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 defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corresponding Eilenberg–MacLane space K, via the natural homomorphismwhere 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 are essential with the exception of the 2-sphere S2.
- Real projective space RPn is essential since the inclusion
- :
- All compact aspherical manifolds are essential
- *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.