Gromov's systolic inequality for essential manifolds

Last updated April 05, 2019

In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;^[1] it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.

Essential manifolds

A closed manifold is called essential if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corresponding Eilenberg–MacLane space. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.

In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M of dimension n, which corresponds to the generator of the homology group $. The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.$

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.

Examples of essential manifolds include aspherical manifolds, real projective spaces, and lens spaces.

In mathematics, real projective space, or RPⁿ or $, is the topological space of lines passing through the origin 0 in R n +1 . It is a compact, smooth manifold of dimension n, and is a special case Gr (1, R n +1) of a Grassmannian space.$

A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.

Proofs of Gromov's inequality

Gromov's original 1983 proof is about 35 pages long. It relies on a number of techniques and inequalities of global Riemannian geometry. The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm. The imbedding is defined by mapping a point p of X, to the real function on X given by the distance from the point p. The proof utilizes the coarea inequality, the isoperimetric inequality, the cone inequality, and the deformation theorem of Herbert Federer.

In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on Rⁿ is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems.

Herbert Federer was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.

Filling invariants and recent work

One of the key ideas of the proof is the introduction of filling invariants, namely the filling radius and the filling volume of X. Namely, Gromov proved a sharp inequality relating the systole and the filling radius,

In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.

\mathrm {sys\pi } _{1}\leq 6\;\mathrm {FillRad} (X),

valid for all essential manifolds X; as well as an inequality

\mathrm {FillRad} (X)\leq C_{n}\mathrm {vol} _{n}{}^{\tfrac {1}{n}}(X),

valid for all closed manifolds X.

It was shown by Brunnbauer (2008) that the filling invariants, unlike the systolic invariants, are independent of the topology of the manifold in a suitable sense.

Guth (2011) and Ambrosio & Katz (2011) developed approaches to the proof of Gromov's systolic inequality for essential manifolds.

Inequalities for surfaces and polyhedra

Stronger results are available for surfaces, where the asymptotics when the genus tends to infinity are by now well understood, see systoles of surfaces. A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the Grushko decomposition theorem.

Notes

↑ see Gromov (1983)

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References

Ambrosio, Luigi; Katz, Mikhail (2011), "Flat currents modulo p in metric spaces and filling radius inequalities", Commentarii Mathematici Helvetici , 86 (3): 557–592, arXiv: 1004.1374  , doi:10.4171/CMH/234, MR 2803853 .
Brunnbauer, M. (2008), "Filling inequalities do not depend on topology", J. Reine Angew. Math., 624: 217–231
Gromov, M. (1983), "Filling Riemannian manifolds", J. Diff. Geom., 18: 1–147, MR 0697984, Zbl 0515.53037, PE euclid.jdg/1214509283
Guth, Larry (2011), "Volumes of balls in large Riemannian manifolds", Annals of Mathematics , 173 (1): 51–76, arXiv: math/0610212  , doi:10.4007/annals.2011.173.1.2, MR 2753599
Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, 137, Providence, R.I.: American Mathematical Society, p. 19, ISBN 978-0-8218-4177-8

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[1] see Gromov (1983)

v t e Systolic geometry
1-systoles of surfaces	Loewner's torus inequality Pu's inequality Filling area conjecture Bolza surface Systoles of surfaces Eisenstein integers
1-systoles of manifolds	Gromov's systolic inequality for essential manifolds Essential manifold Filling radius Hermite constant
Higher systoles	Gromov's inequality for complex projective space Systolic freedom Systolic category