The concept of size homotopy group is analogous in
size theory In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance b ...
of the classical concept of
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
. In order to give its definition, let us assume that a
size pair In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance b ...
is given, where
is a
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
of class
and
is a
continuous function. Consider the
lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...
on
defined by setting
if and only if
. For every
set
.
Assume that
and
. If
,
are two paths from
to
and a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
from
to
, based at
, exists in the
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, then we write
. The first size homotopy group of the
size pair In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance b ...
computed at
is defined to be the
quotient set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of the set of all
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire ...
s from
to
in
with respect to the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
, endowed with the operation induced by the usual composition of based
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, ...
s.
[Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.]
In other words, the first size homotopy group of the
size pair In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance b ...
computed at
and
is the image
of the first
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
with base point
of the
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, when
is the
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
induced by the inclusion of
in
.
The
-th size homotopy group is obtained by substituting the loops based at
with the
continuous functions
taking a fixed point of
to
, as happens when higher
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s are defined.
See also
*
Size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain
*
Size functor
*
Size pair In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance b ...
*
Natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ t ...
References
Algebraic topology
{{topology-stub