Essential manifold

In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.

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 (π, 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
${ℝ ℙ}^{n} \to {ℝ ℙ}^{\infty}$

is injective in homology, where

{ℝ ℙ}^{\infty} = K (ℤ_{2}, 1)

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

All compact aspherical manifolds are essential (since being aspherical means the manifold itself is already a K(π, 1))
- In particular all compact hyperbolic manifolds are essential.
All lens spaces are essential.

Properties

The connected sum of essential manifolds is essential.
Any manifold which admits a map of nonzero degree to an essential manifold is itself essential.

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

Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem

Essential manifold

Definition

Examples

Properties

See also