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
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.
The loop space is naturally itself a dotted topological space, the constant loop for all is taken as the base point .
Loop space as functor
If and are dotted topological spaces and is a continuous mapping, then is through
explains a continuous mapping between the loop spaces. If a third dotted topological space is continuous, then obviously holds
-
.
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
-
, so that
-
for all
-
for all
-
for all
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.
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 .
If there are two loops , a new loop can be formed from them, which runs through first and then , more precisely
-
.
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.
Relationship to the mount
The suspension of the dotted topological space is a quotient space
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
one obtains a continuous mapping
and thus a continuous mapping
-
.
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
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 .
Individual evidence
-
↑ Tammo tom Dieck : Algebraic Topology , European Mathematical Society (2008), ISBN 978-3-03719-048-7 , Section 4.4: Loop Space
-
^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture , Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , paragraph 7.1, sentence 1
-
↑ Tammo tom Dieck: Algebraic Topology , European Mathematical Society (2008), ISBN 978-3-03719-048-7 , Section 4.4: Loop Space