Essential manifold

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Essential manifold a special type of closed manifolds. The notion was first introduced explicitly by Mikhail Gromov.[1]

Definition[edit]

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[edit]

  • 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.

Properties[edit]

References[edit]

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

See also[edit]