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 $\pi$, or more precisely in the homology of the corresponding Eilenberg–MacLane space K(π, 1), via the natural homomorphism
 * $$H_n(M)\to H_n(K(\pi,1)),$$

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
 * $$\mathbb{RP}^n \to \mathbb{RP}^\infty$$
 * is injective in homology, where
 * $$\mathbb{RP}^\infty = K(\mathbb{Z}_2, 1)$$
 * 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.