arXiv subjects
Publications and source records attributed to Mikhail G. Katz.
We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category.
We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our earlier result in the orientable case. Finally, we prove the homotopy invariance of systolic category.
We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3-bound syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
We show that the existence of a nontrivial Massey product in the cohomology ring H^*(X) imposes global constraints upon the Riemannian geometry of a manifold X. Namely, we exhibit a suitable systolic inequality, associated to such a product. This generalizes an inequality proved in collaboration with Y. Rudyak, in the case when X has unit Betti numbers, and realizes the next step in M. Gromov's program for obtaining geometric inequalities associated with nontrivial Massey products. The inequality is a volume lower bound, and depends on the metric via a suitable isoperimetric quotient. The proof relies upon W. Banaszczyk's upper bound for the successive minima of a pair of dual lattices. Such an upper bound is applied to the integral lattices in homology and cohomology of X. The possibility of applying such upper bounds to obtain volume lower bounds was first exploited in joint work with V. Bangert. The latter work deduced systolic inequalities from nontrivial cup-product relations, whose role here is played by Massey products.
Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko's decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space ``minimal model'' for X, reminiscent of the ``chopping off long fingers'' construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.
We prove an optimal systolic inequality for CAT(0) metrics on a genus~2 surface. We use a Voronoi cell technique, introduced by C.~Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y^2=x^5-x. Thus, among all CAT(0) metrics, the one with the best systolic ratio is composed of six flat regular octagons centered at the Weierstrass points of the Bolza surface.
We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.
We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns out to be an equality, e.g. in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality, and that both categories are sensitive to Massey products. The comparison with the value of cat(M) leads us to prove or conjecture new systolic inequalities on M.
We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.
We prove that C. Loewner's inequality for the torus is satisfied by all hyperelliptic surfaces X, as well. We first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from the Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromov's area estimates for epsilon-regular metrics on X.
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X,g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T^b, which are area-decreasing (on b-dimensional areas), with respect to suitable norms. These norms are the stable norm of g, the conformally invariant norm, as well as other L^p-norms. The case of equality is characterized in terms of the criticality of the lattice of deck transformations of T^b, while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, in the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics.
We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in collaboration with D. Burago. It turns out that one can successfully combine this construction with the coarea formula, to yield new optimal inequalities.
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in χ(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of X, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima λ_i, as deforming a λ_2-bound into a λ_1-bound.
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary, allows the survey to be reasonably short. Although, even in this limited case the authors have left out many interesting results.
Given a pair of integers m and n such that 1 < m < n, we show that every n-dimensional manifold admits metrics of arbitrarily small total volume, and possessing the following property: every m-dimensional submanifold of less than unit m-volume is necessarily torsion in homology. This result is different from the case of a pair of complementary dimensions, for which a direct geometric construction works and gives the analogous theorem in complete generality. In the present paper, we use Sullivan's telescope model for the rationalisation of a space to observe systolic freedom.
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X.
Let X be a closed manifold of dimension 2m >= 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.