the hyperdeterminant reduced modulo 2 of the multilinear map induced by the fundamental class of the manifold on its first Z2 -cohomology group using the cup product. 1. I NTRODUCTION Minkowski’s second theorem asserts that the product of the successive minima of a symmetric convex body in Euclidean space is bounded in terms of its volume. More precisely, if n denotes a positive integer, the successive minima of a given symmetric convex body K of Rn , defined for k = 1, . . . , n as the numbers λk := min{t | tK contains k linearly-independent vectors of Zn }, satisfy the sharp inequality n Y λk · Vol(K) ≤ 2n . k=1 If we analyze the situation in terms of metric geometry, this inequality can be reformulated as the following upper bound on a length product: any flat Finsler torus of dimension n with Busemann- Hausdorff volume V admits a family of closed geodesics (γ1 , . . . , γn ) inducing a basis of its first real homology group—in short, an R-homology basis—whose product of lengths satisfies n Y 2n (1.1) ℓ(γk ) ≤ · V. bn k=1 Indeed symmetric convex bodies naturally parametrize (up to isometry) the space of flat Finsler tori through the correspondence K 7→ (Rn /Zn , ∥ · ∥K ). Here ∥ · ∥K denotes the norm induced by the symmetric convex K. In this setting, each successive minima λk is realized as the Finsler length of some closed geodesic γk such that the real homology classes of these loops induce a R-homology basis. Busemann-Hausdorff volume is defined by integrating the multiple of Lebesgue measure in 2020 Mathematics Subject Classification. 53C23. Key words and phrases. Asymptotic volume, cup product, homology basis, hyperdeterminant, length product inequality, Minkowski’s theorem, surface, systolic geometry. The first author acknowledges support from the European Social Fund and the Agencia Estatal de Investigación through the Ramón y Cajal grant RYC-2016-19334 ”Local and global systolic geometry and topology”, as well as from the grant ANR-12-BS01-0009-02. The second author acknowledges support from grant ANR CEMPI (ANR-11-LABX- 0007-01). The third author acknowledges support from the ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033. 1 2 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER each tangent space for which K—the unit ball in each tangent space of the Finsler metric—has volume equal to the Lebesgue measure (denoted by bn ) of the Euclidean unit ball. The purpose of this article is to study analogs of this length product inequality for Riemannian and Finsler manifolds. Of course, a first natural direction is to ask whether inequality (1.1) still holds for non-flat tori. Indeed, in [BK16] the first two authors proved that Riemannian n-dimensional tori with unit volume have their minimal length product among Z2 -homology basis induced by n closed geodesics uniformly bounded from above by nn , answering a question by Gromov (see [Gro96, p.339]). Using comparison procedures explained in [ABT16, Section 4.4], it implies that inequality (1.1) still holds for non-flat Finsler tori—albeit with a weaker constant—even for non-reversible ones if we use the Holmes-Thompson definition of volume instead of the Busemann-Hausdorff one. The question of the best constant in such length product inequalities for Finsler or Riemannian tori is still open. In a perhaps surprising direction, we present here a length product inequality similar to (1.1) but valid for closed manifolds whose first Z2 -Betti number does not necessarily coincide with the dimension. Let M be a closed connected manifold of dimension n (and from now on all our manifolds will be connected and closed). Denote by H 1 (M ; Z2 ) the first cohomology group of M with Z2 -coefficients and suppose that its dimension b is positive. The Z2 -fundamental class of M induces a n-multilinear symmetric form on H 1 (M ; Z2 ) denoted by FM as follows: FM : H 1 (M ; Z2 ) × . . . × H 1 (M ; Z2 ) → Z2 (β1 , . . . , βn ) 7→ β1 ∪ . . . ∪ βn [M ]. Among the algebraic invariants of multilinear forms over complex vector spaces, the hyperdeter- minant denoted by Det was introduced by Cayley [Cay45] and extensively studied by Gelfand, Kapranov and Zelevinsky [GKZ92]. We consider its reduction modulo 2 which we denote by Det2 (see Section 2.2 for more details). Theorem 1.1. Consider a closed Riemannian manifold M of first Z2 -Betti number b > 0 and of dimension n. Suppose that Det2 FM ̸= 0. Then there exists a Z2 -homology basis induced by closed geodesics γ1 , . . . , γb whose length product satisfies the following inequality: b Y (1.2) ℓ(γk ) ≤ nb · Vol(M )b/n . k=1 Although the closed geodesics obtained here are also Z-homologically independent, they do not necessarily form a Z-homology basis. We do not know if an analog of Theorem 1.1 still holds in the orientable case for Z-homology basis except in the case of closed surfaces, see Theorem 1.5. Indeed the proof of Theorem 1.1 does not seem to easily adapt from Z2 to Z-coefficients, see Remark 2.6. Applying this theorem to M diffeomorphic to the n-dimensional torus, we recover the result of [BK16] that Riemannian n-dimensional tori with unit volume have their minimal length product among Z2 -homology basis uniformly bounded from above by nn . Minkowski’s second theorem thus appears as a principle which holds even when the first Betti number and the dimension do not necessarily coincide. Again, by comparison procedures explained in [ABT16, Section 4.4], Theorem 1.1 implies that a similar inequality to (1.2) holds—albeit with a worst constant—for LENGTH PRODUCTS FOR MANIFOLDS 3 Finsler manifolds M with Det2 FM ̸= 0. Here are some direct applications of Theorem 1.1. First we derive the following Minkowski’s second principle for surfaces. Corollary 1.2. Consider a closed Riemannian surface S of first Z2 -Betti number b > 0. Then there exists a Z2 -homology basis induced by closed geodesics γ1 , . . . , γb whose length product satisfies the following inequality: b Y ℓ(γi ) ≤ 2b · Area(S)b/2 . i=1 Indeed we will check in Section 2 that for such a surface S the reduction modulo 2 of the hyper- determinant of FS is always non-zero. When m = 3 and b = 2 the hyperdeterminant was explicitly described by Cayley (see [Cay45]), and so using this, we are also able to deduce the following from Theorem 1.1: Corollary 1.3. Consider a closed Riemannian manifold M diffeomorphic to RP 3 #RP 3 . Then there exists a Z2 -homology basis induced by two closed geodesics γ1 , γ2 whose length product satisfies the following inequality: ℓ(γ1 ) · ℓ(γ2 ) ≤ 9 · Vol(M )2/3 . Using the notion of homological asymptotic volume introduced in [Bab92] and its connection to the stable norm (see Section 2.6 for more details), we derive the following asymptotic estimate from Theorem 1.1. Corollary 1.4. Consider a closed Riemannian manifold M of first Z2 -Betti number b > 0 and of dimension n. Suppose that its integral homology has no 2-torsion and that Det2 FM ̸= 0. Then the number NM (t) of homologically distinct closed geodesics of length ≤ t satisfies the following inequality: 2b NM (t) ≥ tb + o(tb−1 ). Vol(M )b/n · nb · b! This improves a previous lower bound due to Babenko [Bab92, Theorem 7.1.a]. Note that the condition that integral homology has no 2-torsion is equivalent to having the first R-Betti number and first Z2 -Betti number coincide. For an orientable Riemannian surface S of genus g ≥ 1 of unit area, this gives the following asymptotic estimate: NS (t) ≥ t2g /(2g)! + o(t2g−1 ). Now we switch our focus to surfaces. As already observed, the closed geodesics obtained in Corollary 1.2 are also Z-homologically independent, but do not necessarily form a Z-homology basis. Nonetheless, by using techniques specific to dimension 2, we exhibit Z-homology basis whose length product is uniformly bounded above as follows. 