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The antipodal map preserves orientation (is homotopic to the identity map) [2] when is odd, and reverses it when is even. Its degree is ( − 1 ) n + 1 . {\displaystyle (-1)^{n+1}.} If antipodal points are identified (considered equivalent), the sphere becomes a model of real projective space .
The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a ...
the inclusion, a retraction is a continuous map r such that =, that is, the composition of r with the inclusion is the identity of A. Note that, by definition, a retraction maps X onto A. A subspace A is called a retract of X if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (any constant map ...
Two maps , are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy : [,] such that, for each p in and t in [,], the element (,) is in A. Note that ordinary homotopy groups are recovered for the special case in which A = { x 0 } {\displaystyle A=\{x_{0}\}} is the singleton containing the base point.
The degree of a map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps ,: are homotopic if and only if = (). In other words, degree is an isomorphism between [ S n , S n ] = π n S n {\displaystyle \left[S^{n},S^{n}\right]=\pi _{n}S^{n}} and Z {\displaystyle ...
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.)
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group). n -connectivity for spaces can in turn be defined in terms of n -connectivity of maps: a space X with basepoint x 0 is an n -connected space if and only if the inclusion of the ...
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 .