Search results
Results from the WOW.Com Content Network
A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex.
We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology). Then the map E f → B {\displaystyle E_{f}\to B} given by ( a , γ ) ↦ γ ( 1 ) {\displaystyle (a,\gamma )\mapsto \gamma (1)} is a fibration.
There is a cofibration (A, X), if and only if there is a retraction from to () ({}), since this is the pushout and thus induces maps to every space sensible in the diagram. Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.
The homotopy extension property is depicted in the following diagram If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map f ~ ∙ {\displaystyle {\tilde {f}}_{\bullet }} which makes the diagram commute.
The normal complement, specifically, is the kernel of the retraction. Every direct factor is a retract. [1] Conversely, any retract which is a normal subgroup is a direct factor. [5] Every retract has the congruence extension property. Every regular factor, and in particular, every free factor, is a retract.
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: : where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map : in the category theory sense.
For example, a sphere has two cells: one 0-cell and one -cell, since can be obtained by collapsing the boundary of the n-disk to a point. 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.
For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.