geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.

Definition

A closed

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''M'' is called essential if its

fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...

'M''defines a nonzero element in the homology of its

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...

, or more precisely in the homology of the corresponding

Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...

''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 manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, r ...

s are essential. *All lens spaces are essential.

Properties

*The

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

of essential manifolds is essential. *Any manifold which admits a map of nonzero degree to an essential manifold is itself essential.

Definition

Examples

Properties

References

See also