Filling Radius - Homological Definition

Homological Definition

Denote by A the coefficient ring or, depending on whether or not X is orientable. Then the fundamental class, denoted , of a compact n-dimensional manifold X, is a generator of the homology group, and we set

where is the inclusion homomorphism.

To define an absolute filling radius in a situation where X is equipped with a Riemannian metric g, Gromov proceeds as follows. One exploits an imbedding due to Kazimierz Kuratowski (the first name is sometimes spelled with a "C"). One imbeds X in the Banach space of bounded Borel functions on X, equipped with the sup norm . Namely, we map a point to the function defined by the formula for all, where d is the distance function defined by the metric. By the triangle inequality we have 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 X is the Riemannian circle (the distance between opposite points must be π, not 2!). We then set in the formula above, and define