4 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER Theorem 1.5. Consider an orientable closed Riemannian surface S of genus g. Then there exists a Z-homology basis induced by closed geodesics γ1 , . . . , γ2g whose length product satisfies the following inequality: 2g Y ℓ(γk ) ≤ C g · Area(S)g k=1 for some positive constant C < 231 . We also prove a similar result for non-orientable surfaces and conjecture the following. Conjecture 1.6. There exists a positive constant c > 0 such that any closed Riemannian surface S of first Z2 -Betti number b > 0 admits a Z2 -homology basis induced by closed geodesics γ1 , . . . , γb whose length product satisfies the following inequality: b  b/2 Y b Area(S) ℓ(γi ) ≤ (c log b) · . b i=1 There exist infinite sequences of hyperbolic surfaces of growing genus g whose shortest homolog- ically non-trivial closed geodesic (also known as the homological systole) has length growth rate at least 4/3 · log g. The first known construction is due to Buser and Sarnak [BS94]. As a consequence, the minimal length product of a Z2 -homology basis cannot have an upper bound with asymptotic growth in terms of the genus any less than (4/3 · log g)2g for area equal to 4π(g − 1). So Conjecture 1.6, if true, would be optimal up to some exponential factor. We now present various partial results supporting this conjecture. First, the analogous result for graphs is true. By graph we mean a finite simplicial complex of dimension 1 and for which the notion of length ℓ of a subgraph (such as a cycle) is simply the number of its edges. Theorem 1.7. Consider a connected graph Γ of first Z2 -Betti number b ≥ 2. Then there exists a Z2 -homology basis induced by cycles γ1 , . . . , γb whose length product satisfies the following inequality: b ℓ(Γ) b Y   b ℓ(γk ) ≤ (4e log2 b) · . b k=1 This result is a straightforward consequence of previous work [BT97, BS02] and an induction argument. Secondly, the conjecture for surfaces with large enough systole can be deduced from [BPS12, Theorem 1.1]. Proposition 1.8. Consider an orientable closed Riemannian surface S of genus g. Suppose that its homological systole satisfies s Area(S) sys(S) ≥ . 4π(g − 1) LENGTH PRODUCTS FOR MANIFOLDS 5 Then there exists a Z2 -homology basis induced by closed geodesics γ1 , . . . , γ2g whose length product satisfies the following inequality: 2g Area(S) g Y   2g ℓ(γi ) ≤ (C log g) · g i=1 for some positive constant C < 231 . Next we can ask for the minimal length product over Z2 -homologically independent families of closed geodesics of cardinality less than 2g. In that direction, we are able to prove the analog of Conjecture 1.6 up to cardinality g. Proposition 1.9. Consider an orientable closed Riemannian surface S of genus g. Then there exists Z2 -homologically independent closed geodesics γ1 , . . . , γg whose length product satisfies the following inequality: g  g/2 g Area(S) Y ℓ(γk ) ≤ (C log g) . g k=1 for some positive constant C < 231 . The family of loops appearing in the previous proposition could possibly belong to the same Lagrangian subspace with respect to the intersection form. In the transversal direction and in the particular case of hyperbolic metrics, we are able to prove the following. Theorem 1.10. There exists a positive constant c > 0 such that any orientable genus g ≥ 2 closed hyperbolic surface S0 has a pair of simple closed geodesics γ and δ pairwise intersecting once and whose length product satisfies the following inequality: ℓ(γ) · ℓ(δ) ≤ (c log(2g))2 . The paper is organized as follows. The next section is dedicated to the proof of Theorem 1.1 and its Corollaries 1.2, 1.3 and 1.4. In the last section we treat in detail the case of surfaces, mainly focusing on the orientable case. Acknowledgements. We are grateful to A. Abdesselam, I. Babenko, B. Kahn and J. Milnor for valu- able exchanges. We are also indebted to J. Gutt whose proof of a symplectic analog of Minkowski’s first theorem (see Lemma 3.10) inspired a mechanism used in the proof of Theorem 2.5. Finally we would like to thank the referee for valuable comments. From now on, all manifolds (including surfaces) are assumed to be closed. 2. M INIMAL LENGTH PRODUCT OVER Z2 - HOMOLOGY BASES FOR MANIFOLDS In this section, we prove Theorem 1.1 and derive Corollaries 1.2, 1.3 and 1.4. 2.1. From cup product to length product. Let M be a Riemannian manifold of dimension n whose first cohomology group with Z2 -coefficients is non-trivial. Set b := dim H 1 (M ; Z2 ) > 0. Given a non-zero cohomological class α ∈ H 1 (M ; Z2 ), we define its length by L(α) := inf{ℓ(γ) | γ is a closed curve with α[γ] ̸= 0} 6 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER where [γ] denotes the homology class in H1 (M ; Z2 ) corresponding to the curve γ. The following result was proved in [BK16]. Theorem 2.1. Let M be a Riemannian manifold of dimension n and suppose that there exist (not necessarily distinct) cohomology classes α1 , . . . , αn in H 1 (M ; Z2 ) whose cup product α1 ∪ . . . ∪ αn ̸= 0 ∈ H n (M ; Z2 ). Then Yn L(αk ) ≤ nn · Vol (M ). k=1 For the reader’s convenience, we sketch the proof in the 3-dimensional case and refer to [BK16] for the general case and further details. Sketch of proof. The proof is inspired by Guth’s proof of isosystolic inequality, see [Gut10]. So we assume in the sequel n = 3. Furthermore suppose that we have ordered our cohomological classes such that L(α1 ) ≤ L(α2 ) ≤ L(α3 ). Choose a hypersurface Z2 of M 3 with the following properties: i) Z2 is Poincaré dual to α3 . In particular: · for any z ∈ H1 (M ; Z2 ), α3 (z) = z ∩ [Z2 ] where · ∩ · denotes the intersection number modulo 2 between two cycles of complementary dimension; · for any ω ∈ H 2 (M ; Z2 ), ω[Z2 ] = ω ∪ α3 [M ]; ii) Z2 is (almost) minimal1 for the area among hypersurfaces in the class [Z2 ]. Now observe that the restrictions of α1 , α2 to Z2 (denoted by α1′ , α2′ ) satisfy · α1′ ∪ α2′ [Z2 ] = α1 ∪ α2 ∪ α3 [M ] = 1; · L(αi′ ) ≥ L(αi ) for i = 1, 2. Define Z1 as a minimal closed geodesic of Z2 Poincaré dual to α2′ . In particular α2 (z) = z ∩ [Z1 ] for any z ∈ H1 (Z2 ; Z2 ). Pick some point z0 ∈ Z1 and define the following nested sets: D1 := {z ∈ Z1T | d(z, z0 ) ≤ R1 } D2 := {z ∈ Z2 T | d(z, D1 ) ≤ R2 } D3 := {z ∈ M | d(z, D2 ) ≤ R3 } with Ri := L(αi )/6 for i = 1, 2, 3. Obviously ℓ(D1 ) ≥ 2R1 . By the coarea formula we have Z R2 Area(D2 ) = ℓ(∂D2 (r))dr 0 where D2 (r) := {z ∈ Z2 | d(z, D1 ) ≤ r}. 