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Wikipedia / 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 loopss 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.

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

Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs". 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.

Definition

called the base-point 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

, a loop is defined to be a continuous function (also known as a continuous map)

.

A homotopy such that

  • (which is often thought of as a time parameter).

  • .

  • .

are said to be homotopic" is an equivalence relation This set (with the group structure described below) is called the fundamental group. 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) is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above quotient is, in many cases, more manageable and computable.

Group structure

is just a set. It becomes a group (and therefore deserves the name fundamental group, their product is defined as the loop

with "twice the speed".

. It can be shown that this product does not depend on the choice of representatives and therefore gives a well-defined into a group. Its neutral element (i.e. this class consists of all loops that can be continuously deformed into the constant loop; intuitively speaking of all the loops that do not "wrap around a hole"). The inverse of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction (which is in a different homotopy class). More formally,

the product

with double speed. By comparison,

with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The associativity axiom

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 is path-connected: more precisely, one obtains an isomorphism by pre- and post-concatenating with a path between the two basepoints. This isomorphism is, in general, not unique: it depends on the choice of path up to homotopy. However changing the path results in changing the isomorphism between the two fundamental groups only by composition with an inner automorphism when the choice of basepoint does not matter.

Concrete examples

This section lists some basic examples of fundamental groups. To begin with, in Euclidean space) or any convex subset 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 2-sphere depicted on the right, and also all the higher-dimensional spheres that is (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert–van Kampen theorem or the cellular approximation theorem.

The circle

The circle (also known as the 1-sphere)

times. Therefore, the fundamental group of the circle is isomorphic the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

The figure eight

The fundamental group of the figure eight is the free group where and , the fundamental group of the figure eight is not abelian

More generally, the fundamental group of a r circles letters.

The fundamental group of a wedge sum of two path connected spaces can be computed as the free product This generalizes the above observations since the figure eight is the wedge sum of two circles.

-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 are the cycles. be a spanning tree is a concatenation of such simple loops. Therefore, the fundamental group of a graph is a free group.

has 24 edges overall, and the number of edges in each spanning tree is is the free group with 9 generators. 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. 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. Thus the trefoil knot cannot be continuously transformed into the circle (also known as the unknot. There are, however, knots that cannot be deformed into each other, but have isomorphic knot groups.

Oriented surfaces

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

This includes the torus

Topological groups

The fundamental group of a topological group (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a Lie group:

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.

is abelian for any H-space, 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 is abelian. Related ideas lead to Heinz Hopf's computation of the cohomology of a Lie group.

Functoriality

is a continuous map This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphism, called the induced homomorphism 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. 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 are path connected, then

and if they are also locally contractible, then

  1. ,

denotes the wedge sum the free product of groups. The latter formula is a special case of the Seifert–van Kampen theorem, which states that the fundamental group functor takes pushouts along inclusions to pushouts.

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 abelianization of the fundamental group can be identified with the first homology group of the space.

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

is path-connected, this homomorphism is surjective and its kernel is the commutator subgroup is isomorphic to the abelianization of the fundamental group.

Gluing topological spaces

is the free product This fact is a special case of the Seifert–van Kampen theorem can be obtained by gluing two copies of slightly overlapping half-spheres along a neighborhood of the equator 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).

Coverings

, a continuous map

is called a covering is called a covering space admits an open neighborhood such that there is a homeomorphism between the preimage and a disjoint union),

Universal covering

A covering is called a universal covering is, in addition to the preceding condition, simply connected.. Knowing a universal covering

identifies with the group of deck transformations, i.e., the group of homeomorphisms For example, the map

Any path connected, locally path connected and locally simply connected admits a universal covering.. For example, the uniformization theorem shows that any simply connected Riemann surface or the upper half-plane. General Riemann surfaces then arise as quotients of group actions on these three surfaces.

The quotient of a free action of a discrete has fundamental group

-dimensional real projective space.

Lie groups

be a connected, simply connected compact Lie group, for example, the special unitary group. Then the homogeneous space. Among the many variants of this construction, one of the most important is given by locally symmetric spaces, where

is actually contractible (by the Cartan decomposition for Lie groups).

any torsion-free congruence subgroup of the modular group.

From the explicit realization, it also follows that the universal covering space of a path connected topological group. Moreover, the covering map is a continuous open, a closed discrete normal subgroup:

has to be a subgroup of the center is an abelian group is called the universal covering group.

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

Fibrations the so-called total space is path-connected. can be regarded as a "twisted product" of the base space and the fiber The great importance of fibrations to the computation of homotopy groups stems from a long exact sequence

is path-connected. is the second homotopy group

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

happens to be connected and simply connected, it reduces to an isomorphism

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 classical Lie groups such as the special unitary group This group acts transitively The stabilizer It then can be shown that this yields a fiber sequence

has dimension at least 3, which implies

The long exact sequence then shows an isomorphism

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. These methods give the following results: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 be the Lie algebra The exponential map

The map

can be shown to be surjective of integer linear combination of coroots. This leads to the computation

This method shows, for example, that any connected compact Lie group for which the associated root system is of G_2 is simply connected.; 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.

is a connected simplicial complex, an edge-path. Two edge-paths are said to be edge-equivalent, an edge-loop. The edge-path group, with product and inverse defined by concatenation and reversal of edge-loops.

, the fundamental group of the geometric realisation. Since it depends only on the 2-skeleton are isomorphic.

The edge-path group can be described explicitly in terms of generators and relations is a maximal spanning tree in the 1-skeleton is replaced by any simply connectedin particular contractible. 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.

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; various other authors such as Lorenzo Calabi, Wu Wen-tsün 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.

Realizability

  • 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.

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. For example, the Hurewicz theorem the n-th homotopy group of the n-sphere is

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 pointed space, endowed with the compact open topology, is known as the loop space is in bijection with the set of path components of its loop space:

Fundamental groupoid

The fundamental groupoid is undesirable. It is defined by first considering the category of paths i.e., continuous functions

  1. ,

is variable in this approach, such paths can be concatenated as is (i.e., not up to homotopy) and therefore yield a category.

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

  1. .

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

Local systems

Generally speaking, representationss 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 is a constant sheaf) gives rise to the so-called monodromy representation-dimensional-vector space. Conversely arises in this manner. This equivalence of categories 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. Since the Zariski topology on an algebraic variety or scheme is much coarser than, say, the topology it is no longer meaningful to consider continuous maps from an interval. Instead, the approach developed by Grothendieck by considering all finite étale covers. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situations 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 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.

Fundamental group of algebraic groups

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

Fundamental group of simplicial sets

The homotopy relation between 1-simplices of a simplicial set is a Kan complex but not necessarily so in general. are defined to be the homotopy group of its topological realization.

See also

Notes

References