Loop space

The loop space is a construction from the mathematical subfield of topology , in particular homotopy theory .

definition

Let it be a dotted topological space . It is the space of all continuous functions , provided with a compact, open topology . The loop space of is the subspace ${\displaystyle (X,x_{0})}$ ${\displaystyle C([0,1],X)}$ ${\displaystyle w:[0,1]\rightarrow X}$ ${\displaystyle (X,x_{0})}$

${\displaystyle \Omega (X,x_{0}):=\{w\in C([0,1],X)\,\mid \,w(0)=w(1)=x_{0}\}}$

with the subspace topology .

The "points" of Thus are closed paths with start and end points , so-called grinding on . This explains the term loop space. ${\displaystyle \Omega (X,x_{0})}$ ${\displaystyle w}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

The loop space is naturally itself a dotted topological space, the constant loop for all is taken as the base point . ${\displaystyle \Omega (X,x_{0})}$ ${\displaystyle k:[0,1]\rightarrow X,k(t)=x_{0}}$ ${\displaystyle t\in [0,1]}$

Loop space as functor

If and are dotted topological spaces and is a continuous mapping, then is through ${\displaystyle (X,x_{0})}$ ${\displaystyle (Y,y_{0})}$ ${\displaystyle f:(X,x_{0})\rightarrow (Y,y_{0})}$

${\displaystyle \Omega f:\Omega (X,x_{0})\rightarrow \Omega (Y,y_{0}),w\mapsto f\circ w}$

explains a continuous mapping between the loop spaces. If a third dotted topological space is continuous, then obviously holds ${\displaystyle (Z,z_{0})}$ ${\displaystyle g:(Y,y_{0})\rightarrow (Z,z_{0})}$

${\displaystyle \Omega (g\circ f)=\Omega g\circ \Omega f}$.

In this way one obtains a functor on the category of dotted topological spaces.

Homotopias and fundamental group

A homotopy between two loops is a continuous mapping ${\displaystyle v,w\in \Omega (X,x_{0})}$

${\displaystyle H:[0,1]\times [0,1]\rightarrow X}$, so that

${\displaystyle H(s,0)=v(s)}$   for all ${\displaystyle s\in [0,1]}$

${\displaystyle H(s,1)=w(s)}$   for all ${\displaystyle s\in [0,1]}$

${\displaystyle H(0,t)=H(1,t)=x_{0}}$   for all ${\displaystyle t\in [0,1]}$

One imagines it in such a way that the loops and are constantly "deformed" into one another. The last of the mentioned conditions ensures that the loops are also on . Such homotopias, which fix the base point of the dotted topological space, are more precisely called dotted homotopias. ${\displaystyle v=H(\cdot ,0)}$ ${\displaystyle w=H(\cdot ,1)}$ ${\displaystyle H(\cdot ,t)}$ ${\displaystyle H(\cdot ,t)}$ ${\displaystyle x_{0}}$

Homotopy between loops is an equivalence relation , the set of equivalence classes of is often denoted by. The equivalence class of a loop is denoted by and called the homotopy class . ${\displaystyle \Omega (X,x_{0})}$ ${\displaystyle \pi _{1}(X,x_{0})}$ ${\displaystyle w}$ ${\displaystyle [w]}$

If there are two loops , a new loop can be formed from them, which runs through first and then , more precisely ${\displaystyle v,w\in \Omega (X,x_{0})}$ ${\displaystyle v*w}$ ${\displaystyle v}$ ${\displaystyle w}$

${\displaystyle (v*w)(t)={\begin{cases}v(2t)&{\text{ für }}t\in [0,{\textstyle {\frac {1}{2}}}]\\w(2t-1)&{\text{ für }}t\in [{\textstyle {\frac {1}{2}}},1]\end{cases}}}$.

This link is compatible with the homotopy of loops, thus induces a shortcut on the set of homotopy classes: . It can be shown that this linkage to a group makes, which are the fundamental group of names, neutral element is that the constant homotopy loop. The loop space itself is not a group with the link *, so it is necessary to move on to the homotopy classes. ${\displaystyle \pi _{1}(X,x_{0})}$ ${\displaystyle [v]*[w]:=[v*w]}$ ${\displaystyle \pi _{1}(X,x_{0})}$ ${\displaystyle (X,x_{0})}$ ${\displaystyle [k]}$

Relationship to the mount

The suspension of the dotted topological space is a quotient space ${\displaystyle \Sigma (X,x_{0})}$ ${\displaystyle (X,x_{0})}$

${\displaystyle \Sigma (X,x_{0})=(X\times [0,1])/(X\times \{0\}\cup X\times \{1\}\cup \{x_{0}\}\times [0,1])}$

defined, let the quotient mapping be defined, whereby the image of is taken as the base point in as usual . Let it be another dotted topological space. To a continuous mapping ${\displaystyle q:X\times [0,1]\rightarrow \Sigma (X,x_{0})}$ ${\displaystyle X\times \{0\}\cup X\times \{1\}\cup \{x_{0}\}\times [0,1]}$ ${\displaystyle \Sigma (X,x_{0})}$ ${\displaystyle (Y,y_{0})}$

${\displaystyle f:\Sigma (X,x_{0})\rightarrow (Y,y_{0})}$

one obtains a continuous mapping

${\displaystyle f\circ q:X\times [0,1]\rightarrow Y}$

and thus a continuous mapping

${\displaystyle {\tilde {f}}:(X,x_{0})\rightarrow \Omega (Y,y_{0}),\,({\tilde {f}}(x))(t):=(f\circ q)(x,t),\quad x\in X,t\in [0,1]}$.

Since and below are mapped to the base point of and receive base points is, that is , is actually an element of the loop space . We thus get a bijective mapping ${\displaystyle (x,0)}$ ${\displaystyle (x,1)}$ ${\displaystyle q}$ ${\displaystyle \Sigma (X,x_{0})}$ ${\displaystyle f}$ ${\displaystyle (f\circ q)(x,0)=(f\circ q)(x,1)=y_{0}}$ ${\displaystyle {\tilde {f}}(x)}$ ${\displaystyle \Omega (Y,y_{0})}$

${\displaystyle C(\Sigma (X,x_{0}),(Y,y_{0}))\rightarrow C((X,x_{0}),\Omega (Y,y_{0})),\,f\mapsto {\tilde {f}}}$

in the category of dotted topological spaces, this mapping is compatible with dotted homotopies, and therefore induces a bijection between the sets of homotopy classes. In this sense the functors and are adjoint . ${\displaystyle \Omega }$ ${\displaystyle \Sigma }$

Individual evidence

1. ↑ Tammo tom Dieck : Algebraic Topology , European Mathematical Society (2008), ISBN 978-3-03719-048-7 , Section 4.4: Loop Space
2. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture , Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , paragraph 7.1, sentence 1
3. ↑ Tammo tom Dieck: Algebraic Topology , European Mathematical Society (2008), ISBN 978-3-03719-048-7 , Section 4.4: Loop Space