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

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

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.

Properties

References

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

See also