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| {{For|the fundamental group of a factor|von Neumann algebra}}
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| In [[mathematics]], more specifically [[algebraic topology]], the '''fundamental group''' (defined by [[Henri Poincaré]] in his article ''Analysis Situs'', published in 1895) is a [[Group (mathematics)|group]] associated to any given [[pointed space|pointed topological space]] that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. Intuitively, it records information about the basic shape, or ''holes'', of the topological space. The fundamental group is the first and simplest of the [[homotopy group]]s. It is a [[topological invariant]]: homeomorphic topological spaces have the same fundamental group.
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| Fundamental groups can be studied using the theory of [[covering space]]s, since a fundamental group coincides with the group of [[deck transformation]]s of the associated [[covering space#Universal covers|universal covering space]]. Its [[abelianisation]] can be identified with the first [[homology group]] of the space. When the topological space is homeomorphic to a [[simplicial complex]], its fundamental group can be described explicitly in terms of [[Presentation of a group|generators and relations]].
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| Historically, the concept of fundamental group first emerged in the theory of [[Riemann surface]]s, in the work of [[Bernhard Riemann]], [[Henri Poincaré]] and [[Felix Klein]], where it describes the [[monodromy]] properties of [[complex function]]s, as well as providing a complete topological [[classification of surfaces|classification of closed surfaces]].
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| == Intuition ==
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| Start with a space (e.g. a surface), and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second.
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| Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.
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| ==Definition==
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| Let ''X'' be a topological space, and let ''x''<sub>0</sub> be a point of ''X''. We are interested in the following set of [[continuous function (topology)|continuous]] functions called '''[[Path (topology)|loops]]''' with '''base point''' ''x''<sub>0</sub>.
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| :<math>\{f:[0,1]\to X : \ f(0)=x_0=f(1)\}</math>
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| Now the '''fundamental group of X with base point x<sub>0</sub>''' is this set modulo [[homotopy]] ''h''
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| :<math>\{f:[0,1]\to X : \ f(0)=x_0=f(1)\} / h</math>
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| equipped with the group multiplication defined by
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| :<math> (f * g) (t) = \begin{cases} f(2t) & 0 \leq t \leq \tfrac{1}{2} \\ g(2t-1) & \tfrac{1}{2} \leq t \leq 1 \end{cases}</math>
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| Thus the loop ''f'' ∗ ''g'' first follows the loop ''f'' with "twice the speed" and then follows ''g'' with twice the speed. The product of two homotopy classes of loops [''f''] and [''g''] is then defined as [''f'' ∗ ''g''], and it can be shown that this product does not depend on the choice of representatives.
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| With the above product, the set of all homotopy classes of loops with base point ''x''<sub>0</sub> forms the '''fundamental group''' of ''X'' at the point ''x''<sub>0</sub> and is denoted
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| :<math>\pi_1(X,x_0),</math>
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| or simply π(''X'', ''x''<sub>0</sub>). The identity element is the constant map at the basepoint, and the inverse of a loop ''f'' is the loop ''g'' defined by ''g''(t) = ''f''(1 − ''t''). That is, ''g'' follows ''f'' backwards.
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| Although the fundamental group in general depends on the choice of base point, it turns out that, [[up to]] [[isomorphism]] (actually, even up to ''inner'' isomorphism), this choice makes no difference as long as the space ''X'' is [[Path connected|path-connected]]. For path-connected spaces, therefore, we can write π<sub>1</sub>(''X'') instead of π<sub>1</sub>(''X'', ''x''<sub>0</sub>) without ambiguity whenever we care about the [[isomorphism class]] only.
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| == Examples ==
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| ===Trivial Fundamental Group===
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| In [[Euclidean space]] '''R'''<sup>''n''</sup>, or any [[convex set|convex subset]] of '''R'''<sup>''n''</sup>, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. A path-connected space with a trivial fundamental group is said to be [[Simply connected space|simply connected]].
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| ===Infinite Cyclic Fundamental Group===
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| The [[circle]]. Each homotopy class consists of all loops which wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop which winds around ''m'' times and another that winds around ''n'' times is a loop which winds around ''m'' + ''n'' times. So the fundamental group of the circle is [[isomorphic]] to ('''Z''', +), the additive group of [[integer]]s. This fact can be used to give proofs of the [[Brouwer fixed point theorem]] and the [[Borsuk–Ulam theorem]] in dimension 2.
