Definitions

# Essential manifold

In mathematics, in algebraic topology and differential geometry, the notion of an essential manifold seems to have been first introduced explicitly in Mikhail Gromov's classic text in '83 (see below).

## 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

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

$mathbb\left\{RP\right\}^n to mathbb\left\{RP\right\}^\left\{infty\right\}$

is injective in homology, where

$mathbb\left\{RP\right\}^\left\{infty\right\} = K\left(mathbb\left\{Z\right\}_2, 1\right)$

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

Further examples of essential manifolds include aspherical manifolds and lens spaces.

## References

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