homotopy
التعريفات والمعاني
== English ==
=== Etymology ===
From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.
=== Pronunciation ===
(US) IPA(key): /həˈmɑtəpi/, /ˈhoʊmoʊˌtoʊpi/, /ˈhoʊmə-/
=== Noun ===
homotopy (countable and uncountable, plural homotopies)
(topology) A continuous deformation of one continuous function or map to another.
2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,
In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X.
(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident.
(informal) Ellipsis of homotopy theory (“the systematic study of homotopies and their equivalence classes”).
(topology) A theory associating a system of groups with each topological space.
(topology) A system of groups associated with a topological space.
==== Usage notes ====
Formally, there are two alternative formulations:
Given topological spaces
X
,
Y
{\displaystyle X,Y}
and continuous maps
f
,
g
:
X
→
Y
{\displaystyle f,g:X\rightarrow Y}
A continuous map
H
:
[
0
,
1
]
×
X
→
Y
{\displaystyle H:[0,1]\times X\rightarrow Y}
such that
H
(
x
,
0
)
=
f
(
x
)
{\displaystyle H(x,0)=f(x)}
and
H
(
x
,
1
)
=
g
(
x
)
{\displaystyle H(x,1)=g(x)}
∀
x
∈
X
{\displaystyle \forall x\in X}
.
A family of continuous maps
h
t
:
X
→
Y
,
t
∈
[
0
,
1
]
{\displaystyle h_{t}:X\rightarrow Y,t\in [0,1]}
such that
h
0
=
f
,
{\displaystyle h_{0}=f,}
h
1
=
g
{\displaystyle h_{1}=g}
and the map
(
x
,
t
)
↦
h
t
{\displaystyle (x,t)\mapsto h_{t}}
is continuous from
X
{\displaystyle X}
to
Y
{\displaystyle Y}
. (Note that it is not sufficient to require that each map
h
t
(
x
)
{\displaystyle h_{t}(x)}
be continuous.)
Replacing the unit interval
[
0
,
1
]
{\displaystyle [0,1]}
with the affine line A¹ leads to A¹ homotopy theory.
The adjective homotopic is used specifically in the sense, with respect to two functions, of "having the relationship of being in homotopy".
Being homotopic is an equivalence relation on the class of all continuous functions between given topological spaces. An equivalence class of such a relation is called a homotopy class.
==== Hyponyms ====
(continuous deformation): isotopy, regular homotopy
==== Derived terms ====
==== Related terms ====
homotopic
==== Translations ====
=== See also ===
homeotopy group
homology
=== References ===
=== Further reading ===
Homotopy group on Wikipedia.Wikipedia
Fundamental group on Wikipedia.Wikipedia
Homotopy theory on Wikipedia.Wikipedia
A¹ homotopy theory on Wikipedia.Wikipedia
Homeotopy on Wikipedia.Wikipedia
Fiber-homotopy equivalence on Wikipedia.Wikipedia
Poincaré conjecture on Wikipedia.Wikipedia
Homotopy on Encyclopedia of Mathematics