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

$

$

**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__: 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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