Fundamental Group of a Topological Space  

In General > s.a. homotopy.
* History: It was introduced by Poincaré in the 1890s.
$ Def 1: The fundamental group of (X, τ) is it first homotopy group, the set of equivalence classes of loops in X, where the equivalence relation is homotopy of paths.
$ Def 2: The group of covering transformations of the universal covering space of X; When the universal covering space exists, e.g., for topological groups, the fundamental group can also be defined as the group of homeomorphisms f : X*X * such that f \(\circ\) φ = f, where (X*, f) is the universal covering space of X.
$ Def 3: π1(X, x0) is the set of path-connected components of the space of loops at x0, with the compact-open topology.

Calculating Theorem
$ Def: For a polyhedron K, π1(K, a0) ≅ G, presented by the generator set H = {gij | [ai, aj] is an ordered 1-simplex of K}, and relations D = {gij gjk gik–1 for all ordered 2-simplices [ai, aj, ak] of K \ L, and gij = 1 if [ai, aj] ∈ L, where L is a 1D subpolyhedron which is contractible and contains all the vertices ai of K}.
* Applications: Can be used to calculate the fundamental group of a topological space whose triangulation is K, but it is not theoretically satisfying (> see Presentations); Notice that it involves 1 and 2-simplices only.

Properties and Results
* For a product: π1(X × Y, (x0, y0)) ≅ π1(X, x0) ⊕ π1(Y, y0).
* For a topological group: It is always Abelian (but π1(X) is not always Abelian, for example it is not for the genus-2 2D compact manifold).
* For a compact manifold: It is finite if the manifold has constant positive R.
* Seifert-Van Kampen theorem: If X = UV, where U and V are open and pathwise connected, and U ∩ V ≠ Ø, then π1(X) is the "amalgamated sum" of π1(U) and π1(V), i.e., the free product π1(U) * π1(V) with the extra relations that, for all z ∈ π1(UV), i*(z) = j*(z), where i and j are the inclusion maps of UV in U and V; Special cases:
- If UV is simply connected, then π1(X) = π1(U) * π1(V).
- If U is simply connected, then π1(X) = π1(V) / {smallest normal subgroup containing j*1(UV)]}.
- If U and V are simply connected, then X is simply connected.
@ Seifert-Van Kampen: Crowell & Fox 63; Massey 77; in Armstrong 83, p138.

Examples
* π1(S1) = π1(U(1)) = \(\mathbb Z\); π1(Sn) = {0}, for n > 1.
* π1(Pn) = \(\mathbb Z\)2, for n > 1.
* π1(SU(n)) = {0}.
* π1(O(n)) = \(\mathbb Z\)2, for n > 2.
* π1(SO(n)) = \(\mathbb Z\)2.
* π1(SO(3,1)) = \(\mathbb Z\)2.
* π1(Sp(n)) = {0}.

References
@ General: in Chevalley 46; in Massey 77; in Choquet-Bruhat et al 89; in Nash & Sen 83.
@ Special spaces: Cannon & Conner T&A(06) [1D]; Fabel T&A(07) [locally path-connected metric spaces]; Yeganefar DG&A(07) [open Riemannian manifolds].
@ Representations: in Nelson & Picken a0903-conf [deformed, and 2+1 quantum geometry].
@ Related topics: Conner & Eda T&A(05) [recovering spaces].
> Generalizations: see topology in physics [fundamental grupoid].


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