Image torus

In mathematics , mapping tori are topological spaces with which topological maps are described.

definition

The Möbius strip is the image torus of the image defined by . ${\displaystyle f(x)=-x}$ ${\displaystyle f\colon \left[-1,1\right]\rightarrow \left[-1,1\right]}$

The Klein bottle is a nontrivial bundle over S ¹ with fiber S ¹ and monodromy . ${\displaystyle f(z)={\frac {1}{z}}}$

Be a topological space and a homeomorphism . The mapping torus of is defined as the quotient ${\displaystyle X}$ ${\displaystyle f\colon X\rightarrow X}$ ${\displaystyle f}$

${\displaystyle T_{f}:=X\times \left[0,1\right]/\sim }$

of regarding the equivalence relation for all . ${\displaystyle X\times \left[0,1\right]}$ ${\displaystyle (x,1)\sim (f(x),0)}$ ${\displaystyle x\in X}$

Bundles of fibers over the circle

The circuit can be used as quotient space with be construed so defines the projection on the second factor , a fiber bundle ${\displaystyle S^{1}}$ ${\displaystyle S^{1}=\left[0,1\right]/\sim }$ ${\displaystyle 0\sim 1}$ ${\displaystyle p\colon X\times \left[0,1\right]\rightarrow \left[0,1\right]}$

${\displaystyle p\colon T_{f}\rightarrow S^{1}}$.

Conversely, each fiber bundle above the circle can be represented as a mapping torus of a homeomorphism . The mapping is called the monodromy of the fiber bundle. ${\displaystyle f\colon X\rightarrow X}$ ${\displaystyle f}$

Mapping gate in the 3-dimensional topology

Mapping tori play an important role in Thurston's approach to the geometrization of 3-manifolds .

Homeomorphisms of compact surfaces fall into one of three categories: periodic, reducible, or pseudo-anosov. Thurston has proven that a 3-dimensional mapping torus is hyperbolic if and only if the monodromy is pseudo-Anosov.

In 2012, Ian Agol showed that every compact 3-manifold has a finite overlay , which can be represented as a mapping torus.

Group theory

In group theory, mapping gates are defined for endomorphisms of free groups . Let be the free group created by a set and be an endomorphism. Then the mapping torus is defined by the presentation ${\displaystyle F(X)=\langle X\mid -\rangle }$ ${\displaystyle X}$ ${\displaystyle \phi \colon F(X)\rightarrow F(X)}$

${\displaystyle T_{\phi }:=\langle t,X\mid t^{-1}xt=\phi (x)\ \forall x\in X\rangle }$.

Web links

1. ↑ Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
2. ^ The virtual hook conjecture Documenta Math. 18 (2013) 1045-1087