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In
geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...
, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.Gromov, M.: "Filling Riemannian manifolds," ''J. Diff. Geom.'' 18 (1983), 1–147.


Definition

A closed
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics, a manifold is a topological space that locally resembles Euclidean space ...
''M'' is called essential if its
fundamental class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'M''defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding
Eilenberg–MacLane space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
''K''(, 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^n \to \mathbb^\infty :is injective in homology, where ::\mathbb^\infty = K(\mathbb_2, 1) :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''(, 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


See also

*Gromov's systolic inequality for essential manifolds *Systolic geometry {{Systolic geometry navbox Algebraic topology Riemannian geometry Differential geometry Systolic geometry Manifolds