1The notion of almost minimality is that Z is minimal for area among hypersurfaces in its cohomological class up to 2 some choosen small additive constant. Because this constant can be fixed as small as wanted, the proof remains the same if we actually suppose that Z2 is minimal. We do so and refer the reader to the original proof in [BK16] for more details. LENGTH PRODUCTS FOR MANIFOLDS 7 But Z1 ∩ D2 (r) = 0 ∈ H1 (D2 (r), ∂D2 (r); Z2 ) for r ∈ (0, R2 ). If not, by Lefschetz-Poincaré duality, there exists a closed geodesic γ with non-zero homology class in H1 (D2 (r); Z2 ) such that [γ] ∩ Z1 = α2 [γ] = 1. Observe that by construction d(z, z0 ) < R1 + R2 ≤ L(α2 )/2 for any z ∈ D2 (r).P According to Gromov’s curve factoring lemma (see [Gut10]), the loop γ decomposes into a sum i γi of loops of length less than L(α2 ). But for some index i0 we must have α2 [γi0 ] = 1 which gives a contradiction. Thus ℓ(∂D2 (r)) ≥ 2ℓ(Z1 ∩ D2 (r)) ≥ 4R1 for any r ∈ (0, R2 ) by minimality of Z1 from which we derive by integration that Area(D2 ) ≥ 4R1 R2 . In exactly the same way we have Z R3 Vol(D3 ) = Area(∂D3 (r))dr 0 with D3 (r) := {z ∈ M | d(z, D2 ) ≤ r}. Because Z2 ∩ D3 (r) = 0 ∈ H2 (D3 (r), ∂D3 (r); Z2 ) for r ∈ (0, R3 ), the minimality of Z2 implies the lower bound Area(∂D3 (r)) ≥ 2 Area(Z2 ∩ D3 (r)) ≥ 8R1 R2 from which we derive by integration that Vol(D3 ) ≥ 8R1 R2 R3 . This implies the desired inequality L(α1 ) · L(α2 ) · L(α3 ) ≤ 33 · Vol (M ). □ To fully exploit this result, we introduce the following map. As the dimension b of the first cohomology group H 1 (M ; Z2 ) is positive, the Z2 -fundamental class of M induces a map FM : H 1 (M ; Z2 ) × . . . × H 1 (M ; Z2 ) → Z2 (β1 , . . . , βn ) 7→ β1 ∪ . . . ∪ βn [M ]. This map is n-multilinear and symmetric. 2.2. Algebraic invariants of multilinear forms over Z2 -vector spaces. We now introduce the class of invariants of multilinear maps which will be relevant to our purpose. Let V be a Z2 -vector space of dimension b and n a positive integer. We denote by Fn (V ) the set of n-multilinear forms on V . If we fix a basis B := (e1 , . . . , eb ) of V , an element F ∈ Fn (V ) decomposes as X F = Fi1 ,...,in e∗i1 ⊗ . . . ⊗ e∗in (i1 ,...,in )∈J1,bKn where Fi1 ,...,in := F (ei1 , . . . , ein ) ∈ Z2 . The n-dimensional matrix ÂB (F ) := (Fi1 ,...,in ) with Z2 -entries thus encodes the multilinear map F . Definition 2.2. An algebraic invariant over Fn (V ) is a map κ : Fn (V ) → Z2 F 7→ κ(F ) which coincides for some homogeneous polynomial with its evaluation on the entries of the matrix ÂB (F ) for any basis B of V . 8 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER An example of such an algebraic invariant over F2 (V ) is given by the reduction modulo 2 of the determinant. More precisely, the expression defined by b X Y Fk,σ(k) σ∈Sn k=1 does not depend on the basis in which we have decomposed F . This defines an algebraic invariant we will denote by Det2 . If F is symmetric, Milnor [Mil70] remarks that the term corresponding to σ cancels the term corresponding to σ −1 unless σ = σ −1 . Therefore for symmetric bilinear forms we can write X Yq Det2 F = F (Pj ) P1 ∪...∪Pq ={1,...,b} j=1 where the sum is taken over all partitions of the set {1, . . . , b} into one or two elements subsets Pj with the convention that F ({i}) = Fii and F ({i, j}) = Fij . This remark will be helpful in the sequel. This example generalizes to higher dimensions using the hyperdeterminant as introduced by Cayley [Cay45] and later studied by Gelfand, Kapranov and Zelevinsky (see [GKZ92] and the references therein). More precisely, consider the Segre embedding of the product X = Pb−1 × . . . × Pb−1 of n n complex projective spaces of dimension b − 1 into the projective space Pb −1 . According to [GKZ92, n Theorem 1.3], the projective dual variety X ∨ consisting of all hyperplanes in Pb −1 tangent to X n ∗ at some point is a hypersurface in the dual projective space Pb −1 . In particular, there exists a  (unique up to sign) homogeneous polynomial with integer coefficients and irreducible over Z which defines X ∨ . This homogeneous polynomial is called hyperdeterminant and denoted by Det. The hyperdeterminant is thus a homogeneous polynomial function on the space (Cb )∗ ⊗ . . . ⊗ (Cb )∗ of n-multilinear maps over Cb . If Cb is equipped by a basis C = (v1 , . . . , vb ) an element f ∈ (Cb )∗ ⊗ . . . ⊗ (Cb )∗ is represented by a n-dimensional complex matrix AB (f ) = (ai1 ,...,in ) where ai1 ,...,in = f (ei1 , . . . , ein ), and so Det(f ) = Det(AB (f )) is a homogeneous polynomial function of matrix entries. An important property of the hyperdeterminant is its relative invariance under the action of the group GL(Cb ) × . . . × GL(Cb ), and thus its invariance by SL(Cb ) × . . . × SL(Cb ), according to [GKZ92, Proposition 1.4]. In particular Det AB (f ) remains unchanged up to sign if we commute two elements vi and vj , or if we replace vi by vi + vj in the basis C. Now fix an element F ∈ Fn (V ). Given a basis B := (e1 , . . . , eb ) of V , any n-multilinear form F over V is represented by an unique n-dimensional real matrix AB (F ) = (ai1 ,...,in )(i1 ,...,in )∈J1,bKn where each coefficient is either 0 or 1 according to the equation F (ei1 , . . . , ein ) = ai1 ,...,in mod 2. As Det is a homogeneous polynomial function with integers coefficient, the expression Det AB (F ) belongs to Z. Its invariance under the action of the group SL(Cb ) × . . . × SL(Cb ) implies that Det AB (F ) = Det AB′ (F ) mod 2 if B ′ is obtained from B by commuting two elements ei and ej or by replacing ei by ei + ej . Because these transformations span the linear group GL(V ) we can set Det2 F := Det AB (F ) mod 2 LENGTH PRODUCTS FOR MANIFOLDS 9 which does not depend on the chosen basis B. This reduction modulo 2 of the hyperdeterminant Det2 : Fn (V ) → Z2 is thus an algebraic invariant as obviously ÂB (F ) = AB (F ) mod 2. In dimension n = 2, the hyperdeterminant coincides with the usual determinant so the algebraic invariant Det2 coincides with the one previously defined on F2 (V ). In dimensions n = 3 and b = 2, the hyperdeterminant has an explicit formula first discovered by Cayley [Cay45, p.89] and