Partial residual plot

Essential manifold a special type of closed manifolds. 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

$H_n(M)\to H_n(K(\pi ,1))$,

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

  ${\mathbb{ℝ}\mathbb{\mathbb{P}}}^n\to {\mathbb{ℝ}\mathbb{\mathbb{P}}}^{\mathrm{\infty }}$

is injective in homology, where

${\mathbb{ℝ}\mathbb{\mathbb{P}}}^{\mathrm{\infty }}=K({\mathbb{\mathbb{Z}}}_2,1)$

is the Eilenberg-MacLane space of the finite cyclic group of order 2.

• All compact aspherical manifolds are essential;
  • In particular all compact hyperbolic manifolds are essential.
• All lens spaces are essential.

Properties

• Connected sum of essential manifolds is essential.

References

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See also

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