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: $H_n(M) o H_n(K(pi,1))$ , 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

: $mathbb{RP}^n o mathbb{RP}^{infty}$

is injective in homology, where

: $mathbb{RP}^{infty} = K(mathbb{Z}_2, 1)$

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.

ee also

*Gromov's systolic inequality for essential manifolds
*Systolic geometry

Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

Manifold — For other uses, see Manifold (disambiguation). The sphere (surface of a ball) is a two dimensional manifold since it can be represented by a collection of two dimensional maps. In mathematics (specifically in differential geometry and topology),… … Wikipedia
Gromov's systolic inequality for essential manifolds — In Riemannian geometry, M. Gromov s systolic inequality for essential n manifolds M dates from 1983. It is a lower bound for the volume of an arbitrary metric on M, in terms of its homotopy 1 systole. The homotopy 1 systole is the least length of … Wikipedia
3-manifold — In mathematics, a 3 manifold is a 3 dimensional manifold. The topological, piecewise linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3… … Wikipedia
CR manifold — In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formally, a CR manifold is a… … Wikipedia
Whitehead manifold — In mathematics, the Whitehead manifold is an open 3 manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture.A contractible manifold is one… … Wikipedia
gauge manifold — [1] Chamber device constructed to hold both compound and high pressure gauges. Valves control flow of fluids through it. [2] The one essential diagnostic tool required for every air conditioner service procedure. A typical gauge set includes high … Dictionary of automotive terms
List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… … Wikipedia
List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… … Wikipedia
Aspherical space — In topology, an aspherical space is a topological space with all higher homotopy groups equal to {0}. If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is… … Wikipedia
Introduction to systolic geometry — Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C , and the length or perimeter of C . Since the area A may be small while the… … Wikipedia

Academic Dictionaries and Encyclopedias

Essential manifold

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Essential manifold

Look at other dictionaries:

Share the article and excerpts

Direct link