whose reduction modulo 2 gives the following expression: Det2 F = F111 · F222 + F122 · F211 + F212 · F121 + F221 · F112 . In particular, we deduce that Det2 F = F111 · F222 + F122 · F211 in the case where F is symmetric. We now introduce the following notion. Definition 2.3. An algebraic invariant κ over Fn (V ) is said balanced if · there exist q, m ∈ N∗ such that qn = mb, · the homogeneous polynomial function associated to κ can be written as X Y (2.1) κ(F ) = Fi1 ,...,in , I (i1 ,...,in )∈I where each I can be described as the subset of families of indices of length n I = {(π(1), . . . , π(n)), (π(n + 1), . . . , π(2n)), . . . , (π((q − 1)n + 1), . . . , π(qn))}. for some map π : {1, . . . , qn} → {1, . . . , b} satisfying |π −1 (i)| = m for i = 1, . . . , b. In other terms, we require the Is in the sum above to be obtained by taking m times the indices {1, . . . , b} and rearranging them into q families of length n. For example, it is easy to check from its expression that Det2 is balanced in the case n = 2, as well as in the case when n = 3 and b = 2. This fact indeed generalizes as follows. Proposition 2.4. The algebraic invariant Det2 is balanced for any n and b. Proof. This is a straightforward consequence of the main result [Cra27] which proves, using tensorial algebra, that any algebraic relative invariant over the space of multilinear forms of a real vector space has this property. □ Note that we did not find examples of unbalanced algebraic invariants in the litterature, but whether or not they exist is possibly well known to experts. 2.3. From a non-zero balanced algebraic invariant to a length product inequality. We now prove the main result of this section. Theorem 2.5. Consider a Riemannian manifold M of first Betti number b > 0 and of dimension n. Suppose that κ(FM ) = 1 for some balanced algebraic invariant κ. 10 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER Then there exists a Z2 -homology basis induced by closed geodesics γ1 , . . . , γb whose length product satisfies the following inequality: b Y ℓ(γk ) ≤ nb · Vol(M )b/n . k=1 Theorem 1.1 is the direct application of this result with κ = Det2 . Proof. Choose a sequence of Z2 -homologically independent closed geodesics γ1 , . . . , γb in M corresponding to the successive minima in the sense that ℓ(γk ) = min{L | there exist k closed curves Z2 -homologically independent of length at most L} for k = 1, . . . , b. The corresponding family of homology classes ([γ1 ], . . . , [γb ]) is a basis of H1 (M ; Z2 ) and we denote by B := (α1 , . . . , αb ) the dual basis in H 1 (M ; Z2 ) defined by αi [γj ] = δij . By construction, L(αk ) = ℓ(γk ) for k = 1, . . . , b. Now consider a balanced homogeneous polynomial function associated to the algebraic invariant κ as in formula (2.1). As κ(FM ) = 1, we have that q Y FM (απ((j−1)n+1) , . . . , απ(jn) ) = 1 j=1 for some map π : {1, . . . , qn} → {1, . . . , b} satisfying |π −1 (i)| = m. In particular for all j = 1, . . . , q απ((j−1)n+1) ∪ . . . ∪ απ(jn) [M ] = FM (απ((j−1)n+1) , . . . , απ(jn) ) = 1 which implies according to Theorem 2.1 that L(απ((j−1)n+1) ) . . . L(απ(jn) ) ≤ nn · Vol (M ). We deduce that b !m q Y Y L(αk ) = L(απ((j−1)n+1) ) . . . L(απ(jn) ) ≤ nqn · Vol (M )q . k=1 j=1 As such b Y L(αk ) ≤ nqn/m · Vol (M )q/m k=1 and we can conclude using the relations qn = mb and ℓ(γk ) = L(αk ) for all k = 1, . . . , b. □ Remark 2.6. Even if an analog of Theorem 2.1 might well be true with Z-coefficients —a version of the Macroscopic stability inequality on which it relies having been recently proved for Z-coefficients in [Alp17]—, we do not see how to adapt the proof above to Z-coefficients and derive an upperbound on the minimal length product over Z-homology bases for orientable manifolds. The fact that Z2 is a field is used here in a critical way when deducing from their homological independence that the family of closed geodesics corresponding to the successive minima forms a homology basis. LENGTH PRODUCTS FOR MANIFOLDS 11 2.4. Minimal length product over Z2 -homology bases for surfaces. In the case where M is an orientable Riemannian surface of genus g, the map FM is alternating. Indeed if a ∈ H1 (M ; Z2 ) is the Poincaré dual of a class α ∈ H 1 (M ; Z2 ), we have FM (α, α) = α ∪ α[M ] = α(a) = a ∩ a = 0. This implies that in any basis of H 1 (M ; Z2 ) we can write X Det2 FM = FM (P1 ) . . . FM (Pg ) where the sum is taken over partitions of {1, . . . , 2g} into two elements subsets. This is exactly the reduction of the Pfaffian modulo 2 which is consistent with the fact that x2 = x in characteristic 2. It is well known that we can choose a basis (a1 , b1 , . . . , ag , bg ) of H1 (M ; Z2 ) such that ai ∩ bi = 1 for all i = 1, . . . , g, the other intersection numbers between two elements of the basis being zero. If we denote by (α1 , β1 , . . . , αg , βg ) the basis of H 1 (M ; Z2 ) such that αi (resp. βi ) is the Poincaré dual class of ai (resp. bi ), then we get that αi ∪ βi [M ] = ai ∩ bi = 1 while the other cup products are zero. This implies that there exists only one partition for which the term FM (P1 ) . . . FM (Pg ) is non zero and thus that Det2 FM = 1. This proves Corollary 1.2 when the surface is orientable. In the case where M is a non-orientable Riemannian surface we can argue similarly. First recall that in any basis of H 1 (M ; Z2 ) we can write X q Y Det2 FM = FM (Pj ) P1 ∪...∪Pq ={1,...,b} j=1 where the sum is taken over partitions of {1, . . . , 2g} into one or two elements subsets. Suppose that the first Betti number is odd and set b = 2g + 1 (possibly g can be 0). We can always find a basis (a1 , b1 , . . . , ag , bg , c) of H1 (M ; Z2 ) such that ai ∩ bi = c ∩ c = 1 for all i = 1, . . . , g while the other intersection numbers are zero. If we denote by (α1 , β1 , . . . , αg , βg , ζ) the basis of H 1 (M ; Z2 ) obtained by considering the Poincaré dual classes of (a1 , b1 , . . . , ag , bg , c), we get that αi ∪ βi [M ] = ζ ∪ ζ[M ] = 1, while the other cup products are zero. It implies that there exists only one partition for which the term FM (P1 ) . . . FM (Pq ) is not zero and thus Det2 FM = 1. If the first Betti number is even, we can find a basis (a1 , b1 , . . . , ag , bg , c1 , c2 ) of H1 (M ; Z2 ) such that ai ∩ bi = cj ∩ cj = 1 and the other intersection numbers are zero. In a similar way we can check that Det2 FM = 1. This proves the non-orientable case of Corollary 1.2. 