Essential manifold

Essential manifold a special type of closed manifolds. The notion was first introduced explicitly by Mikhail Gromov.^[1]

Definition

A closed manifold M is called essential if its fundamental class [M] defines a nonzero element in the homology of its fundamental group π, or more precisely in the homology of the corresponding Eilenberg–MacLane space K(π, 1), via the natural homomorphism

H_n(M)\to H_n(K(\pi,1))

where n is the dimension of M. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.

Examples

All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere S².
Real projective space RPⁿ is essential since the inclusion
$\mathbb{RP}^n \to \mathbb{RP}^{\infty}$

is injective in homology, where

\mathbb{RP}^{\infty} = K(\mathbb{Z}_2, 1)

is the Eilenberg-MacLane space of the finite cyclic group of order 2.

All compact aspherical manifolds are essential;
- In particular all compact hyperbolic manifolds are essential.
All lens spaces are essential.

Properties

Connected sum of essential manifolds is essential.

References

↑ Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147.

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

Essential manifold

Definition

Examples

Properties

References

See also