In the mathematical field of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a
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 ...
. More generally, the simplicial norm measures the complexity of
homology class
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromo ...
es.
Given a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The simplicial volume is the simplicial norm of the
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 fundamen ...
.
[.]
It is named after
Mikhail Gromov, who introduced it in 1982. With
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
, he proved that the simplicial volume of a finite volume
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, res ...
is proportional to the
hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...
.
The simplicial volume is equal to twice the
Thurston norm In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces.
Definitio ...
Thurston also used the simplicial volume to prove that hyperbolic volume decreases under
hyperbolic Dehn surgery In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperb ...
.
[, pp. 196ff.]
References
* Michael Gromov
''Volume and bounded cohomology.'' Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
External links
Simplicial volumeat the Manifold Atlas.
Homology theory
Norms (mathematics)
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