Fundamental group

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups.

The abelianization of the fundamental group 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 generators and relations.

Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs".[1] The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.

IntuitionEdit

Start with a space (for example, 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. 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.

DefinitionEdit

Throughout this article, X is a topological space. A typical example is a surface such as the one depicted at the right. Moreover,   is a point in X called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on X can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

Homotopy of loopsEdit

Given a topological space X, a loop based at   is defined to be a continuous function (also known as a continuous map)

 

such that both the starting point   and the end point   are both equal to  

 
Homotopy of loops

A homotopy is a continuous interpolation between two loops. More precisely, a homotopy between two loops   (based at the same point  ) is a continuous map

 

such that

  for all   that is, the starting point of the homotopy is   for all t (which is often thought of as a time parameter).
  for all   that is, similarly the end point stays at   for all t.
  for all  

If such a homotopy h exists,   and   are said to be homotopic. The relation "  is homotopic to  " is an equivalence relation so that the set of equivalence classes can be considered:

 

It is called the fundamental group of the topological space X and the base point   The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops (the so-called loop space of X) is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above quotient has, in many cases, a manageable and computable size.

Group structureEdit

 
Addition of loops

By the above definition,   is just a set. It becomes a group (and therefore deserves the name fundamental group) using the concatenation of loops. More precisely, given two loops   their product is defined as the loop

 

Thus the loop   first follows the loop   with "twice the speed" and then follows   with "twice the speed".

The product of two homotopy classes of loops   and   is then defined as   It can be shown that this product does not depend on the choice of representatives and therefore gives a well-defined operation on the set   This operation turns   into a group. Its neutral element is the constant loop, which stays at   for all times t. The inverse of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction. More formally,

 

Given three based loops   the product

 

is the concatenation of these loops, traversing   and then   with quadruple speed, and then   with double speed. By comparison,

 

traverses the same paths (in the same order), but   with double speed, and   with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The associativity axiom

 

therefore crucially depends on the fact that paths are considered up to homotopy. Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops   with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn   into a group.

Dependence of the base pointEdit

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. For path-connected spaces, therefore, many authors therefore write :  instead of  

Concrete examplesEdit

 
x0.

This section lists some basic examples of fundamental groups. To begin with, in Euclidean space ( ) or any convex subset of   there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. More generally, any star domain and, yet more generally any contractible space has a trivial fundamental group. Thus, the fundamental group does not distinguish between such spaces.

 
The 2-sphere (that is, the surface of a ball) is simply connected: any loop on the surface can be contracted to a point.

A path-connected space whose fundamental group is trivial is called simply connected. For example, the 2-sphere   depicted at the left, and also all the higher-dimensional spheres are simply-connected. The figure illustrates a homotopy contracting one particular loop to the constant loop. This idea can be adapted to all loops   such that there is a point   that is not in the image of   Since there are, however, loops such that   constructed from the Peano curve, a rigorous complete proof requires some methods of algebraic topology, in this case the Seifert–van Kampen theorem. A similar phenomenon occurs with many of the following examples: it is often straightforward to exhibit certain elements in   and show they satisfy certain relations, but proving that they yield a complete description of the fundamental group often requires some more technique. Such techniques are explained further below in the article.

The circleEdit

 
Elements of the homotopy group of the circle

The circle (also known as the 1-sphere)

 

is not simply connected. Instead, each homotopy class consists of all loops that 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 that winds around m times and another that winds around n times is a loop that winds around   times. Therefore, the fundamental group of the circle is isomorphic to   the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem[2] and the Borsuk–Ulam theorem in dimension 2.[3]

The figure eightEdit

 
The fundamental group of the figure eight is the free group on two generators a and b.

The fundamental group of the figure eight is the free group on two letters. The idea to prove this is as follows: choosing the base point to be the point where the two circles meet (dotted in black in the picture at the right), any loop   can be decomposed as

 

where a and b are the two loops winding around each half of the figrue as depicted, and the exponents   are integers. Unlike   the fundamental group of the figure eight is not abelian: the two ways of composing a and b are not homotopic to each other:

 

More generally, the fundamental group of a bouquet of r circles is the free group on r letters.

The fundamental group of a wedge sum of two path connected spaces X and Y can be computed as the free product of the individual fundamental groups:

 

This generalizes the above observations since the figure eight is the wedge sum of two circles.

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.

GraphsEdit

The fundamental group of any 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 spanning tree.

Knot groupsEdit

Knot groups are, by definition the fundamental group of the complement of a knot K embedded in   For example, the knot group of the trefoil knot is known to be the braid group   which gives another example of a non-abelian fundamental group. The Wirtinger presentation explicitly describes knot groups in terms of generators and relations based on a diagram of the knot. Therefore knot groups have some usage in knot theory to distinguish between knots: if   is not isomorphic to some other knot group   of another knot K', then K can not be transformed into   Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group Z. There are, however, knots that can not be deformed into each other, but have isomorphic knot groups.