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| Since the fundamental group is a homotopy invariant, the theory of the [[winding number]] for the complex plane minus one point is the same as for the circle.
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| ===Free Groups of Higher Rank===
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| Unlike the [[homology group]]s and higher homotopy groups associated to a topological space, the fundamental group need not be [[abelian group|abelian]]. For example, the fundamental group of the [[Rose (topology)|figure eight]] is the [[free group]] on two letters. More generally, the fundamental group of any [[graph (mathematics)|graph]] is a [[free group]]. If the graph ''G'' is connected, then the rank of the free group is equal to the number of edges not in a [[minimum spanning tree]].
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| The fundamental group of the plane punctured at ''n'' points is also the free group with ''n'' generators. The ''i''-th generator is the class of the loop that goes around the ''i''-th puncture without going around any other punctures.
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| ===Knot Theory===
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| {{main|knot group}}
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| A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a [[trefoil knot]] in '''R'''<sup>3</sup>, as known, whose fundamental group is the braid group <math>B_3</math>.
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| == Functoriality ==
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| If ''f'' : ''X'' → ''Y'' is a continuous map, ''x''<sub>0</sub> ∈ ''X'' and ''y''<sub>0</sub> ∈ ''Y'' with ''f''(''x''<sub>0</sub>) = ''y''<sub>0</sub>, then every loop in ''X'' with base point ''x''<sub>0</sub> can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''<sub>0</sub>. This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting [[group homomorphism]], called the [[Induced homomorphism (fundamental group)|induced homomorphism]], is written as π(''f'') or, more commonly,
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| :<math>f_* : \pi_1(X, x_0) \to \pi_1(Y,y_0).</math>
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| This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In other words, we have a [[functor]] from the [[category of pointed spaces|category of topological spaces with base point]] to the [[category of groups]].
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| It turns out that this functor cannot distinguish maps which are [[homotopic]] relative to the base point: if ''f'', ''g'' : ''X'' → ''Y'' are continuous maps with ''f''(''x''<sub>0</sub>) = ''g''(''x''<sub>0</sub>) = ''y''<sub>0</sub>, and ''f'' and ''g'' are homotopic relative to {''x''<sub>0</sub>}, then ''f''<sub>∗</sub> = ''g''<sub>∗</sub>. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:
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| :<math>X \simeq Y \Rightarrow \pi_1(X,x_0) \cong \pi_1(Y,y_0).</math>
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| As an important special case, if ''X'' is [[path-connected]] then any two basepoints give isomorphic fundamental groups, with isomorphism given by a choice of path between the given basepoints.
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| The fundamental group functor takes [[product (category theory)|products]] to [[direct product of groups|products]] and [[coproduct]]s to coproducts. That is, if ''X'' and ''Y'' are path connected, then
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| :<math>\pi_1 (X\times Y) \cong \pi_1(X) \times \pi_1(Y)</math>
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| and
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| :<math>\pi_1 (X\vee Y) \cong \pi_1(X) * \pi_1(Y).</math>
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| (In the latter formula, <math>\vee</math> denotes the [[wedge sum]] of topological spaces, and * the [[free product]] of groups.) Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the [[Seifert–van Kampen theorem]] which states that the fundamental group functor takes [[Pushout (category theory)|pushout]]s along inclusions to pushouts.
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| ==Fibrations==
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| {{Main|Fibration}}
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| A generalization of a product of spaces is given by a [[fibration]],
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| :<math>F \to E \to B. </math>
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| Here the [[fibration|total space]] ''E'' is a sort of "[[Twist (mathematics)|twisted]] product" of the [[fibration|base space]] ''B'' and the [[fibration|fiber]] ''F''. In general the fundamental groups of ''B'', ''E'' and ''F'' are terms in a [[Homotopy group#Long exact sequence of a fibration|long exact sequence]] involving [[homotopy group|higher homotopy groups]]. When all the spaces are connected, this has the following consequences for the fundamental groups:
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| * π<sub>1</sub>(''B'') and π<sub>1</sub>(''E'') are isomorphic if ''F'' is simply connected
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| * π<sub>''n''+1</sub>(''B'') and π<sub>''n''</sub>(''F'') are isomorphic if ''E'' is contractible
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| The latter is often applied to the situation ''E'' = [[path space]] of ''B'', ''F'' = [[loop space]] of ''B'' or ''B'' = [[classifying space]] ''BG'' of a [[topological group]] ''G'', ''E'' = universal ''G''-bundle ''EG''.
