Loop (topology)

A free loop in the plane.

A loop on the torus with a base point . ${\displaystyle p}$

In topology , a branch of mathematics, closed curves are also known as loops .

Free loops and loops

A free loop in a topological space is a continuous mapping from the unit interval to , where applies . This means that the start point is the same as the end point. A free loop can also serve as continuous mapping of the unit circle to be seen because as the ratio of can be seen under the identification of 0 at 1. ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle I=[0,1]}$ ${\displaystyle X}$ ${\displaystyle f(0)=f(1)}$ ${\displaystyle S^{1}}$ ${\displaystyle X}$ ${\displaystyle S^{1}}$ ${\displaystyle I}$

In space when a base point is set, then is called the loop is a continuous mapping of the unit interval on , where the following applies . This means that the start point and end point are the same as the fixed base point. A loop can also be considered continuous mapping of the unit circle to be seen, with a fixed base point selected by the fixed selected base point is imaged. ${\displaystyle X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle f}$ ${\displaystyle I=[0,1]}$ ${\displaystyle X}$ ${\displaystyle f(0)=f(1)=x_{0}}$ ${\displaystyle S^{1}}$ ${\displaystyle X}$ ${\displaystyle \left[1\right]}$ ${\displaystyle S^{1}}$ ${\displaystyle x_{0}\in X}$

Homotopy classes

Two free loops are called (free) homotopic if there is a homotopy between the two images . ${\displaystyle S^{1}\to X}$

Two loops are called homotopic if there is a free homotopy that additionally fulfills the condition for all . The sets of the homotopy classes of loops form the important concept of the fundamental group . ${\displaystyle H\colon S^{1}\times \left[0,1\right]\to X}$ ${\displaystyle H(\left[1\right],t)=x_{0}}$ ${\displaystyle t\in \left[0,1\right]}$ ${\displaystyle \pi _{1}(X,x_{0})}$

The set of all loops in a topological space is called loop space and denoted by. The set of free loops is called the free loop space. ${\displaystyle X}$ ${\displaystyle \Omega X}$ ${\displaystyle LX}$

Individual evidence

1. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture , Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , section 7.1: Paths and homotopy of paths