Information AboutHomotopy Group |
| CATEGORIES ABOUT HOMOTOPY GROUP | |
| homotopy theory | |
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HOMOTOPY GROUPS In the sphere ''S''''n'' we choose a base point ''a''. For a space ''X'' with base point ''b'', we define π''n''(''X'') to be the set of homotopy classes of maps ''f'' : ''S''''n'' → ''X'' that map the base point ''a'' to the base point ''b''. In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, we can define π''n''(X) to be the group of homotopy classes of maps ''g'' : {Link without Title} ''n'' → ''X'' from the ''n''-cube to ''X'' that take the boundary of the ''n''-cube to ''b''. For ''n'' ≥ 1, the homotopy classes form a of two ''n''-spheres that collapses the equator and ''h'' is the map from the wedge sum of two ''n''-spheres to ''X'' that is defined to be ''f'' on the first sphere and ''g'' on the second. If ''n'' ≥ 2, then π''n'' is Abelian . (For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other.) The Long Exact Sequence of a Fibration Let ''p'' : ''E'' → ''B'' be a basepoint-preserving Serre Fibration with fiber ''F'', that is, a map possessing the Homotopy Lifting Property with respect to CW Complex es. Then there is a long Exact Sequence of homotopy groups :… → π''n''(''F'') → π''n''(''E'') → π''n''(''B'') → π''n''−1(''F'') → … → π0(''E'') → π0(''B'') → 0 Here the maps involving π0 are not group Homomorphism s because the π0 are not groups, but they are exact in the sense that the image equals the kernel. Example: the Hopf Fibration . Let ''B'' equal ''S''2 and ''E'' equal S3. Let ''p'' be the Hopf Fibration , which has fiber S1. From the long exact sequence :… → π''n''(''S''1) → π''n''(''S''3) → π''n''(''S''2) → π''n''−1(''S''1) → … and the fact that π''n''(''S''1) = 0 for ''n'' ≥ 2, we find that π''n''(''S''3) = π''n''(''S''2) for ''n'' ≥ 3. In particular, π3(S2) = π3(S3) = Z. RELATIVE HOMOTOPY GROUPS There are also relative homotopy groups π''n''(''X'',''A'') for a pair (''X'',''A''). The elements of such a group are relative homotopy classes of maps ''Sn → X''. Two maps ''f, g'' are called homotopic relative to ''A'' if they are homotopic by a homotopy ''F'' : ''Sn'' × {Link without Title} → ''X'' such that, for each ''a'' in ''A'', the map ''F''(''a,t'') is constant (independent of ''t''). The ordinary homotopy groups are the special case in which ''A'' is the base point. There is a long exact sequence of relative homotopy groups. SEE ALSO |
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