2.5. Minimal length product over Z2 -homology bases for RP 3 #RP 3 . For M = RP 3 #RP 3 we have Det2 FM = (FM )111 · (FM )222 + (FM )122 · (FM )211 as FM is symmetric, and the dimension n of the manifold and the dimension b of its first homology group are respectively n = 3 and b = 2. Now consider a decomposition of M into two copies of (RP 3 \ B 3 )i for i = 1, 2 glued along their boundaries via a diffeomorphism. According to the Mayer-Vietoris sequence H1 (M ; Z2 ) ≃ H1 (RP 3 ; Z2 ) ⊕ H1 (RP 3 ; Z2 ) where the generator of the i-th component in this direct composition 12 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER can be realized by a closed curve ai ⊂ (RP 3 \B 3 )i for i = 1, 2. If we denote by B = (α1 , α2 ) the ba- sis of H 1 (M ; Z2 ) formed by considering the Poincaré dual class of (a1 , a2 ) in M , a straightforward calculation shows that (FM )111 = (FM )222 = 1, and the other terms are all zero. Thus Det2 FM = 1. This proves Corollary 1.3. 2.6. Homological asymptotic volume and asymptotic number of closed geodesics. The Rie- mannian universal abelian covering space M̄ of a Riemannian manifold M is the normal covering space associated to the commutator subgroup [π1 M, π1 M ] of the fundamental group of M endowed with the Riemannian pullback metric. If the first integral homology group of M has free rank d > 0, it is well known that the limit Vol B(x̄, R) Ωab (M ) := lim R→∞ Rd exists, is positive and does not depend on the choice of the base point x̄ ∈ M̄ . This Riemannian invariant appears in [Bab92] under the name of homological asymptotic volume and is related to the stable norm as follows. First recall that the stable norm ∥ · ∥ is defined on H1 (M, R) by n X ∥h∥ := inf { |ri | · ℓ(γi )} i=1 Pn where the infimum is taken over all real Lipschitz 1-cycles i=1 ri γi realising a given class h. If we denote by B(M ) the unit ball of the stable norm, then (2.2) Ωab (M ) = µ (B(M )) · Vol(M ) where µ denotes the Haar measure on H1 (M, R) ≃ Rd for which the fundamental parallelepiped of the lattice i H1 (M, Z) has unit volume. Here i : H1 (M, Z) → H1 (M, R) denotes the homology mapping induced by the inclusion i : Z ,→ R of coefficients. The volume of the unit stable ball turns out to be related to length products through the following. Proposition 2.7. Let (γ1 , . . . , γd ) be a family of closed geodesics inducing a basis of H1 (M, R). Then d Y 2d µ(B(M )) · ℓ(γk ) ≥ . d! k=1 This inequality still holds true for finite simplicial complexes endowed with Riemannian poly- hedral metrics. In such a generality this inequality then turns out to be optimal, the equality being reached for the wedge product of d circles (compare with [BB06]). Proof. First observe that Conv {±[γ1 ]/∥[γ1 ]∥, . . . , ±[γd ]/∥[γd ]∥} ⊂ B(M ) where Conv denotes the convex hull of a finite collection of points. Because for any closed geodesic γ ℓ(γ) ≥ ∥[γ]∥, LENGTH PRODUCTS FOR MANIFOLDS 13 this implies that d Y d Y µ(B(M )) · ℓ(γk ) ≥ µ(B(M )) · ∥[γk ]∥ k=1 k=1 ≥ µ(Conv{±[γ1 ], . . . , ±[γd ]}) 2d ≥ . d! The last inequality is a consequence of the observation that Z[γ1 ] ⊕ . . . ⊕ Z[γd ] is a sublattice of i (H1 (M, Z)). □ Using this proposition we can now bound from below the homological asymptotic volume as follows. Theorem 2.8. Consider a Riemannian manifold M of first Z2 -Betti number b > 0 and of dimension n. Suppose that the integral homology has no 2-torsion and that Det2 FM ̸= 0. Then  b 2 Vol(M )1−b/n Ωab (M ) ≥ · . n b! Proof. Remember that for any abelian group G H1 (M, G) ≃ H1 (M, Z) ⊗ G according to the so-called “universal coefficient Theorem” (see [Hat02, Theorem 3A.3 and Propo- sition 3A.5]. This implies that if H1 (M, Z) has no 2-torsion then the rank of H1 (M, Z2 ) and H1 (M, R) coincide. By applying Theorem 1.1 we find a Z2 -homology basis of M made of closed geodesics (γ1 , . . . , γb ) such that Yb ℓ(γk ) ≤ nb · Vol(M )b/n . k=1 Because the family of closed geodesics (γ1 , . . . , γb ) induces a basis of H1 (M, R), Proposition 2.7 implies that  b 2 Vol(M )−b/n µ(B(M )) ≥ · n b! and we can conclude using equality (2.2). □ In particular we get the following lower bound for surfaces. Corollary 2.9. Consider an orientable Riemannian surface S of genus g. Then 1 Ωab (S) · Area(S)g−1 ≥ . (2g)! This improves the previous known lower bound due to Babenko [Bab92, Corollary 5.5]. To conclude this section let us denote by NM (t) the number of R-homologically distinct closed geodesics of length ≤ t. This number has polynomial growth of order the first R-Betti number b′ which is captured by the asymptotic volume of the covering space corresponding to the natural map π1 M → H1 (M ; Z)/Tors ≃ i (H1 (M ; Z)). Because this asymptotic volume coincides with 14 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER the homological asymptotic volume and because the stable norm is the unique norm enclosing the asymptotic geometry of this covering space (see [Bur92] and [CS16] for details), one obtains the equality NM (t) lim = µ(B(M )). t→∞ tb′ From this we can deduce the following. Corollary 2.10. Consider a Riemannian manifold M satisfying the hypothesis of Theorem 2.8. Then 2b NM (t) ≥ tb + o(tb−1 ). Vol(M )b/n · nb · b! A similar lower bound can be deduced for the number of R-homologically distinct and non- multiple closed geodesics of length ≤ t, compare with [Bab92, Theorem 7.1 and Corollary 7.2]. 3. M INIMAL LENGTH PRODUCT OF CLOSED GEODESICS FOR SURFACES In this section, after a discussion about graphs, we focus on surfaces. 3.1. Minimal length product over Z2 -homology bases for graphs. In this subsection we show that a Minkowski’s second principle also holds for graphs. This result is a straightforward consequence of Bollobás-Szemerédi-Thomason’s systolic inequality and paves the way for our conjecture on length product inequalities for surfaces. Consider a finite and connected 1-dimensional simplicial complex Γ endowed with a metric. Here by metric we mean a function which associates to every edge e a positive real value ℓ(e) encoding its length. In short, we will call such a pair (Γ, ℓ) a connected metric graph. We denote by E(Γ) its set of edges and V (Γ) its set of vertices. We introduce several quantities. First its total length X ℓ(Γ) = ℓ(e), e∈E(Γ) which can be interpreted as its 1-dimensional volume. Secondly, we associate to any path P = (e1 , . . . , en ) made of consecutive edges e1 , . . . , en ∈ E(Γ) its length ℓ(P ) defined by n X ℓ(P ) = ℓ(ei ). i=1 A cycle will mean a closed embedded path. The first Betti number of the graph (sometimes called its cyclic number) can be computed using the formula b1 (Γ) = |E(Γ)| − |V (Γ)| + 1, where | · | denotes cardinality of finite sets. Minkowski’s second principle for graphs takes the following form. Theorem 3.1. Consider a connected metric graph (Γ, ℓ) with first Betti number b ≥ 2. Then there exists a Z2 -homology basis induced by cycles γ1 , . . . , γb whose length product satisfies the following inequality: b b Y 4b−1 Y (3.1) ℓ(γk ) ≤ · log2 k · ℓ(Γ)b . (b − 1)! k=1 k=2 LENGTH PRODUCTS FOR MANIFOLDS 15 In particular b !1/b Y 4 ℓ(Γ) ℓ(γk ) ≤ √ b · log2 b · ℓ(Γ) ≃ 4e · log2 b · . b! b→∞ b k=1 Proof. Bollobás-Szemerédi-Thomason’s