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Other inequalities due to Gromov

There is a number of inequalities in the systolic literature that could be described as Gromov's inequality.

The deepest result in systolic geometry is Gromov's inequality for the homotopy 1-systole of essential manifolds. Gromov's original definition of an essential manifold $ {M}$ depends on the choice of the coefficient ring $ A$, taken to be  $ {\mathbb{Z}}$ if $ {M}$ is orientable, or  $ {\mathbb{Z}}_2$ otherwise. We then have a nonzero fundamental homology class  $ [{M}]\in H_n({M},A)$.

Definition 6.1   A closed $ n$-dimensional manifold $ {M}$ is called essential if there exists a map from $ {M}$ to a suitable Eilenberg-MacLane space $ K(\pi,1)$ such that the induced homomorphism

$\displaystyle h: H_n({M},A)\to H_n(K(\pi,1),A) $

maps the fundamental class $ [{M}]$ to a nonzero class in the homology group  $ H_n(K(\pi,1),A)$, i.e.   $ h([{M}])\not= 0$.

A more general definition of an $ n$-essential space $ X$, in the context of an arbitrary polyhedron $ X$, can be defined in terms of arbitrary local coefficients.

The following theorem was proved in [Gro83, Section 0] and [Gro83, Appendix 2, p. 139, item $ B_1'$].

Theorem 6.2 (M. Gromov)   Every $ n$-essential, compact, $ n$-dimensional polyhedron $ X$ satisfies the inequality

$\displaystyle \pisys _1(X)^n \leq C_n \vol _n(X).$ (6.1)

where the constant $ C_n$ depends only on $ n$. If $ X$ is a manifold, the constant $ C_n$ can be chosen to be on the order of $ n^{2n^2}$.

In other words, the quotient

$\displaystyle \frac{\vol _n}{(\pisys _1)^n} > 0$ (6.2)

is bounded away from zero for such polyhedra.

A summary of a proof appears in [Ka07, Section 12.2].

next up previous contents
Next: Bibliography Up: A sixty second introduction Previous: Gromov's inequality for complex   Contents
Mikhail Katz 2007-09-19