Essential manifold explained

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, or more precisely in the homology of the corresponding Eilenberg–MacLane space K( 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

RP ^{n} \to RP ^{infty}

is injective in homology, where

RP ^{infty} = K( 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( 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.

See also

• Gromov's systolic inequality for essential manifolds
• Systolic geometry

Notes and References

1. Gromov . M. . Filling Riemannian manifolds . J. Diff. Geom. . 18 . 1983 . 1–147 . 10.1.1.400.9154.