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 π, or more precisely in the homology of the corresponding Eilenberg–MacLane space K(π, 1), via the natural homomorphism

[math]\displaystyle{ H_n(M)\to H_n(K(\pi,1)), }[/math]

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

  [math]\displaystyle{ \mathbb{RP}^n \to \mathbb{RP}^\infty }[/math]

is injective in homology, where

[math]\displaystyle{ \mathbb{RP}^\infty = K(\mathbb{Z}_2, 1) }[/math]

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.

See also

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