Systolic geometry
Gromov's systolic inequality

The deepest result in the field is Gromov's inequality for the homotopy 1-systole of an essential n -manifold M:

where C is a universal constant only depending on the dimension of M. Here the homotopy systole sysπ is by definition the least length of a noncontractible loop in M. A manifold is called essential if its fundamental class represents a nontrivial class in the homology of its fundamental group. The proof involves a new invariant called the filling radius, introduced by Gromov, defined as follows.

Denote by A the coefficient ring Z or Z, depending on whether or not M is orientable. Then the fundamental class, denoted, of a compact n -dimensional manifold M is a generator of H n (M; A) = A {\displaystyle H_{n}(M;A)=A}. Given an imbedding of M in Euclidean space E, we set

where ι is the inclusion homomorphism induced by the inclusion of M in its ε-neighborhood U M in E.

To define an absolute filling radius in a situation where M is equipped with a Riemannian metric g, Gromov proceeds as follows. One exploits an imbedding due to C. Kuratowski. One imbeds M in the Banach space L (M) of bounded Borel functions on M, equipped with the sup norm ‖ ‖ {\displaystyle \|\;\|}. Namely, we map a point x ∈ M to the function f ∈ L (M) defined by the formula f (y) = d(x,y) for all y ∈ M, where d is the distance function defined by the metric. By the triangle inequality we have d (x, y) = ‖ f x − f y ‖, {\displaystyle d(x,y)=\|f_{x}-f_{y}\|,} and therefore the imbedding is strongly isometric, in the precise sense that internal distance and ambient distance coincide. Such a strongly isometric imbedding is impossible if the ambient space is a Hilbert space, even when M is the Riemannian circle (the distance between opposite points must be π, not 2!). We then set E = L (M) in the formula above, and define

Namely, Gromov proved a sharp inequality relating the systole and the filling radius,

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

valid for all closed manifolds M.

A summary of a proof, based on recent results in geometric measure theory by S. Wenger, building upon earlier work by L. Ambrosio and B. Kirchheim, appears in Section 12.2 of the book "Systolic geometry and topology" referenced below. A completely different approach to the proof of Gromov's inequality was recently proposed by Larry Guth.
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