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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, ... where t varies with time from 0 to 1 over each cycle of the animation loop.
A CW complex is a space that has a filtration whose union is and such that . is a discrete space, called the set of 0-cells (vertices) in .; Each is obtained by attaching several n-disks, n-cells, to via maps ; i.e., the boundary of an n-disk is identified with the image of in .
Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
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.
Higher homotopy groups are sometimes difficult to compute. For instance, the homotopy groups of spheres are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups. For an = example, suppose is the figure eight.
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 ...
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.
The term () is the second homotopy group of B, which is defined to be the set of homotopy classes of maps from to B, in direct analogy with the definition of . If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism