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In geometry, an **essential manifold** is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.^{[1]}

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.

- 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 (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.

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