Gromov norm
Encyclopedia
In mathematics
, the Gromov norm (or simplicial volume) of a compact oriented n-manifold
is a norm on the homology
(with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class
.
It is named after Mikhail Gromov, who with William Thurston
, proved that the Gromov norm of a finite volume hyperbolic n-manifold
is proportional to the hyperbolic volume. Thurston also used the Gromov norm to prove that hyperbolic volume decreases under hyperbolic Dehn surgery
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Gromov norm (or simplicial volume) of a compact oriented n-manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is a norm on the homology
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
(with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class
Fundamental class
In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
.
It is named after Mikhail Gromov, who with William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, proved that the Gromov norm of a finite volume hyperbolic n-manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
is proportional to the hyperbolic volume. Thurston also used the Gromov norm to prove that hyperbolic volume decreases under hyperbolic Dehn surgery
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...
.
External links
- Simplicial volume at the Manifold Atlas.