# 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

${\displaystyle H_{n}(M)\to H_{n}(K(\pi ,1))}$,

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
${\displaystyle \mathbb {RP} ^{n}\to \mathbb {RP} ^{\infty }}$
is injective in homology, where
${\displaystyle \mathbb {RP} ^{\infty }=K(\mathbb {Z} _{2},1)}$
is the Eilenberg–MacLane space of the finite cyclic group of order 2.

## 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.

## References

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