systolic inequality for metric graphs (see [BT97, BS02]), says that there exists a non-trivial cycle γ1 ∈ H1 (Γ; Z2 ) such that log2 b ℓ(γ1 ) ≤ 4 · ℓ(Γ). b−1 Now by deleting one edge of γ1 we get a connected subgraph Γ1 with b1 (Γ1 ) = b − 1 and ℓ(Γ1 ) ≤ ℓ(Γ). If b = 2, we choose the only remaining minimal cycle γ2 ⊂ Γ1 whose length satisfies ℓ(γ2 ) ≤ ℓ(Γ1 ) ≤ ℓ(Γ). Then we are done as the cycles γ1 and γ2 induce a Z2 -homology basis by construction, and their length product satisfies the desired upper bound. If b > 2 we argue by induction: because Γ1 together with the metric induced by ℓ is also a connected metric graph, there exists a family of cycles (γ2 , . . . , γb ) which form a homology basis of H1 (Γ1 ; Z2 ) and whose length product satisfies b b−1 Y 4b−2 Y ℓ(γk ) ≤ · log2 k · ℓ(Γ1 )b−1 . (b − 2)! k=2 k=2 In turn, the cycles γ1 , . . . , γb form a Z2 -homology basis of Γ by construction. Furthermore their length product satisfies the desired bound. □ This inequality cannot be substantially improved as one can construct sequences of p-regular graphs whose girth—defined as the shortest length of a non-trivial cycle—grows logarithmically in the number of vertices. These examples are metric graphs with all edges of length 1. The best known constructions have asymptotic girth 4 logp−1 n 3 as the number of vertices n goes to infinity (see [Wei84] and [LPS88]). The 3-regular case of these examples provides a sequence of graphs whose first Betti number b → ∞ and for which every homology basis (γ1 , . . . , γb ) satisfies the asymptotic lower bound b !1/b Y 4 ℓ(Γ) ℓ(γk ) ≳ · log2 b · . 9 b k=1 By analogy we conjecture the following for surfaces. Conjecture 3.2. There exists a positive constant c > 0 such that the following holds. Consider a Riemannian surface S of first Z2 -Betti number b > 0. Then there exists a Z2 -homology basis induced by closed geodesics γ1 , . . . , γb and whose length product satisfies the following inequality: b Area(S) b/2 Y   b ℓ(γi ) ≤ (c log b) · . b i=1 16 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER 3.2. Minimal length product over Z-homology bases for orientable surfaces. Observe that Theorem 3.1 and its proof are still valid for Z-coefficients. On an orientable surface S we could try to use the intersection form on H1 (S, Z) to deduce an analog of Corollary 1.5 for Z-coefficients. Indeed, the closed geodesics composing a minimal Z2 -homology basis are straight, by which we mean that between any two points on the curve the distance equals the length of the shortest segment of the curve joining them. This implies that any two closed geodesics in a minimal Z2 -homology basis (for successive lengths) have pairwise geometric intersection 0 or 1. Combining this with the fact that they are linearly independent in H1 (S, Z), one could hope to deduce that they indeed form a Z-homology basis but at the moment, it is unclear to us how to make this strategy work. Nonetheless, we prove using a different strategy the following Minkowski’s second principle for orientable Riemannian surfaces and in any event, this different approach will also be useful in the sequel. Theorem 3.3. Consider an orientable Riemannian surface S of genus g. Then there exists a Z-homology basis induced by closed geodesics γ1 , . . . , γ2g whose length product satisfies the following inequality: 2g Y ℓ(γk ) ≤ C g · Area(S)g i=k for some constant C < 231 . Remark 3.4. This result for genus g = 1 is already well-known. More precisely, in [Kee67] the following optimal inequality was proved: on any Riemannian 2-torus of unit area, we can always find a Z-homology basis induced by two closed geodesics γ1 , γ2 whose length product satisfies 2 ℓ(γ1 ) · ℓ(γ2 ) ≤ √ . 3 Furthermore there is equality if and only if the torus is isometric to a flat hexagonal torus (the torus obtained by pairing the opposite sides of a flat regular hexagon). Proof. The homological systole is defined by sys(S) := min{ℓ(γ) | [γ] ̸= 0 ∈ H1 (S; Z)} where the minimum is taken over loops whose homological class is non trivial. A closed geodesic γ is said to realize the homological systole if [γ] ̸= 0 and ℓ(γ) = sys(S). Let · ∩ · denote the intersection form on H1 (S; Z) (this notation will be used throughout the sequel). If γ is a closed geodesic with [γ] ̸= 0 we can now define L(γ) := min{ℓ(δ) | [γ] ∩ [δ] ̸= 0}. Observe that L(γ) depends only on the homological class induced by γ and is realized by the length of some closed geodesic δ with non zero intersection with γ. Furthermore if γ is the homological systole, then it is easy to see by a standard surgery argument that in this case [γ] ∩ [δ] = 1. Lemma 3.5. If a closed geodesic γ realizes the homological systole, then ℓ(γ) · L(γ) ≤ 2 Area(S). LENGTH PRODUCTS FOR MANIFOLDS 17 This is a sort of dual version of Theorem 2.1 for Z-coefficients and specialized to the systole. Proof of the lemma. Pick a point p on γ and denote by D(p, R) the set of points at a distance at most R from p. For any R ∈ (0, L(γ)/2) the curve γ ∩ D(p, R) is homologically trivial in the relative homology H1 (D(p, R), ∂D(p, R); Z). The argument is the same as in the corresponding part of the proof of Theorem 2.1 but with Z-coefficients. Thus ℓ(∂D(p, R)) ≥ 2ℓ(γ ∩ D(p, R)) = 4R and, by the coarea formula, Z L(γ) 2 Area(S) ≥ Area(D(p, L(γ)/2)) = ℓ(∂D(p, R)) dR ≥ L(γ)2 /2 ≥ ℓ(γ) · L(γ)/2. 0 □ Suppose in first instance that the homological systole satisfies s π Area(S) sys(S) ≥ . g−1 By [BPS12, Theorem 1.1] there exists a family (γ1 , . . . , γ2g ) of loops on S that generate H1 (S; Z) and whose lengths satisfy the following inequalities: s log(2g − k + 2) Area(S) (3.2) length(γk ) ≤ 216 · g · · . 2g − k + 1 4π(g − 1) for k = 1, . . . , 2g. Thus 2g 2g 232g g 2g Y Area(S) g Y   length(γk ) ≤ · log(2g − k + 2) · . (2g)! 4π(g − 1) i=k k=1 From the following refinement of Stirling’s formula (see [Fel57, p. 54]) √  n n n! > 2πn e1/(12n+1) e we can deduce that 2g Area(S) g Y   g 2g ℓ(γk ) ≤ C · (log 2g) · g k=1 231 e2 with C = 4π < 231 . In particular we get 2g Y ℓg (γk ) ≤ C g · Area(S)g . i=k Suppose now that the homological systole satisfies s π Area(S) sys(S) ≤ g−1 and pick a class γ realizing the homological systole. By Lemma 3.5 ℓ(γ) · L(γ) ≤ 2 Area(S). Because γ realizes the systole, we can find a closed geodesic δ of length L(γ) and such that [γ] ∩ [δ] = 1. Set γ2g = γ and γ2g−1 = δ. We cut S along γ2g and glue two round hemispheres as 18 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER caps along the two components of the boundary ∂(S \ γ) to get a new Riemannian closed surface S ′ of genus g − 1. Its area satisfies ℓ(γ)2 g Area(S ′ ) ≤ Area(S) + ≤ Area(S) π g−1 by assumption on the systole. By induction (and using Remark 3.4) we know that there exists a family of closed geodesics γ1 , . . . , γ2g−2 on S ′ inducing a basis of H1 (S ′ ; Z) and whose length product satisfies 2g−2 Y ℓ(γk ) ≤ C g−1 · Area(S ′ )g−1 . k=1 We can homotope all these curves in S \ γ ⊂ S ′ without increasing their length. The corresponding curves in S define, together with γ2g and γ2g−1 , a family (γ1 , . . . , γ2g ) of closed geodesics. It is straightforward to check that this family induces a basis of H1 (S; Z) using the symplectic nature of · ∩ ·. Furthermore 2g Y ℓ(γk ) ≤ 2 Area(S) · C g−1 · Area(S ′ )g−1 k=1  g−1 g−1 g ≤ 2C Area(S)g g−1 ≤ 2e · C g−1 · Area(S)g which concludes the argument. □ Observe that, as a byproduct of the proof, we have the following. Proposition 3.6. Let S be an orientable Riemannian surface of genus g whose homological systole satisfies s Area(S) sys(S) ≥ . 