Editing Homotopy (section)

== Variants ==

===Relative homotopy===

In order to define the [[fundamental group]], one needs the notion of '''homotopy relative to a subspace'''. These are homotopies which keep the elements of the subspace fixed. Formally: if ''f'' and ''g'' are continuous maps from ''X'' to ''Y'' and ''K'' is a [[subset]] of ''X'', then we say that ''f'' and ''g'' are homotopic relative to ''K'' if there exists a homotopy {{nowrap|1=''H'' : ''X'' &times; [0,&thinsp;1] → ''Y''}} between ''f'' and ''g'' such that {{nowrap|1=''H''(''k'',&thinsp;''t'') = ''f''(''k'') = ''g''(''k'')}} for all {{nowrap|1=''k'' ∈ ''K''}} and {{nowrap|1=''t'' ∈ [0,&thinsp;1].}} Also, if ''g'' is a [[retraction (topology)|retraction]] from ''X'' to ''K'' and ''f'' is the identity map, this is known as a strong [[deformation retract]] of ''X'' to ''K''.
When ''K'' is a point, the term '''pointed homotopy''' is used.

=== Isotopy ===
{{multiple image
| total_width = 320
| image1 = Blue Unknot.png
| image2 = Blue Trefoil Knot.png
| footer = The [[unknot]] is not equivalent to the [[trefoil knot]] since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. Thus they are not ambient-isotopic.
}}
When two given continuous functions ''f'' and ''g'' from the topological space ''X'' to the topological space ''Y'' are [[embedding]]s, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of '''isotopy''', which is a homotopy, ''H'', in the notation used before, such that for each fixed ''t'', ''H''(''x'',&thinsp;''t'') gives an embedding.<ref>{{MathWorld|Isotopy|Isotopy}}</ref>

A related, but different, concept is that of [[ambient isotopy]].

Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1,&thinsp;1] into the real numbers defined by ''f''(''x'') = &minus;''x'' is ''not'' isotopic to the identity ''g''(''x'') = ''x''.  Any homotopy from ''f'' to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, ''f'' has changed the orientation of the interval and ''g'' has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from ''f'' to the identity is ''H'':&nbsp;[−1,&thinsp;1]&nbsp;&times;&nbsp;[0,&thinsp;1]&nbsp;→&nbsp;[−1,&thinsp;1] given by ''H''(''x'',&thinsp;''y'')&nbsp;=&nbsp;2''yx''&nbsp;−&nbsp;''x''.

Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using [[Alexander's trick]]. For this reason, the map of the [[unit disc]] in <math>\mathbb{R}^2</math> defined by ''f''(''x'',&thinsp;''y'') = (&minus;''x'',&nbsp;&minus;''y'') is isotopic to a 180-degree [[rotation]] around the origin, and so the identity map and ''f'' are isotopic because they can be connected by rotations.

In [[geometric topology]]&mdash;for example in [[knot theory]]&mdash;the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same?  We take two knots, ''K''<sub>1</sub> and ''K''<sub>2</sub>, in three-[[dimension]]al space. A knot is an [[embedding]] of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one can ''deform'' one embedding to another through a path of embeddings: a continuous function starting at ''t''&nbsp;=&nbsp;0 giving the ''K''<sub>1</sub> embedding, ending at ''t''&nbsp;=&thinsp;1 giving the ''K''<sub>2</sub> embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An [[ambient isotopy]], studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots ''K''<sub>1</sub> and ''K''<sub>2</sub> are considered equivalent when there is an ambient isotopy which moves ''K''<sub>1</sub> to ''K''<sub>2</sub>. This is the appropriate definition in the topological category.

Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence.  For example, a path between two smooth embeddings is a '''smooth isotopy'''.

=== Timelike homotopy ===
On a [[Lorentzian manifold]], certain curves are distinguished as [[timelike]] (representing something that only goes forwards, not backwards, in time, in every local frame). A [[timelike homotopy]] between two [[timelike curve]]s is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No [[closed timelike curve]] (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be [[multiply connected]] by timelike curves. A manifold such as the [[3-sphere]] can be [[simply connected]] (by any type of curve), and yet be [[timelike multiply connected]].<ref>{{Cite journal|last=Monroe|first=Hunter|date=2008-11-01|title=Are Causality Violations Undesirable?|journal=Foundations of Physics|language=en|volume=38|issue=11|pages=1065–1069|doi=10.1007/s10701-008-9254-9|issn=0015-9018|arxiv=gr-qc/0609054|bibcode=2008FoPh...38.1065M|s2cid=119707350 }}</ref>