Oriented surfacesEdit

The fundamental group of an genus n orientable surface can be computed in terms of generators and relations as

 

This includes the torus, being the case of genus 1, whose fundamental group is

 

Topological groupsEdit

The fundamental group of a topological group X (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a Lie group is commutative. In fact, the group structure on X endows   with another group structure: given two loops   and   in X, another loop   can defined by using the group multiplication in X:

 

This binary operation   on the set of all loops is a priori independent from the one described above. However, the Eckmann–Hilton argument shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.[4][5]

An inspection of the proof shows that, more generally,   is abelian for any H-space X, i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a loop space of another topological space Y,   is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

FunctorialityEdit

If   is a continuous map,   and   with   then every loop in X with base point   can be composed with f to yield a loop in Y with base point   This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as   or, more commonly,

 

This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In the parlance of category theory, the formation of associating to a topological space its fundamental group is therefore a functor

 

from the category of topological spaces together with a base point to the category of groups. It turns out that this functor does not distinguish maps that are homotopic relative to the base point: if f, g : XY are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f = g. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:

 

For example, the inclusion of the circle in the punctured plane

 

is a homotopy equivalence and therefore yields an isomorphism of their fundamental groups.

The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then

 

Abstract resultsEdit

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of algebraic topology.

Relationship to first homology groupEdit

A special case of the Hurewicz theorem asserts that the first singular homology group   is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a group homomorphism

 

from the fundamental group of a topological space X to its first singular homology group   This homomorphism is not in general an isomorphism since the fundamental group may be non-abelian, but the homology group is, by definition, always abelian. This difference is, however, the only one: if X is path-connected, this homomorphism is surjective and its kernel is the commutator subgroup of the fundamental group, so that   is isomorphic to the abelianization of the fundamental group.[6]

Glueing topological spacesEdit

Generalizing the statement above, for a family of path connected spaces   the fundamental group   is the free product of the fundamental groups of the  [7] This fact is a special case of the Seifert–van Kampen theorem, which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces. For example, the 2-sphere   can be obtained by glueing two copies of slightly overlapping half-spheres along a neighborhood of the equator. In this case the theorem yields   is trivial, since the two half-spheres are contractible and therefore have trivial fundamental group. The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem.

In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes pushouts (in the category of topological spaces) along inclusions to pushouts (in the category of groups).[8]

CoveringsEdit

 
The map   is a covering: the preimage of U (highlighted in gray) is a disjoint union of copies of U. Moreover, it is a universal covering since   is contractible and therefore simply connected.

Given a topological space B, a continuous map

 

is called a covering or E is called a covering space of B if every point b in B admits an open neighborhood U such that there is a homeomorphism between the preimage of U and a disjoint union of copies of U (indexed by some set I),

 

in such a way that   is the standard projection map  [9]

Universal coveringEdit

A covering is called a universal covering of E is, in addition to the preceding condition, simply connected.[10] It is universal in the sense that all other coverings can be constructed by suitably identifying points in E. Knowing a universal covering

 

of a topological space X is helpful in understanding its fundamental group in several ways: first,   identifies with the group of deck transformations, i.e., the group of homeomorphisms   that commute with the map to X, i.e.,   Another relation to the fundamental group is that   can be identified with the fiber   For example, the map

 

(or, equivalently,  ) is a universal covering. The deck transformations are the maps   for   This is in line with the identification   in particular this proves the above claim  

Any path connected, locally path connected and locally simply connected topological space X admits a universal covering.[11] An abstract construction proceeds 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 x0 to x. The passage from a topological space to its universal covering can be used in understanding the geometry of X. For example, the uniformization theorem shows that any simply connected Riemann surface is (isomorphic to) either     or the upper half plane.[12] General Riemann surfaces then arise as quotients of group actions on these three surfaces.

The quotient of an action of a (discrete) group G on a simply connected space Y has fundamental group

 

As an example, the real n-dimensional real projective space   is obtained as the quotient of the n-dimensional sphere   by the antipodal action of the group   sending   to   As   is simply connected for n ≥ 2, it is a universal cover of   in these cases, which implies   for n ≥ 2.

Lie groupsEdit

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 spaces X = Γ\G/K, where

  • G is a non-compact simply connected, connected Lie group (often semisimple),
  • K is a maximal compact subgroup of G
  • Γ is a discrete countable torsion-free subgroup of G.

In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups).

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).

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 Γ, a closed discrete normal subgroup of G:

 

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 of G. In particular π1(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.

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 Lattice of covering groups.

