Search results
Results from the WOW.Com Content Network
A homotopy between two embeddings of the torus into : as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such that ...
In general, every manifold has the homotopy type of a CW complex; [3] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex. [citation needed] Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
One can likewise define a homotopy of loops keeping the base point fixed. The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path f {\displaystyle f} under this relation is called the homotopy class of f , {\displaystyle f,} often denoted [ f ] . {\displaystyle [f].}
Tables of homotopy groups of spheres are most conveniently organized by showing π n+k (S n). The following table shows many of the groups π n+k (S n). The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following ...
Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path space objects. These notions coincide when the domain is cofibrant and the codomain is fibrant. In that case, homotopy defines an equivalence relation on the hom sets in the model category giving rise to homotopy classes.
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
These homotopy classes form a group, called the n-th homotopy group, (), of the given space X with base point. Topological spaces with differing homotopy groups are never homeomorphic , but topological spaces that are not homeomorphic can have the same homotopy groups.