enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Antipodal point - Wikipedia

    en.wikipedia.org/wiki/Antipodal_point

    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 .

  3. Homotopy - Wikipedia

    en.wikipedia.org/wiki/Homotopy

    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 ...

  4. Homotopy category of chain complexes - Wikipedia

    en.wikipedia.org/wiki/Homotopy_category_of_chain...

    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.)

  5. Degree of a continuous mapping - Wikipedia

    en.wikipedia.org/wiki/Degree_of_a_continuous_mapping

    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 ...

  6. Retraction (topology) - Wikipedia

    en.wikipedia.org/wiki/Retraction_(topology)

    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 ...

  7. Chain complex - Wikipedia

    en.wikipedia.org/wiki/Chain_complex

    The map hd A + d B h is easily verified to induce the zero map on homology, for any h. It immediately follows that f and g induce the same map on homology. One says f and g are chain homotopic (or simply homotopic), and this property defines an equivalence relation between chain maps. Let X and Y be topological spaces.

  8. Homotopy groups of spheres - Wikipedia

    en.wikipedia.org/wiki/Homotopy_groups_of_spheres

    The null homotopic class acts as the identity of the group addition, and for X equal to S n (for positive n) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some i all maps are null homotopic, then the group π i consists of one element, and is called the trivial group.

  9. Homotopy group - Wikipedia

    en.wikipedia.org/wiki/Homotopy_group

    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.