FibrationsEdit

Fibrations provide a very powerful means to compute homotopy groups. A fibration f the so-called total space, and the base space B has, in particular, the property that all its fibers   are homotopy equivalent and therefore can not be distinguished using fundamental groups (and higher homotopy groups), provided that B is path-connected.[13] Therefore, the space E can be regarded as a "twisted product" of the base space B and the fiber   The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence

 

provided that B is path-connected.[14] The term   is the second homotopy group of B, which is defined to be the set of homotopy classes of maps from   to B, in direct analogy with the definition of  

If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism

 

which generalizes the above fact about the universal covering (which amounts to the case where the fiber F is also discrete). If instead F happens to be connected and simply connected, it reduces to an isomorphism

 

What is more, the sequence can be continued at the left with the higher homotopy groups   of the three spaces, which gives some access to computing such groups in the same vein.

Classical Lie groupsEdit

Such fiber sequences can be used to inductively compute fundamental groups of compact classical Lie groups such as the special unitary group   with   This group acts transitively on the unit sphere   inside   The stabilizer of a point in the sphere is isomorphic to   It then can be shown[15] that this yields a fiber sequence

 

Since   the sphere   has dimension at least 3, which implies

 

The long exact sequence then shows an isomorphism

 

Since   is a single point, so that   is trivial, this shows that   is simply connected for all  

The fundamental group of noncompact Lie groups can be reduced to the compact case, since such a group is homotopic to its maximal compact subgroup.[16] These methods give the following results:[17]

compact classical Lie group G non-compact Lie group  
special unitary group     1
unitary group      
special orthogonal group       for   and   for  
compact symplectic group     1

A second method of computing fundamental groups applies to all connected compact Lie groups and uses the machinery of the maximal torus and the associated root system. Specifically, let   be a maximal torus in a connected compact Lie group   and let   be the Lie algebra of   The exponential map

 

is a fibration and therefore its kernel   identifies with   The map

 

can be shown to be surjective[18] with kernel given by the set I of integer linear combination of coroots. This leads to the computation

 [19]

This method shows, for example, that any connected compact Lie group for which the associated root system is of type   is simply connected.[20] Thus, there is (up to isomorphism) only one connected compact Lie group having Lie algebra of type  ; this group is simply connected and has trivial center.

Edge-path group of a simplicial complexEdit

If X is a 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(Xv) 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.

The edge-path group is naturally isomorphic to π1(|X|, v), the fundamental group of the geometric realisation |X| of X.[21] Since it depends only on the 2-skeleton X2 of X (that is, the vertices, edges, and triangles of X), the groups π1(|X|,v) and π1(|X2|, v) are isomorphic.

The edge-path group can be described explicitly in terms of generators and relations. If T is a maximal spanning tree in the 1-skeleton of X, then E(Xv) 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 surfaces, which are classified by their fundamental groups.

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 γu from v to u. The points (w,γ) and (u, γu) 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.

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 and Jean Leray and explicitly appeared as a remark in a paper by André Weil;[22] various other authors such as Lorenzo Calabi, Wu Wen-tsün, and Nodar 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 the covering.

RealizabilityEdit

  • Every group can be realized as the fundamental group of a 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).
  • Every finitely presented group can be realized as the fundamental group of a 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. It can be proved that every group can be realized as the fundamental group of a compact Hausdorff space if and only if there is no measurable cardinal.[23]

Related conceptsEdit

Higher homotopy groupsEdit

Roughly speaking, the fundamental group detects the 1-dimensional hole structure of a space, but not holes in higher dimensions such as for the 2-sphere. Such "higher-dimensional holes" can be detected using the higher homotopy groups  , which are defined to consist of homotopy classes of (basepoint-preserving) maps from   to X. For example, the Hurewicz theorem implies that the n-th homotopy group of the n-sphere is (for all  ) are

 [24]

As was mentioned in the above computation of   of classical Lie groups, higher homotopy groups can be relevant even for computing fundamental groups.

Loop spaceEdit

The set of based loops (as is, i.e., not taken up to homotopy) in a pointed space X, endowed with the compact open topology, is known as the loop space, denoted   The fundamental group of X is in bijection with the set of path components of its loop space:[25]

 

Fundamental groupoidEdit

The fundamental groupoid is a variant of the fundamental group that is useful in situations where the choice of a base point   is undesirable. It is defined by first considering the category of paths in   i.e., continuous functions

 

where r is an arbitrary non-negative real number. Since the length r is variable in this approach, such paths can be concatenated as is (i.e., not up to homotopy) and therefore yield a category.[26] Two such paths   with the same endpoints and length r, resp. r' are considered equivalent if there exist real numbers   such that   and   are homotopic relative to their end points, where  [27][28] The category of paths up to this equivalence relation is denoted   Each morphism in   is an isomorphism, with inverse given by the same path traversed in the opposite direction. Such a category is called a groupoid. It reproduces the fundamental group since

 

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 van Kampen theorem admits a version for fundamental groupoids which gives, for example, another way to compute the fundamental group(oid) of  [29]

Local systemsEdit

Generally speaking, representations may serve to exhibit features of a group by its actions on other mathematical objects, often vector spaces. Representations of the fundamental group have a very geometric significance: any local system (i.e., a sheaf   on X with the property that locally in a sufficiently small neighborhood U of any point on X, the restriction of F is a constant sheaf of the form  ) gives rise to the so-called monodromy representation, a representation of the fundamental group on an n-dimensional  -vector space. Conversely, any such representation on a path-connected space X arises in this manner.[30] This equivalence of categories between representations of   and local sytems is used, for example, in the study of differential equations.

