Thurston norm

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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.

3-manifold 3-dimensional manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

William Thurston mathematician

William Paul Thurston was an American mathematician. He was 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. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

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Definition

Let be a differential manifold and . Then can be represented by a smooth embedding where is a (generally not connected) surface which is compact and without boundary. The Thurston norm of is then defined to be [1]

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

Surface (topology) two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

where the minimum is taken over all embedded surfaces (the being the connected components) representing as above and is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres).

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .

This function satisfies the following properties:

These properties imply that extends to a function on which can then be extended by continuity to a seminorm on . [2] By Poincaré duality one can define the Thurston norm on .

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors.

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold, then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k

When is compact with boundary the Thurston norm is defined in a similar manner on the relative homology group and its Poincaré dual .

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

It follows from further work of Gabai [3] that one can also define the Thurston norm using only immersed surfaces. This implies that the Thurston norm is also equal to half the Gromov norm on homology.

David Gabai, a mathematician, is the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.

Immersion (mathematics) differentiable function whose derivative is everywhere injective

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : MN is an immersion if

In mathematics, the Gromov norm of a compact oriented n-manifold is a norm on the homology 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.

Topological applications

The Thurston norm was introduced in view of its applications to fiberings and foliations of 3-manifolds.

The unit ball of the Thurston norm of a 3-manifold is a polytope with integer vertices. It can be used to describe the structure of the set of fiberings of over the circle: if can be written as the mapping torus of a diffeomorphism of a surface then the embedding represents a class in a top-dimensional (or open) face of : moreover all other integer points on the same face are also fibers in such a fibration. [4]

Embedded surfaces which minimise the Thurston norm in their homology class are exactly the closed leaves of foliations of . [3]

Notes

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In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group. These manifolds include some particularly beautiful or remarkable examples.

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