Homotopy 212250 221181747 2008-06-23T11:50:43Z 81.98.164.114 /* See also */ {{Otheruses4|topology|chemistry|Homotopic groups}} [[Image:Homotopy between two paths.png|thumb|The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.]] In [[topology]], two [[continuous function (topology)|continuous]] [[Function (mathematics)|functions]] from one [[topological space]] to another are called '''homotopic''' ([[Greek language|Greek]] ''homos'' = identical and ''topos'' = place) if one can be "continuously deformed" into the other, such a deformation being called a '''homotopy''' between the two functions. An outstanding use of homotopy is the definition of [[homotopy groups]] and [[cohomotopy groups]], important [[invariant (mathematics)|invariant]]s in [[algebraic topology]]. In practice, there are technical difficulties in using homotopies with certain pathological spaces. Consequently most algebraic topologists work with [[compactly generated space]]s, [[CW complex]]es, or [[Spectrum (homotopy theory)|spectra]]. ==Formal definition== [[Image:Mug and Torus morph.gif|thumb|right|250px|A homotopy of a coffee cup into a doughnut ([[torus]]).]] Formally, a homotopy between two [[continuous function (topology)|continuous function]]s ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function ''H'': ''X'' &times; [0,1] &rarr; ''Y'' from the [[product topology|product]] of the space ''X'' with the [[unit interval]] [0,1] to ''Y'' such that, for all points ''x'' in ''X'', ''H''(''x'',0)=''f''(''x'') and ''H''(''x'',1)=''g''(''x''). If we think of the second [[parameter]] of ''H'' as "time", then ''H'' describes a "continuous deformation" of ''f'' into ''g'': at time 0 we have the function ''f'', at time 1 we have the function ''g''. ===Properties=== Continuous functions ''f'' and ''g'' (both from topological space ''X'' to ''Y'') are said to be homotopic iff there is a homotopy ''H'' taking ''f'' to ''g'' as described above. Being homotopic is an [[equivalence relation]] on the set of all continuous functions from ''X'' to ''Y''. This homotopy relation is compatible with [[function composition]] in the following sense: if ''f''<sub>1</sub>, ''g''<sub>1</sub>: ''X'' &rarr; ''Y'' are homotopic, and ''f''<sub>2</sub>, ''g''<sub>2</sub>: ''Y'' &rarr; ''Z'' are homotopic, then their compositions ''f''<sub>2</sub> o ''f''<sub>1</sub> and ''g''<sub>2</sub> o ''g''<sub>1</sub>: ''X'' &rarr; ''Z'' are homotopic as well. ==Homotopy equivalence and null-homotopy== Given two spaces ''X'' and ''Y'', we say they are '''homotopy equivalent''' or of the same '''homotopy type''' if there exist continuous [[map (mathematics)|map]]s ''f'': ''X'' &rarr; ''Y'' and ''g'': ''Y'' &rarr; ''X'' such that ''g'' o ''f'' is homotopic to the [[identity function|identity map]] id<sub>''X''</sub> and ''f'' o ''g'' is homotopic to id<sub>''Y''</sub>. The maps ''f'' and ''g'' are called '''homotopy equivalences''' in this case. Clearly, every [[homeomorphism]] is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent. Intuitively, two spaces ''X'' and ''Y'' are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and '''R'''<sup>2</sup> - {(0,0)} is homotopy equivalent to the [[unit circle]] ''S''<sup>1</sup>. Those spaces that are homotopy equivalent to a point are called [[contractible]]. A function ''f'' is said to be '''null-homotopic''' if it is [[homotopic]] to a constant function. (The homotopy from ''f'' to a constant function is then sometimes called a '''null-homotopy'''.) For example, it is simple to show that a map from the [[circle]] ''S''<sup>1</sup> is null-homotopic precisely when it can be extended to a map of the disc ''D''<sup>2</sup>. It follows from these definitions that a space ''X'' is contractible if and only if the identity map from ''X'' to itself&mdash;which is always a homotopy equivalence&mdash;is null-homotopic. ==Homotopy invariance== Homotopy equivalence is important because in [[algebraic topology]] many concepts are '''homotopy invariant''', that is, they respect the relation of homotopy equivalence. For example, if ''X'' and ''Y'' are homotopy equivalent spaces, then: * if ''X'' is [[connected space|path-connected]], then so is ''Y'' * if ''X'' is [[simply connected]], then so is ''Y'' * the (singular) [[homology (mathematics)|homology]] and [[cohomology