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

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
is injective in homology, where
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))
  • 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.
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References

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

See also

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