Gromov's systolic inequality for essential manifolds

In Riemannian geometry, M. Gromov's systolic inequality for essential $n$-manifolds $M$ dates from 1983. It is a lower bound for the volume of an arbitrary metric on $M$, in terms of its homotopy 1-systole. The homotopy 1-systole is the least length of a non-contractible loop. We will denote the homotopy 1-systole by the symbol $operatorname\{syspi\}\_1$, so as to avoid confusion with the homology 1-systole. Then Gromov's inequality takes the form

:$operatorname\{syspi\}\_1\{\}^n\; leq\; C\_n\; operatorname\{vol\}(M),$

where $C\_n$ is a universal constant only depending on the dimension of $M$. Thus Gromov's systolic inequality for essential manifolds can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.

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.

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

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.

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,:$mathrm\{syspi\}\_1\; leq\; 6;\; mathrm\{FillRad\}(X),$valid for all essential manifolds X; as well as an inequality

:$mathrm\{FillRad\}\; leq\; C\_n\; mathrm\{vol\}\_n\{\}^\{\; frac\{1\}\{n(X),$valid for all closed manifolds X.

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

Recently, an alternative approach to the proof of Gromov's systolic inequality for essential manifolds has been proposed by L. Guth.

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. A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the Grushko decomposition theorem.

References

* Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1-147.
*Citation | last1=Katz | first1=Mikhail G. | title=Systolic geometry and topology|pages=19 | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137

ee also

*Filling area conjecture
*Gromov's inequality
*Gromov's inequality for complex projective space
*Loewner's torus inequality
*Pu's inequality
*Systolic geometry