Fundamental Group of a Topological Space| 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 its 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 \(f : X^* \to X^*\) such that \(f \circ \phi
= f\), 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: It 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 = U ∪ V, 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(U
∩ V), i*(z)
= j*(z), where i
and j are the inclusion maps of U ∩ V in
U and V; Special cases:
- If U ∩ V 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*[π1(U ∩ V)]}.
- 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 IJMPA(09)-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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