4π(g − 1) Then there exist closed geodesics γ1 , . . . , γ2g inducing a basis of H1 (S; Z) and whose length product satisfies the following inequality: 2g Area(S) g Y   g 2g ℓ(γk ) ≤ C · (log 2g) · g k=1 with C < 231 . Because H1 (M, Z2 ) ≃ H1 (M, Z) ⊗ Z2 , we deduce that the closed geodesics γ1 , . . . , γ2g above also induce a Z2 -homology basis, thus proving Proposition 1.8. Remark 3.7. A similar result to Theorem 3.3 and Proposition 3.6 holds for non-orientable Riemann- ian surfaces. Indeed the analog of the upper bounds (3.2) on the successive lengths of a Z-homology basis still holds true in the non-orientable case, see [BPS12, Remark 2.4]. LENGTH PRODUCTS FOR MANIFOLDS 19 3.3. Length product of g homologically independent closed geodesics. Using the same approach as in proof of Theorem 3.3, we can prove the following. Proposition 3.8. Let S be an orientable Riemannian surface of genus g. Then there exist closed geodesics γ1 , . . . , γg whose homology classes are independent in H1 (S; Z) and whose length product satisfies the following inequality: g  g/2 g Area(S) Y g ℓ(γk ) ≤ C log(2g) g k=1 with C < 231 . Proof. The proof is quite similar to proof of Theorem 3.3. First notice that if the homological systole satisfies s Area(S) sys(S) ≥ 4π(g − 1) the result follows from Proposition 3.6. Suppose now that the homological systole satisfies s π Area(S) sys(S) ≤ g−1 and pick a class γ realizing the homological systole. We cut S along γ and glue two round hemisphere caps to get a new Riemannian closed surface S ′ of genus g − 1 and area g Area(S ′ ) ≤ Area(S). g−1 By induction we know that there exist a family of closed geodesics γ1 , . . . , γg−1 on S ′ which are Z-homologically independent and whose length product satisfies g−1  ′  g−1 g−1 Area(S ) Y 2 g−1 ℓ(γk ) ≤ C log(2(g − 1)) . g−1 k=1 We can homotope all these curves in S \ γ ⊂ S ′ without increasing their length. The corresponding curves in S define together with γg := γ a family (γ1 , . . . , γg ) of Z-homologically independent closed geodesics. Furthermore g  ′)  g−1 s Y Area(S 2 π Area(S) ℓ(γk ) ≤ C g−1 log(2(g − 1))g−1 · g−1 g−1 k=1 2 ! g−1 r s  2 g Area(S) πg Area(S) ≤ C g−1 log(2g)g−1 · · g−1 g g−1 g g−1  g √  g Area(S) 2 ≤ 2π · C g−1 log(2g)g−1 · · g−1 g g √  Area(S) 2 ≤ 2π · e · C g−1 log(2g)g−1 · g and the result follows. □ The proof also works with Z2 -coefficients thus proving Proposition 1.9. 20 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER 3.4. Length product of two intersecting loops for hyperbolic surfaces. If Conjecture 3.2 is true, this would imply the following length product inequality for two intersecting closed geodesics. Conjecture 3.9. There exists a constant c > 0 such that the following holds. On any orientable closed Riemannian surface S of genus g there are two closed geodesics γ and δ with [γ] ∩ [δ] ̸= 0 and whose length product satisfies the following inequality: Area(S) ℓ(γ) · ℓ(δ) ≤ (c log(2g))2 · . g Let us explain how to deduce Conjecture 3.9 from Conjecture 3.2. If Conjecture 3.2 is true, S contains a family (γ1 , . . . , γ2g ) of closed geodesics which form a basis of H1 (S; Z) and with length product 2g Area(S) g Y   2g ℓ(γk ) ≤ (c log(2g)) · . g k=1 The intersection form on H1 (S, ; Z) induces a symplectic form. This implies that we can reorder the elements of this basis such that [γ2i−1 ] ∩ [γ2i ] ̸= 0 for i = 1, . . . , g according to the following lemma due to Gutt. Lemma 3.10. [Gut18] Let ω a symplectic form and (e1 , . . . , e2n ) a basis of R2n . Then there exists a permutation σ ∈ S2n such that ω(eσ(2k−1) , eσ(2k) ) ̸= 0 for k = 1, . . . , n. Proof. The result is equivalent to finding a permutation σ ∈ S2n such that n Y ω(eσ(2k−1) , eσ(2k) ) ̸= 0. k=1 Recall that if η and ω are respectively an alternating k-form and m-form then X η ∧ ω(e1 , . . . , ek+m ) = sgn(σ) · η(eσ(1) , . . . , eσ(k ) · ω(eσ(k+1) , eσ(k+m) ) σ∈Shk,m where Shk,m ⊂ Σk+m is the subset of (k, m) shuffles, i.e., all permutations σ of {1, . . . , k + m} such that σ(1) < . . . < σ(k) and σ(k + 1) < . . . < σ(k + m)). Then we compute ω n (e1 , . . . , en ) = ω n−1 ∧ ω(e1 , . . . , e2n ) X = sgn(σ) · ω n−1 (eσ(1) , . . . , eσ(2n−2) ) · ω(eσ(2n−1) , eσ(2n) ) σ∈Sh2n−2,2 X n−1 Y = sgn(σ) · ω(eσ(2i+1) , eσ(2i+2) ). σ∈Sh2,...,2 i=0 Because ω is symplectic, we know that ω n (e1 , . . . , e2n ) ̸= 0 and thus at least one of the products is non zero. □ Remark 3.11. This proof came up after a discussion between J. Gutt and the first author about how to prove a symplectic analog of Minkowski’s first theorem. Here we can see the germ of one of the mechanisms used in the proof of Theorem 2.5 and we are very grateful to him for showing us this nice argument. LENGTH PRODUCTS FOR MANIFOLDS 21 Returning to the link between Conjecture 3.9 and Conjecture 3.2, we can reorder the family of closed geodesics such that [γ2i−1 ] ∩ [γ2i ] ̸= 0 for i = 1, . . . , g. This implies that 2g !1/g Y Area(S) min{ℓ(γ) · ℓ(δ) | [γ] ∩ [δ] ̸= 0} ≤ ℓ(γk ) ≤ (c log(2g))2 · . g k=1 Observe that it is not possible to directly deduce Conjecture 3.9 from Proposition 3.8: the shortest g homologically independent closed geodesics might lie in a Lagrangian subspace of H1 (M ; Z) for the intersection form. However for hyperbolic metrics we are able establish the result. Theorem 3.12. There exists a constant c > 0 such that the following holds. A genus g closed orientable hyperbolic surface S0 contains two closed geodesics γ and δ with [γ] ∩ [δ] = 1 and whose length product satisfies the following inequality: ℓ(γ) · ℓ(δ) ≤ (c log(2g))2 . Recall that Area(S0 ) = 4π(g − 1) as the metric is hyperbolic. Thus Theorem 3.12 etablishes Conjecture 3.9 for hyperbolic surfaces. Proof. We first collect some information about the local geometry near simple closed geodesics of small length. Let η be a simple closed geodesic. According to the collar lemma (see for instance [Bus92, Theorem 4.1.1]) the collar C(η) := {p ∈ S0 | dist (p, η) ≤ w(η)} of width w(η) := arcsinh (1/ sinh (ℓ(η)/2)) around η is isometric to the Riemannian