Essential manifold

In geometry, an essential manifold is a special type of closed manifold. 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 ) → H n ( K ( π , 1 ) ) , {\displaystyle H_{n}(M)\to H_{n}(K(\pi ,1)),} {\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.

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.

References

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