Fundamental group
In the
Intuition
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 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.
History
Definition
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 loops
Given a topological space X, a loop based at is defined to be a
such that the starting point and the end point are both equal to .
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:
- .
This set (with the group structure described below) is called the fundamental group of the topological space X at the base point . The purpose of considering the equivalence classes of loops
Group structure
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
- .
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
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 point
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 write instead of
Concrete examples
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
A path-connected space whose fundamental group is trivial is called simply connected. For example, the
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 m + n times. Therefore, the fundamental group of the circle is isomorphic to the additive group of
The figure eight
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 figure 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
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.
Graphs
The fundamental group can be defined for discrete structures too. In particular, consider a connected graph G = (V, E), with a designated vertex v0 in V. The loops in G are the cycles that start and end at v0.[4] Let T be a spanning tree of G. Every simple loop in G contains exactly one edge in E \ T; every loop in G is a concatenation of such simple loops. Therefore, the fundamental group of a graph is a free group, in which the number of generators is exactly the number of edges in E \ T. This number equals |E| − |V| + 1.[5]
For example, suppose G has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then G has 24 edges overall, and the number of edges in each spanning tree is 16 − 1 = 15, so the fundamental group of G is the free group with 9 generators.[6] Note that G has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.
Knot groups
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
Oriented surfaces
The fundamental group of a
This includes the
Topological groups
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.[7][8]
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.
Functoriality
If is a
This mapping from continuous maps to group homomorphisms is compatible with composition of maps and
from the
For example, the inclusion of the circle in the punctured plane
is a
The fundamental group functor takes
and if they are also
(In the latter formula, denotes the wedge sum of pointed topological spaces, and the
Abstract results
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 group
The
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
Gluing topological spaces
Generalizing the statement above, for a family of path connected spaces the fundamental group is the free product of the fundamental groups of the
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).[11]
Coverings
Given a topological space B, a
is called a covering or E is called a
in such a way that is the standard projection map [12]
Universal covering
A covering is called a
of a topological space X is helpful in understanding its fundamental group in several ways: first, identifies with the group of
(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,
The
As an example, the real n-dimensional real projective space is obtained as the quotient of the n-dimensional unit 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 groups
Let G be a connected, simply connected
- G is a non-compact simply connected, connected semisimple),
- K is a maximal compact subgroup of G
- Γ is a discrete torsion-freesubgroup of G.
In this case the fundamental group is Γ and the universal covering space G/K is actually
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
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
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.
Fibrations
provided that B is path-connected.[17] 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 groups
Such fiber sequences can be used to inductively compute fundamental groups of compact
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.[19] These methods give the following results:[20]
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
This method shows, for example, that any connected compact Lie group for which the associated root system is of type is simply connected.[23] 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 complex
When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.
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(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.
The edge-path group is naturally isomorphic to π1(|X |, v), the fundamental group of the geometric realisation |X | of X.[24] Since it depends only on the 2-skeleton X 2 of X (that is, the vertices, edges, and triangles of X), the groups π1(|X |,v) and π1(|X 2|, v) are isomorphic.
The edge-path group can be described explicitly in terms of
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
Realizability
- Every group can be realized as the fundamental group of a CW-complex of dimension 2 (or higher). As noted above, though, only free groupscan occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).
- Every 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.[26]
Related concepts
Higher homotopy groups
Roughly speaking, the fundamental group detects the 1-dimensional hole structure of a space, but not higher-dimensional holes 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 for all the n-th homotopy group of the n-sphere is
As was mentioned in the above computation of of classical Lie groups, higher homotopy groups can be relevant even for computing fundamental groups.
Loop space
The set of based loops (as is, i.e. not taken up to homotopy) in a
Fundamental groupoid
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
- ,
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.[29] 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 [30][31]
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
Local systems
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.[33] This equivalence of categories between representations of and local systems is used, for example, in the study of differential equations, such as the Knizhnik–Zamolodchikov equations.
Étale fundamental group
In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group.[34] 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
This yields a theory applicable in situation where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a
Fundamental group of algebraic groups
The fundamental group of a root system is defined, in analogy to the computation for Lie groups.[36] 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.[37]
Fundamental group of simplicial sets
The homotopy relation between 1-simplices of a
See also
Notes
- ^ 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. Archived (PDF) from the original on 2012-03-27.