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| == Relationship to first homology group ==
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| The fundamental groups of a topological space ''X'' are related to its first singular [[homology group]], because a loop is also a singular 1-cycle. Mapping the homotopy class of each loop at a base point ''x''<sub>0</sub> to the homology class of the loop gives a homomorphism from the fundamental group π<sub>1</sub>(''X'', ''x''<sub>0</sub>) to the homology group ''H''<sub>1</sub>(''X''). If ''X'' is path-connected, then this homomorphism is [[surjective]] and its [[Kernel (algebra)|kernel]] is the [[commutator subgroup]] of π<sub>1</sub>(''X'', ''x''<sub>0</sub>), and ''H''<sub>1</sub>(''X'') is therefore isomorphic to the abelianization of π<sub>1</sub>(''X'', ''x''<sub>0</sub>). This is a special case of the [[Hurewicz theorem]] of algebraic topology.
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| ==Universal covering space==
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| {{Main|Covering space}}
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| If ''X'' is a topological space that is path connected, locally [[Connected_space#Path_connectedness|path connected]] and locally simply connected, then it has a simply connected [[covering space|universal covering space]] on which the fundamental group π(''X'',''x''<sub>0</sub>) [[group action|acts]] freely by [[deck transformations]] with [[quotient space]] ''X''. This space can be constructed analogously to the fundamental group by taking pairs (''x'', γ), where ''x'' is a point in ''X'' and γ is a homotopy class of paths from ''x''<sub>0</sub> to ''x'' and the action of π(''X'', ''x''<sub>0</sub>) is by concatenation of paths. It is uniquely determined as a covering space.
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| ===Examples===
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| ====The Circle====
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| The universal cover of a circle '''S'''<sup>1</sup> is the line '''R''', we have '''S'''<sup>1</sup> = '''R'''/'''Z'''. Thus π<sub>1</sub>('''S'''<sup>1</sup>,''x'') = '''Z''' for any base point ''x''.
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| ====The Torus====
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| By taking the Cartesian product of two instances of the previous example we see that the universal cover of a torus ''T'' = '''S'''<sup>1</sup> × '''S'''<sup>1</sup> is the plane '''R'''<sup>2</sup>: we have ''T'' = '''R'''<sup>2</sup>/'''Z'''<sup>2</sup>. Thus π<sub>1</sub>(''T'',''x'') = '''Z'''<sup>2</sup> for any base point ''x''.
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| Similarly, the fundamental group of the ''n''-dimensional torus equals '''Z'''<sup>''n''</sup>.
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| ====Real Projective Spaces====
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| For ''n'' ≥ 1 the real ''n''-dimensional real projective space '''P'''<sup>''n''</sup>('''R''') is obtained by factorizing the ''n''-dimensional sphere '''S'''<sup>''n''</sup> by the central symmetry: '''P'''<sup>''n''</sup>('''R''') = '''S'''<sup>''n''</sup>/'''Z'''<sub>2</sub>. Since the ''n''-sphere '''S'''<sup>''n''</sup> is simply connected for ''n'' ≥ 2, we conclude that it is the universal cover of the real projective space. Thus the fundamental group of '''P'''<sup>''n''</sup>('''R''') is equal to '''Z'''<sub>2</sub> for any ''n'' ≥ 2.
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| ====Lie Groups====
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| Let ''G'' be a connected, simply connected [[compact Lie group]], for example the [[special unitary group]] SU(''n''), and let Γ be a finite subgroup of ''G''. Then the [[homogeneous space]] ''X'' = ''G''/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space ''G''. Among the many variants of this construction, one of the most important is given by [[locally symmetric space]]s ''X'' = Γ\''G''/''K'', where
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| *''G'' is a non-compact simply connected, connected [[Lie group]] (often [[semisimple Lie group|semisimple]]),
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| *''K'' is a maximal compact subgroup of ''G''
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| * Γ is a discrete countable [[torsion-free group|torsion-free]] subgroup of ''G''.
