Information AboutHomotopy |
| CATEGORIES ABOUT HOMOTOPY | |
| homotopy theory | |
| continuous mappings | |
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In Topology , two Continuous Functions from one Topological Space to another are called homotopic ( 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 s in Algebraic Topology . In practice, for technical convenience homotopy theorists do not work with topological spaces directly, but with CW Complex es, Kan Complex es or another “similar” category. FORMAL DEFINITION Formally, a homotopy between two 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'' × → ''Y'' from the 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 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''1, ''g''1 : ''X'' → ''Y'' are homotopic, and ''f''2, ''g''2 : ''Y'' → ''Z'' are homotopic, then their compositions ''f''2 o ''f''1 and ''g''2 o ''g''1 : ''X'' → ''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 id''X'' and ''f'' o ''g'' is homotopic to id''Y''. 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: a solid disk is not homeomorphic to a single point, for example. 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 R2 - {(0,0)} is homotopy equivalent to the Unit Circle ''S''1. 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 is null-homotopic precisely when it can be extended to a map of the disc . It follows from these definitions that a space ''X'' is contractible if and only if the identity map from ''X'' to itself—which is always a homotopy equivalence— is null-homotopic. HOMOTOPY-INVARIANT PROPERTIES AND FUNCTIONS; HOMOTOPY CATEGORY 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:
More abstractly, one can appeal to Category Theory concepts. One can define the homotopy category, whose objects are topological spaces, and whose morphisms are homotopy 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 , then the group homomorphisms induced by ''f'' and ''g'' on the level of Homotopy Group s are also the same: π''n''(''f'') = π''n''(''g'') : π''n''(''X'') → π''n''(''Y''). 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 , and compactification is not homotopy-invariant). RELATIVE HOMOTOPY Especially in order to define the from ''X'' to ''K'' and ''f'' is the identiy map, this is known as a strong Deformation Retract of ''X'' to ''K'' HOMOTOPY EXTENSION PROPERTY Another useful property involving homotopy is the Homotopy Extension Property , which characterises 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''2 defined by ''f''(''x'',''y'') = (''-x'',''-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 {Link without Title} in ''R'' defined by ''f''(''x'') = -''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 - for example in Knot Theory - the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots ''K1'' and ''K2'' 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 ''K1'' to ''K2''. SEE ALSO Mapping Class Group Homeotopy |
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