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.

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 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.

Definition

Let ${\displaystyle M}$ be a differential manifold and ${\displaystyle c\in H_{2}(M)}$. Then ${\displaystyle c}$ can be represented by a smooth embedding ${\displaystyle S\to M}$ where ${\displaystyle S}$ is a (generally not connected) surface which is compact and without boundary. The Thurston norm of ${\displaystyle c}$ 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.

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.

${\displaystyle \|c\|_{T}=\min _{S}\sum _{i=1}^{n}\chi _{-}(S_{i})}$

where the minimum is taken over all embedded surfaces ${\displaystyle S=\bigcup _{i}S_{i}}$ (the ${\displaystyle S_{i}}$ being the connected components) representing ${\displaystyle c}$ as above and ${\displaystyle \chi _{-}(F)=\max(0,-\chi (F))}$ is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres).

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This function satisfies the following properties:

• ${\displaystyle \|kc\|_{T}=|k|\cdot \|c\|_{T}}$ for ${\displaystyle c\in H_{2}(M),k\in \mathbb {Z} }$;
• ${\displaystyle \|c_{1}+c_{2}\|_{T}\leq \|c_{1}\|_{T}+\|c_{2}\|_{T}}$ for ${\displaystyle c_{1},c_{2}\in H_{2}(M)}$.

These properties imply that ${\displaystyle \|\cdot \|}$ extends to a function on ${\displaystyle H_{2}(M,\mathbb {Q} )}$ which can then be extended by continuity to a seminorm ${\displaystyle \|\cdot \|_{T}}$ on ${\displaystyle H_{2}(M,\mathbb {R} )}$. [2] By Poincaré duality one can define the Thurston norm on ${\displaystyle H^{1}(M,\mathbb {R} )}$.

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 ${\displaystyle M}$ is compact with boundary the Thurston norm is defined in a similar manner on the relative homology group ${\displaystyle H_{2}(M,\partial M,\mathbb {R} )}$ and its Poincaré dual ${\displaystyle H^{1}(M,\mathbb {R} )}$.

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.

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 ${\displaystyle B}$ of the Thurston norm of a 3-manifold ${\displaystyle M}$ is a polytope with integer vertices. It can be used to describe the structure of the set of fiberings of ${\displaystyle M}$ over the circle: if ${\displaystyle M}$ can be written as the mapping torus of a diffeomorphism ${\displaystyle f}$ of a surface ${\displaystyle S}$ then the embedding ${\displaystyle S\hookrightarrow M}$ represents a class in a top-dimensional (or open) face of ${\displaystyle B}$: 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 ${\displaystyle M}$. [3]

Notes

1. Thurston 1986, Theorem 1.
2. Thurston 1986, Theorem 5.

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References

• Gabai, David (1983). "Foliations and the topology of 3-manifolds". J. Differential Geom. 18: 445–503. MR   0723813.
• Thurston, William (1986). "A norm for the homology of 3-manifolds". Mem. Amer. Math. Soc. 59 (33): i–vi and 99–130. MR   0823443.