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| In this case the fundamental group is Γ and the universal covering space ''G''/''K'' is actually [[contractible]] (by the [[Cartan decomposition]] for [[Lie group]]s).
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| As an example take ''G'' = SL(2, '''R'''), ''K'' = SO(2) and Γ any torsion-free [[congruence subgroup]] of the [[modular group]] SL(2, '''Z''').
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| From the explicit realization, it also follows that the universal covering space of a path connected [[topological group]] ''H'' is again a path connected topological group ''G''. Moreover the covering map is a continuous open homomorphism of ''G'' onto ''H'' with [[kernel (algebra)|kernel]] Γ, a closed discrete normal subgroup of ''G'':
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| :<math> 1 \to \Gamma \to G \to H \to 1.</math>
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| Since ''G'' is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the [[center (group theory)|center]] of ''G''. In particular π<sub>1</sub>(''H'') = Γ is an [[Abelian group]]; this can also easily be seen directly without using covering spaces. The group ''G'' is called the ''[[universal covering group]]'' of ''H''.
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| As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at [[Covering_group#Lattice_of_covering_groups|Lattice of covering groups]].
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| ==Edge-path group of a simplicial complex==
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| If ''X'' is a [[connected space|connected]] [[simplicial complex]], an ''edge-path'' in ''X'' is defined to be a chain of vertices connected by edges in ''X''. Two edge-paths are said to be ''edge-equivalent'' if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in ''X''. If ''v'' is a fixed vertex in ''X'', an ''edge-loop'' at ''v'' is an edge-path starting and ending at ''v''. The '''edge-path group''' ''E''(''X'', ''v'') is defined to be the set of edge-equivalence classes of edge-loops at ''v'', with product and inverse defined by concatenation and reversal of edge-loops.
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| The edge-path group is naturally isomorphic to π<sub>1</sub>(|''X''|, ''v''), the fundamental group of the [[geometric realisation]] |''X''| of ''X''. Since it depends only on the [[n-skeleton|2-skeleton]] ''X''<sup>2</sup> of ''X'' (i.e. the vertices, edges and triangles of ''X''), the groups π<sub>1</sub>(|''X''|,''v'') and π<sub>1</sub>(|''X''<sup>2</sup>|, ''v'') are isomorphic.
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| The edge-path group can be described explicitly in terms of [[Presentation of a group|generators and relations]]. If ''T'' is a [[spanning tree|maximal spanning tree]] in the [[n-skeleton|1-skeleton]] of ''X'', then ''E''(''X'', ''v'') is canonically isomorphic to the group with generators (the oriented edge-paths of ''X'' not occurring in ''T'') and relations (the edge-equivalences corresponding to triangles in ''X''). A similar result holds if ''T'' is replaced by any [[simply connected]]—in particular [[contractible]]—subcomplex of ''X''. This often gives a practical way of computing fundamental groups and can be used to show that every [[finitely presented group]] arises as the fundamental group of a finite simplicial complex. It is also one of the classical methods used for [[topological space|topological]] [[surface]]s, which are classified by their fundamental groups.
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| The ''universal covering space'' of a finite connected simplicial complex ''X'' can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (''w'',γ) where ''w'' is a vertex of ''X'' and γ is an edge-equivalence class of paths from ''v'' to ''w''. The ''k''-simplices containing (''w'',γ) correspond naturally to the ''k''-simplices containing ''w''. Each new vertex ''u'' of the ''k''-simplex gives an edge ''wu'' and hence, by concatenation, a new path γ<sub>''u''</sub> from ''v'' to ''u''. The points (''w'',γ) and (''u'', γ<sub>''u''</sub>) are the vertices of the "transported" simplex in the universal covering space. The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just ''X''.
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| It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space. This was doubtless known to [[Eduard Čech|Čech]] and [[Jean Leray|Leray]] and explicitly appeared as a remark in a paper by {{harvtxt|Weil|1960}}; various other authors such as L. Calabi, W-T. Wu and N. Berikashvili have also published proofs. In the simplest case of a compact space ''X'' with a finite open covering in which all non-empty finite intersections of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the [[Nerve of an open covering|nerve of the covering]].