Étale fundamental groupEdit

In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group.[31] Since the Zariski topology on an algebraic variety or scheme X is much coarser than, say, the topology of open subsets in   it is no longer meaningful to consider continuous maps from an interval to X. Instead, the approach developed by Grothendieck consists in constructing   by considering all finite étale covers of X. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situation where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a finite field. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties X over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.[32]

Fundamental group of algebraic groupsEdit

The fundamental group of a root system is defined, in analogy to the computation for Lie groups.[33] This allows to define and use the fundamental group of a semisimple linear algebraic group G, which is a useful basic tool in the classification of linear algebraic groups.[34]

Fundamental group of simplicial setsEdit

The homotopy relation between 1-simplices of a simplicial set X is an equivalence relation if X is a Kan complex but not necessarily so in general.[35] Thus,   of a Kan complex can be defined as the set of homotopy classes of 1-simplices. The fundamental group of an arbitrary simplicial set X are defined to be the homotopy group of its topological realization,   i.e., the topological space obtained by glueing topological simplices as prescribed by the simplicial set structure of X.[36]

See alsoEdit

NotesEdit

  1. ^ Poincaré, Henri (1895). "Analysis situs". Journal de l'École Polytechnique. (2) (in French). 1: 1–123. Translated in Poincaré, Henri (2009). "Analysis situs" (PDF). Papers on Topology: Analysis Situs and Its Five Supplements. Translated by John Stillwell. pp. 18–99.
  2. ^ May (1999, Ch. 1, §6)
  3. ^ Massey (1991, Ch. V, §9)
  4. ^ Strom (2011, Problem 9.30, 9.31), Hall (2015, Exercise 13.7)
  5. ^ Proof: Given two loops   in   define the mapping   by   multiplied pointwise in   Consider the homotopy family of paths in the rectangle from   to   that starts with the horizontal-then-vertical path, moves through various diagonal paths, and ends with the vertical-then-horizontal path. Composing this family with   gives a homotopy   which shows the fundamental group is abelian.
  6. ^ Fulton (1995, Prop. 12.22)
  7. ^ May (1999, Ch. 2, §8, Proposition)
  8. ^ May (1999, Ch. 2, §7)
  9. ^ Hatcher (2002, §1.3)
  10. ^ Hatcher (2002, p. 65)
  11. ^ Hatcher (2002, Proposition 1.36)
  12. ^ Forster (1981, Theorem 27.9)
  13. ^ Hatcher (2002, Prop. 4.61)
  14. ^ Hatcher (2002, Theorem 4.41)
  15. ^ Hall (2015, Proposition 13.8)
  16. ^ Hall (2015, Section 13.3)
  17. ^ Hall (2015, Proposition 13.10)
  18. ^ Bump (2013, Prop. 23.7)
  19. ^ Hall (2015, Corollary 13.18)
  20. ^ Hall (2015, Example 13.45)
  21. ^ Singer, Isadore; Thorpe, John A. (1967). Lecture notes on elementary topology and geometry. Springer-Verlag. p. 98. ISBN 0-387-90202-3.
  22. ^ André Weil, On discrete subgroups of Lie groups, Annals of Mathematics 72 (1960), 369-384.
  23. ^ Adam Przezdziecki, Measurable cardinals and fundamental groups of compact spaces, Fundamenta Mathematicae 192 (2006), 87-92 [1]
  24. ^ Hatcher (2002, §4.1)
  25. ^ Adams (1978, p. 5)
  26. ^ Brown (2006, §6.1)
  27. ^ Brown (2006, §6.2)
  28. ^ Crowell & Fox (1963) use a different definition by reparametrizing the paths to length 1.
  29. ^ Brown (2006, §6.7)
  30. ^ El Zein et al. (2010, p. 117, Prop. 1.7)
  31. ^ Grothendieck & Raynaud (2003).
  32. ^ Grothendieck & Raynaud (2003, Exposé XII, Cor. 5.2).
  33. ^ Humphreys (1972, §13.1)
  34. ^ Humphreys (2004, §31.1)
  35. ^ Goerss & Jardine (1999, §I.7)
  36. ^ Goerss & Jardine (1999, §I.11)

ReferencesEdit

External linksEdit