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• ID : Q662830

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1. D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion.
2. one sometimes loosely speaks of ‘the’ fundamental group of a connected space.
3. There is a relation to universal covers: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space.
4. In particular in algebraic geometry and arithmetic geometry this essentially identifies the concept of fundamental group with that of Galois groups.
5. For this reason one also speaks of the algebraic fundamental group in this context.
6. In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space.
7. This set (with the group structure described below) is called the fundamental group of the topological space X at the base point x 0 {\displaystyle x_{0}} .
8. It becomes a group (and therefore deserves the name fundamental group) using the concatenation of loops.
9. The fundamental group of the plane punctured at n points is also the free group with n generators.
10. In fact, the fundamental group only depends on the homotopy type of .
11. A space with a trivial fundamental group (i.e., every loop is homotopic to the constant loop), is called simply connected.
12. The underlying set of the fundamental group of is the set of based homotopy classes from the circle to , denoted .
13. When is a continuous map, then the fundamental group pushes forward.
14. Exercise (i) Show that the fundamental group of the circle S(1) is Z, the additive group of the integers.
15. A Hopf space is a space in which the proof given above for the statement that the fundamental group of a topological group is Abelian still works.
16. The definition of the fundamental group as the space of path components of gives a topology on the fundamental group.
17. It turns out, though, that if the path component of in is a locally path-connected space, then so is , in which case the fundamental group has a discrete topology.
18. Moreover, any path between these two points yields an isomorphism, and any two such isomorphisms have quotient equal to an inner automorphism of the fundamental group at one point.
19. All loops based at , up to homotopies where the intermediaries must be loops but need not be based at conjugacy class set of fundamental group ?
20. \pi _{1}(X,x_0)\rightarrow \check{\pi }_1(X,x_0)\) from the fundamental group to the first shape group is injective (See Sect.
21. The fundamental group is the first of what are known as the homotopy groups of a topological space.
22. An easy way to approach the concept of fundamental group is to start with a concrete example.
23. This type of approach constitutes the base of the definition of fundamental group and explains essential differences between different kinds of topological spaces.
24. Now, in a fundamental group, we will work with loops.
25. This section is dedicated to the calculation of the fundamental group of S 1 {\displaystyle S^{1}} that we can consider contained in the complex topological space.
26. I want to show the fundamental group of a topological group is abelian.
27. I have just learned the fundamental group.
28. For well-behaved topological spaces, there are several equivalent definitions for the fundamental group.
29. On the other hand, we will find that the fundamental group is too unwieldy to compute (and for deep reasons).
30. Since we want to be able to readily compute the number of holes in twisted and tied-up spaces, we will need to scrap the fundamental group and try something else.
31. For now, we will develop the idea of homotopy and the fundamental group.
32. Although we have not proved it here, the fundamental group is a topological invariant.
33. Note that the fundamental group is not in general abelian.
34. In order for the fundamental group to be a useful topological concept, any two spaces that are topologically "the same" must have the same fundamental group.
35. Daniel K. Biss, The topological fundamental group and generalized covering spaces , Topology and its Applications, Vol.

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위키데이터

• ID : Q662830