group]]s of ''X'' and ''Y'' are [[group isomorphism|isomorphic]] * if ''X'' and ''Y'' are path-connected, then the [[fundamental group]]s of ''X'' and ''Y'' are isomorphic, and so are the higher [[homotopy group]]s. Without the path-connectedness assumption, one has π<sub>1</sub>(''X'', x<sub>0</sub>) isomorphic to π<sub>1</sub>(''Y'', ''f''(''x''<sub>0</sub>)) where ''f'': ''X'' → ''Y'' is a homotopy equivalence and ''x''<sub>0</sub> a given point in ''X''. An example of an algebraic invariant of topological spaces which is not homotopy-invariant is [[compactly supported homology]] (which is, roughly speaking, the homology of the [[compactification (mathematics)|compactification]], and compactification is not homotopy-invariant). ==Homotopy category== The idea of homotopy can be turned into a formal category of [[category theory]]. The '''homotopy category''' is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces ''X'' and ''Y'' are isomorphic in this category if and only if they are homotopy-equivalent. Then a [[functor]] on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category. For example, homology groups are a ''functorial'' homotopy invariant: this means that if ''f'' and ''g'' from ''X'' to ''Y'' are homotopic, then the [[group homomorphism]]s induced by ''f'' and ''g'' on the level of [[homology group]]s are the same: H<sub>''n''</sub>(''f'') = H<sub>''n''</sub>(''g'') : H<sub>''n''</sub>(''X'') &rarr; H<sub>''n''</sub>(''Y'') for all ''n''. Likewise, if ''X'' and ''Y'' are in addition [[connectedness|path-connected]], then the group homomorphisms induced by ''f'' and ''g'' on the level of [[homotopy group]]s are also the same: &pi;<sub>''n''</sub>(''f'') = &pi;<sub>''n''</sub>(''g'') : &pi;<sub>''n''</sub>(''X'') &rarr; &pi;<sub>''n''</sub>(''Y''). ==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 ''H'': ''X'' &times; [0,1] &rarr; ''Y'' between ''f'' and ''g'' such that ''H''(''k'',''t'') = ''f''(''k'') = ''g''(''k'') for all ''k''&isin;''K'' and ''t''&isin;[0,1]. Also, if ''g'' is a [[retract]] from ''X'' to ''K'' and ''f'' is the identity map, this is known as a strong [[deformation retract]] of ''X'' to ''K''. ==Timelike homotopy== On a [[Lorentzian manifold]], certain curves are distinguished as [[timelike]]. A [[timelike homotopy]] between two timelike curves is a homotopy such that each intermediate curve is timelike. 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 multiply timelike connected. == Homotopy extension property == Another useful property involving homotopy is the [[homotopy extension property]], which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with [[cofibration]]s. == Isotopy == In case the two given continuous functions ''f'' and ''g'' from the topological space ''X'' to the topological space ''Y'' are [[homeomorphism]]s, one can ask whether they can be connected 'through homeomorphisms'. 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'',''t'') gives a homeomorphism. Requiring that two homeomorphisms be isotopic really is a stronger requirement than that they be homotopic. For example, the map of the [[unit disc]] in ''R''<sup>2</sup> defined by ''f''(''x'',''y'') = (&minus;''x'',&nbsp;&minus;''y'') is equivalent to a 180-degree [[rotation]] around the origin, and so the identity map and ''f'' are isotopic because they can be connected by rotations. However, the map on the interval [&minus;1,1] in ''R'' defined by ''f''(''x'') = &minus;''x'' is ''not'' isotopic to the identity. Loosely speaking, 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, hence it cannot be isotopic to the identity. 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. The intuitive idea of ''deforming'' one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism, ''h'', such that ''h'' moves ''K''<sub>1</sub> to ''K''<sub>2</sub>. An [[ambient isotopy]], studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. ==See also== *[[Mapping class group]] *[[Homeotopy]] *[[Regular homotopy]] *[[Poincaré conjecture]] [[Category:Homotopy theory| ]] [[Category:Continuous mappings]] [[de:Homotopie]] [[es:Homotopía]] [[fr:Homotopie]] [[he:הומוטופיה]] [[it:Omotopia]] [[nl:Homotopie-equivalentie]] [[ja:ホモトピー]] [[pl:Homotopia]] [[pt:Homotopia]] [[ru:Гомотопия]] [[sr:Хомотопија]] [[fi:Homotopia]] [[vi:Đồng luân]]