cylinder ([−w(η), w(η)] × S1 , ds2 = dρ2 + ℓ(η)2 cosh2 ρ dt2 ). For any l ∈]ℓ(η), ℓ(η) cosh w(η)] denote by ηl+ and ηl− the two parallel curves of C(η) above and below η with length l. Note that they are never geodesics but are of constant curvature. Denote by Cl (η) the collar around η with boundary curves ηl− and ηl+ . With this notation C(η) = Cl (η) for l = Lη := ℓ(η) cosh w(η). Lemma 3.13. There exists a positive constant ε0 ≤ 1 such that the following holds. If η is a simple closed geodesic of length less than or equal to ε0 , then (1) ε0 ≤ Lη ; (2) the distance d(η) between ηε+0 and ηL+η satisfies d(η) ≥ Lη /2. Proof of the lemma. Condition (1) is equivalent to ε0 ≤ f (ℓ) := ℓ · cosh (arcsinh (1/ sinh (ℓ/2))) for any ℓ ≤ ε0 . Observe that limℓ→0 f (ℓ) = 2 so define ε1 > 0 as the largest number such that min f (ℓ) ≥ 1 ℓ∈[0,ε1 ] (possibly ε1 = ∞). Condition (1) is then satisfied for any ε0 ≤ min{ε1 , 1}. Because ε0 d(η) = w(η) − arccosh , ℓ(η) Condition (2) is equivalent to ε0 Lη w(η) − arccosh ≥ ℓ(η) 4 which holds if ε0 ≤ g(ℓ) := ℓ cosh (arcsinh (1/ sinh (ℓ/2)) − f (ℓ)/4) 22 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER for any ℓ ≤ ε0 . Again limℓ→0 g(ℓ) = 2 when ℓ → 0. Define ε2 > 0 as the largest number such that min g(ℓ) ≥ 1 ℓ∈[0,ε2 ] (again we allow ε2 = ∞). Condition (2) is satisfied for any ε0 ≤ min{ε2 , 1}. Now set ε0 := min{ε1 , ε2 , 1} and the lemma is proved. □ If sys(S0 ) ≥ ε0 , then Theorem 1.1 from [BPS12] guarantees the existence of a family of loops γ1 , . . . , γ2g in S0 that generate H1 (S0 ; Z) and such that log(2g − k + 2) length(γk ) ≤ C0 · g · 2g − k + 1 for k = 1, . . . , 2g and with C0 := 216 /ε0 . Among the g + 1 first ones, at least two of them have non-zero intersection and their length product is bounded from above by (C0 log(2g))2 . Furthermore they belong to a minimal length homology basis (for successive lengths), and so they pairwise intersect at most once, hence exactly once. Now suppose that sys(S0 ) ≤ ε0 . Let (η1 , . . . , ηp ) be the set of homologically non-trivial simple closed geodesics in S0 of length less than ε0 and ordered such that ℓ(η1 ) ≤ . . . ≤ ℓ(ηp ). As ε0 ≤ 1 these loops do not pairwise intersect and moreover the corresponding collars C(ηi ) are pairewise disjoint for i = 1, . . . , p. In particular p ≤ 3g − 3. For each i = 1, . . . , p we cut off the collar Cε0 (ηi ) and paste (ηi )+ ε0 with − ′ (ηi )ε0 . We denote by ηi the new curve obtained from the pasting of these two curves in the new surface denoted by S0′ . The new surface S0′ is no longer hyperbolic, but does have area less than the area of S0 . Further- more its homological systole satisfies the following lower bound. Lemma 3.14. sys(S0′ ) ≥ ε0 . Proof of the lemma. Observe that the homological systole is realized by a simple closed curve in S0′ . Consider a closed curve γ ′ with non-zero intersection with one of the ηi′ . We denote by C(ηi′ ) the cylinder around ηi′ of width d(ηi ). The intersection of γ ′ with C(ηi′ ) consists of at least one arc which is homotopically non-trivial relative to the boundary of the cylinder. In particular the length of γ ′ is bounded from below by 2d(ηi ) ≥ ε0 . Now let γ ′ denotes a homologically non-trivial simple closed curve which has zero intersection with any of the curves ηi′ and whose homology class is different from any of the classes [ηi′ ] for i := 1, . . . , p. If γ ′ does not intersect any of the ηi′ ’s we are done: γ ′ actually coincides with a homologically non-trivial simple closed curve in S0 different from any of the ηi ’s so its length is at least ε0 . If γ ′ ∩ ηi′ ̸= ∅, consider one of the arcs of γ ′ ∩ C(ηi′ ) intersecting ηi′ . This arc can not be contained in C(ηi′ ) so its length is at least 2d(ηi ) ≥ ε0 . Because any closed curve in the homology class of one of the ηi′ ’s has length at least ε0 , we conclude that sys(S0′ ) ≥ ε0 . □ By applying once again Theorem 1.1 from [BPS12], we can find two homologically independent closed curves γ ′ and δ ′ of S0′ with non-zero intersection and whose length product is bounded above by (C0 log(2g))2 . Moreover, we can assume that γ ′ and δ ′ minimize the length product over pairs of closed simple curves with non-zero intersection number. Suppose first that both γ ′ and δ ′ have zero intersection number with any of the ηi′ ’s. This implies that γ ′ (respectively δ ′ ) is disjoint from all of the ηi′ ’s or coincides with one of the ηi ’s. LENGTH PRODUCTS FOR MANIFOLDS 23 To see this suppose that γ ′ ∩ ηi ̸= ∅ for some i while γ ′ ̸= ηi′ . There exist two points p and q of γ′ ∩ ηi and a subarc ζ of γ ′ from p to q such that · the interior of ζ does not intersect ηi′ ; · ζ starts and ends on the same side of ηi′ . Thus γ ′ decomposes into the sum of two simple loops γ1′ and γ2′ respectively made of the concatena- tion of ζ and of γ ′ \ ζ with the shortest path of ηi′ from p to q. Observe that ℓ(γi′ ) < ℓ(γ ′ ) as ℓ(ηi′ )/2 = ε0 /2 ≤ d(ηi ). Now because [γ ′ ] = [γ1′ ] + [γ2′ ] one of the curves γ1′ or γ2′ has non-zero intersection number with δ ′ and the product of its length with ℓ(δ ′ ) is strictly less than the initial length product ℓ(γ ′ ) · ℓ(δ ′ ). This is a contradiction and we can thus assume that both γ ′ and δ ′ are either disjoint from any of the ηi′ ’s or coincide with one of them. Because γ ′ and δ ′ are either disjoint from any of the ηi′ ’s or coincide with one of them, they actually coincide with closed curves in S0 , and so we have found a pair of curves with the desired properties. Suppose now that γ ′ (after possibly commuting γ ′ and δ ′ ) has non-zero intersection number with one of the ηi′ ’s. Let i0 ∈ {1, . . . , p} be the smallest integer i between 1 and p and such that [γ ′ ] ∩ [ηi′ ] ̸= 0. The curve γ ′ lifts to a multiarc of S0 with extremal points belonging to ∪pi=1 ∂Cε0 (ηi ). This multiarc can be completed into a closed curve γ on S0 by joining two extremal points lying in the boundary of ∂Cε0 (ηi ) and corresponding to the same point of γ ′ by an arc crossing the cylinder Cε0 (ηi ). The length of each such arcs is bounded from above by   1 2 arcsinh . sinh(ℓ(ηi0 )/2) Because γ ′ is minimizing each arc cutting ηi′ must leave the cylinder C(ηi′ ), and the number of such arcs is bounded from above by ℓ(γ ′ )/(2ε0 ). Thus    ′ 1 1 ℓ(γ) ≤ ℓ(γ ) 1 + · arcsinh . ε0 sinh(ℓ(ηi0 )/2) Then we get   1 1 ℓ(γ) · ℓ(ηi0 ) ≤ ℓ(γ ′ ) · ℓ(ηi0 ) + ℓ(γ ′ ) · ℓ(ηi0 ) · · arcsinh ε0 sinh(ℓ(ηi0 )/2) and because   1 ℓ · arcsinh →0 sinh(ℓ/2) when ℓ → 0, there exists a universal constant co such that   1 1 · ℓ · arcsinh ≤ c0 ε0 sinh(ℓ/2) for any ℓ ≤ ε0 . Thus ℓ(γ) · ℓ(ηi0 ) ≤ ℓ(γ ′ ) · ℓ(ηi0 ) + c0 · ℓ(γ ′ ) ≤ (1 + c0 /ε0 ) · ℓ(γ ′ ) · ℓ(δ ′ ) 24 FLORENT BALACHEFF, STEVE KARAM & HUGO PARLIER as ℓ(δ ′ ) ≥ sys(S0′ ) ≥ ε0 . 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