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, 
)
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 
= f,
where (X*, f) is the universal covering space of X.
$ Def 3: 
1(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, 1(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: 1(X × Y,
(x0, y0)) 1(X, x0)
1(Y, y0).
* For a topological group: It is always Abelian.
* 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 1(X)
is the "amalgamated sum" of 1(U)
and 1(V),
i.e., the free product 1(U)
* 1(V)
with the extra relations that, for all z 1(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 1(X)
= 1(U)
* 1(V).
- If U is simply connected, then 1(X)
= 1(V)
/ {smallest normal subgroup containing j*[1(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, 138.
Examples
* 1(S1) = 1(U(1)) = Z; 1(Sn) = {0}, for n > 1.
* 1(Pn) = Z2,
for n > 1.
* 1(SU(n)) = {0}.
* 1(O(n)) = Z2,
for n > 2.
* 1(SO(n))
= Z2.
* 1(SO(3,1))
= Z2.
* 1(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-in
[deformed, and 2+1 quantum geometry].
@ Related topics: Conner & Eda T&A(05) [recovering spaces].
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may
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