# Essential manifold

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**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

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
*S*.^{2} - Real projective space
*RP*is essential since the inclusion^{n}

- is injective in homology, where
- 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.