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| == Realizability ==
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| *Every group can be realized as the fundamental group of a [[connected space|connected]] [[CW-complex]] of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).
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| *Every [[finitely presented group]] can be realized as the fundamental group of a [[Compact space|compact]], connected, [[smooth manifold]] of dimension 4 (or higher). But there are severe restrictions on which groups occur as fundamental groups of low-dimensional manifolds. For example, no [[free abelian group]] of rank 4 or higher can be realized as the fundamental group of a manifold of dimension 3 or less.
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| == Related concepts ==
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| The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the [[homotopy group]]s are used. The elements of the ''n''-th homotopy group of ''X'' are homotopy classes of (basepoint-preserving) maps from '''S'''<sup>''n''</sup> to ''X''.
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| The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the [[loop space]].
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| For [[topological groups]], a different group multiplication may be assigned to the set of loops in the space, with pointwise multiplication rather than concatenation. The resulting group is the [[loop group]].
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| ===Fundamental groupoid===
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| Rather than singling out one point and considering the loops based at that point up to homotopy, one can also consider ''all'' paths in the space up to homotopy (fixing the initial and final point). This yields not a group but a [[groupoid]], the ''fundamental groupoid'' of the space.
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| More generally, one can consider the fundamental groupoid on a set ''A'' of base points, chosen according to the geometry of the situation; for example, in the case of the circle, which can be represented as the union of two connected open sets whose intersection has two components, one can choose one base point in each component. The exposition of this theory was given in the 1968, 1988 editions of the book now available as ''Topology and groupoids'', which also includes related accounts of [[covering space]]s and [[orbit space]]s.
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| ==See also==
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| * [[Homotopy group]], generalization of fundamental group
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| There are also similar notions of fundamental group for [[algebraic varieties]] (the [[étale fundamental group]]) and for [[orbifold]]s (the [[orbifold#Orbifold fundamental group|orbifold fundamental group]]).
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| ==References==
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| * [[Ronald Brown (mathematician)|Ronald Brown]], ''Topology and groupoids'', Booksurge (2006). ISBN 1-4196-2722-8
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| * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', [[Springer Science+Business Media|Springer-Verlag]], ISBN 0-387-96678-1
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| * [[Isadore Singer]] and John A. Thorpe, ''Lecture Notes on Elementary Geometry and Topology'', Springer-Verlag (1967) ISBN 0-387-90202-3
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| * [[Allen Hatcher]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic Topology''], [[Cambridge University Press]] (2002) ISBN 0-521-79540-0
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| * [[Peter Hilton]] and [[Shaun Wylie]], ''Homology Theory'', Cambridge University Press (1967) [warning: these authors use ''contrahomology'' for [[cohomology]]]
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| * Richard Maunder, ''Algebraic Topology'', [[Dover Publications|Dover]] (1996) ISBN 0-486-69131-4
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| * [[Deane Montgomery]] and Leo Zippin, ''Topological Transformation Groups'', Interscience Publishers (1955)
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| * [[James Munkres]], ''Topology'', [[Prentice Hall]] (2000) ISBN 0-13-181629-2
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| * [[Herbert Seifert]] and William Threlfall, ''A Textbook of Topology'' (translated from German by Wofgang Heil), [[Academic Press]] (1980), ISBN 0-12-634850-2
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| * [[Edwin Spanier]], ''Algebraic Topology'', Springer-Verlag (1966) ISBN 0-387-94426-5
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| * [[André Weil]], ''On discrete subgroups of Lie groups'', Ann. Math. '''72''' (1960), 369-384.
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| * {{planetmath reference|id=849|title=Fundamental group}}
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| * {{planetmath reference|id=3941|title=Fundamental groupoid}}
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| *{{Mathworld | urlname=FundamentalGroup | title=Fundamental group }}
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| ==Notes==
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| {{reflist}}
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| ==External links==
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| {{Commons category}}
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| * Dylan G.L. Allegretti, [http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2008.html ''Simplicial Sets and van Kampen's Theorem''] ''(An elementary discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set)''.
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| * [http://www.istia.univ-angers.fr/~delanoue/topo_alg/ Animations to introduce to the fundamental group by Nicolas Delanoue]
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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