Essential manifold
From HandWiki
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
- [math]\displaystyle{ H_n(M)\to H_n(K(\pi,1)), }[/math]
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
- [math]\displaystyle{ \mathbb{RP}^n \to \mathbb{RP}^\infty }[/math]
- is injective in homology, where
- [math]\displaystyle{ \mathbb{RP}^\infty = K(\mathbb{Z}_2, 1) }[/math]
- 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))
- 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.
References
- ↑ Gromov, M. (1983). "Filling Riemannian manifolds". J. Diff. Geom. 18: 1–147.
See also
Original source: https://en.wikipedia.org/wiki/Essential manifold.
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