- ^ May (1999, Ch. 1, §6)
- ^ Massey (1991, Ch. V, §9)
- ^ "Meaning of Fundamental group of a graph". Mathematics Stack Exchange. Retrieved 2020-07-28.
- ^ Simon, J (2008). "Example of calculating the fundamental group of a graph G" (PDF). Archived (PDF) from the original on 2020-07-28.
- ^ "The Fundamental Groups of Connected Graphs - Mathonline". mathonline.wikidot.com. Retrieved 2020-07-28.
- ^ Strom (2011, Problem 9.30, 9.31), Hall (2015, Exercise 13.7)
- ^ 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.
- ^ Fulton (1995, Prop. 12.22)
- ^ May (1999, Ch. 2, §8, Proposition)
- ^ May (1999, Ch. 2, §7)
- ^ Hatcher (2002, §1.3)
- ^ Hatcher (2002, p. 65)
- ^ Hatcher (2002, Proposition 1.36)
- ^ Forster (1981, Theorem 27.9)
- ^ Hatcher (2002, Prop. 4.61)
- ^ Hatcher (2002, Theorem 4.41)
- ^ Hall (2015, Proposition 13.8)
- ^ Hall (2015, Section 13.3)
- ^ Hall (2015, Proposition 13.10)
- ^ Bump (2013, Prop. 23.7)
- ^ Hall (2015, Corollary 13.18)
- ^ Hall (2015, Example 13.45)
- ISBN 0-387-90202-3.
- ^ André Weil, On discrete subgroups of Lie groups, Annals of Mathematics 72 (1960), 369-384.
- ^ Adam Przezdziecki, Measurable cardinals and fundamental groups of compact spaces, Fundamenta Mathematicae 192 (2006), 87-92 [1]
- ^ Hatcher (2002, §4.1)
- ^ Adams (1978, p. 5)
- ^ Brown (2006, §6.1)
- ^ Brown (2006, §6.2)
- ^ Crowell & Fox (1963) use a different definition by reparametrizing the paths to length 1.
- ^ Brown (2006, §6.7)
- ^ El Zein et al. (2010, p. 117, Prop. 1.7)
- ^ Grothendieck & Raynaud (2003).
- ^ Grothendieck & Raynaud (2003, Exposé XII, Cor. 5.2).
- ^ Humphreys (1972, §13.1)
- ^ Humphreys (2004, §31.1)
- ^ Goerss & Jardine (1999, §I.7)
- ^ Goerss & Jardine (1999, §I.11)
References
- MR 0505692
- ISBN 1-4196-2722-8
- ISBN 978-1-4614-8023-5
- Crowell, Richard H.; Fox, Ralph (1963), Introduction to Knot Theory, Springer
- El Zein, Fouad; Suciu, Alexander I.; Tosun, Meral; Uludağ, Muhammed; Yuzvinsky, Sergey (2010), Arrangements, Local Systems and Singularities: CIMPA Summer School, Galatasaray University, Istanbul, 2007, ISBN 978-3-0346-0208-2
- Forster, Otto (1981), Lectures on Riemann Surfaces, ISBN 0-387-90617-7
- ISBN 9780387943275
- Goerss, Paul G.; ISBN 978-3-7643-6064-1
- ISBN 978-2-85629-141-2
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
- ISBN 0-521-79540-0
- Peter Hilton and Shaun Wylie, Homology Theory, Cambridge University Press (1967) [warning: these authors use contrahomology for cohomology]
- ISBN 9780387901084
- Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, ISBN 0-387-90052-7
- Maunder, C. R. F. (January 1996), Algebraic Topology, ISBN 0-486-69131-4
- ISBN 038797430X
- ISBN 9780226511832
- Deane Montgomery and Leo Zippin, Topological Transformation Groups, Interscience Publishers (1955)
- ISBN 0-13-181629-2
- Rotman, Joseph (1998-07-22), An Introduction to Algebraic Topology, ISBN 0-387-96678-1
- Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3
- ISBN 0-12-634850-2
- ISBN 0-387-90202-3
- ISBN 0-387-94426-5
- Strom, Jeffrey (2011), Modern Classical Homotopy Theory, AMS, ISBN 9780821852866
External links
- Weisstein, Eric W. "Fundamental group". MathWorld.
- Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem: A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set
- Animations to introduce fundamental group by Nicolas Delanoue
- Sets of base points and fundamental groupoids: mathoverflow discussion
- Groupoids in Mathematics