호모토피 군

노트

1. Any homotopy system is isomorphic to the homotopy system constructed above, which consists of homotopy groups.
2. Two principal methods are known for the computation of the homotopy groups of specific spaces: the method of killing spaces (cf.
3. In this theory stable homotopy groups arise as the homotopy groups of spectra.
4. Homotopy groups have been generalized in various directions.
5. This tells us that a basketball and a donut are not the same: they don’t have the same first homotopy groups.
6. They’re a bit like the first homotopy group in that they also measure holes.
7. The first homology group is related to the first homotopy group by something called abelianization.
8. It doesn’t matter too much what abelianization is, but it means the first homology group is a little simpler than the first homotopy group.
9. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
10. The first and simplest homotopy group is the fundamental group, which records information about loops in a space.
11. These are related to relative homotopy groups and to n-adic homotopy groups respectively.
12. A higher homotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on homotopy types.
13. Next we consider the properties of the homotopy groups π n + p ( S n ) for p ≥ 0.
14. The homotopy groups of spheres describe the ways in which spheres can be attached to each other.
15. This degree identifies the homotopy group π n ( S n ) with the group of integers under addition.
16. This degree identifies the homotopy group with the group of integers under addition.
17. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64.
18. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory.
19. Lower homotopy groups act on higher homotopy groups; the nonabelian group cohomology of this gives the Postnikov invariants of the space.
20. Accordingly, homotopy groups are defined for all other models of homotopy types, notably for simplicial sets.
21. However, the homotopy groups by themselves, even considering the operations of π 1 \pi_1 , do not characterise homotopy types.
22. Homotopy groups and their properties can naturally be formalized in homotopy type theory.
23. GF \rightarrow {\overline틀:\varOmega}F\) that induces an isomorphism on the level of homotopy groups.
24. The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle.
25. The choice of direction of a loop in the fundamental group corresponds to a manifold orientation of in a homotopy group.
26. As with the fundamental group, the homotopy groups do not depend on the choice of basepoint.
27. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.
28. Much less has been done to consider the homotopy groups of manifolds.
29. On homotopy groups of the suspended classifying spaces.
30. Homotopy groups as centres of finitely presented groups.
31. The definition of homotopy group still gives a set definition for .
32. For a path-connected space, the homotopy groups for all basepoints are isomorphic.
33. In general, the homotopy group may differ for different path components.
34. The homotopy groups depend only on the homotopy type of the based topological space.
35. a Π‐algebra—that is, a graded group G * with a prescribed action of the primary homotopy operations—as the homotopy groups of some space.

메타데이터

위키데이터

• ID : Q1626416

Spacy 패턴 목록

• [{'LOWER': 'homotopy'}, {'LEMMA': 'group'}]