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__:
π_{1}(*X*, *x*_{0}) is
the set of path-connected components of the space of loops at *x*_{0},
with the compact-open topology.

**Calculating Theorem**

$ __Def__: For a polyhedron *K*,
π_{1}(*K*, *a*_{0}) ≅
*G*, presented by the generator set *H* = {*g*_{ij} |
[*a*_{i}, *a*_{j}]
is an ordered 1-simplex of *K*}, and relations *D*
= {*g*_{ij} *g*_{jk}
*g*_{ik}^{−1}
for all ordered 2-simplices [*a*_{i},
*a*_{j}, *a*_{k}]
of *K* \ *L*, and *g*_{ij} = 1 if
[*a*_{i}, *a*_{j}]
∈ *L*, where *L* is a 1D subpolyhedron which is contractible and contains all the vertices
*a*_{i} 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__:
π_{1}(*X* × *Y*,
(*x*_{0}, *y*_{0}))
≅ π_{1}(*X*, *x*_{0})
⊕ π_{1}(*Y*, *y*_{0}).

* __For a topological group__:
It is always Abelian (but π_{1}(*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 π_{1}(*X*) is the
"amalgamated sum" of π_{1}(*U*)
and π_{1}(*V*), i.e.,
the free product π_{1}(*U*)
* π_{1}(*V*)
with the extra relations that, for all *z* ∈ π_{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 π_{1}(*X*)
= π_{1}(*U*)
* π_{1}(*V*).

- If *U* is simply connected, then
π_{1}(*X*)
= π_{1}(*V*) / {smallest normal subgroup containing
*j*_{*}[π_{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, p138.

**Examples**

* π_{1}(S^{1})
= π_{1}(U(1)) = \(\mathbb Z\);
π_{1}(S^{n}) = {0}, for *n* > 1.

* π_{1}(P^{n})
= \(\mathbb Z\)_{2}, for *n* > 1.

* π_{1}(SU(*n*)) = {0}.

* π_{1}(O(*n*))
= \(\mathbb Z\)_{2}, for *n* > 2.

* π_{1}(SO(*n*))
= \(\mathbb Z\)_{2}.

* π_{1}(SO(3,1))
= \(\mathbb Z\)_{2}.

* π_{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 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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