The Fundamental Group
Level: 0 1 2 3 4 5 6 7
MSC Classification: 55 (Algebraic topology)

Getting Oriented

The fundamental group is a tool used to study topological spaces. It is a topological invariant, which means that it is the same for homeomorphic spaces. Because of this, it is frequently used to determine when two spaces are not homeomorphic.

image.svg

On the sphere, two paths looping from the same point must intersect, while on the donut (or torus), the paths need not intersect anywhere but at the starting point.


The fundamental group is made up of objects that are essentially closed paths, or loops in the space. Simple properties of these paths can be used to distinguish spaces. For example, imagine you take two walks starting and ending at point A along the earth's equator. On the first walk, you begin by heading North, circle around, and arrive back at point A from the South. On the second walk, you begin by heading East, circle around, and arrive back at point A from the West. Now, no matter what your paths look like, at some point, the two paths must cross… otherwise the second path couldn't end up "on the other side" of the first path. On the other hand, if one tries the same thing on a donut, it is possible for the two paths never to intersect!

The fundamental group encodes this idea in a mathematical structure. For it to make sense, one must consider in what ways it makes sense to say two paths are "equivalent". The language of homotopy provides the proper equivalence relation: two paths are (homotopically) equivalent if they can be "smoothly deformed" into each other. Armed with just this idea, the space of loops on a topological space takes on the structure of a group.

This group tells a lot about the structure of a topological space. For every space, there is a space with trivial fundamental group, called the universal cover, which is formed by "unrolling" the topological space, almost in the same way that one unwinds a roll of paper towels. And the spaces with this same universal cover are in a one-to-one correspondence with normal subgroups of that fundamental group.

The fundamental group also makes a complete classification of surfaces possible. We will briefly describe how this is done using the Seifert-Van Kampen Theorem, a calculational tool that describes how fundamental groups behave under the "connect sum" operation on surfaces.

The Basics

In what follows, the term map indicates a continuous function between topological spaces. A path in a topological space X is a map $f:I\to X$ from the unit interval [0,1] to X. A loop is a map $f:S^1\to X$ from the circle into the space. A pointed loop is a map $f:I\to X$ such that f(0)=f(1), so it takes the two endpoints to the same point in X. A constant map is any map $f:X\to Y$ whose image is a single point.

A one-parameter family of maps from X to Y is a collection of continuous maps $F_t:X\to Y$ for $t\in[0,1]$. We will generally assume the family is continuous, meaning the map $F : X\times I \to Y$ defined by $F(x,t) = F_t(x)$ is continuous.

Homotopy

Homotopy is the term for continuous deformation in topological spaces, i.e. the idea of transforming part of a topological space in a continuous way into another part.

homotopy.svg

Homotopies are continuous deformations of maps. Contractible spaces are homotopic to a point.

Definition

A homotopy between maps $f:X\to Y$ and $g:X\to Y$ is a continuous family of maps $F_t:X\to Y$ for $t\in[0,1]$ such that F0=f and F1=g. Homotopy equivalence is denoted $f\simeq g$. A function f is null-homotopic if there is a homotopy between f and a constant function.

If two homotopies Ft and Gt are "compatible" in the sense that F1=G0, then they may be concatenated to form the composite homotopy F*G. Under this composition rule, homotopy equivalence is an equivalence relation. The homotopy class $[f]$ of a map $f:X\to Y$ is the equivalence class of maps that are homotopic to f.

The notion of homotopy also extends to the spaces themselves. Topological spaces X and Y are homotopy equivalent (denoted $X\simeq Y$) if there exist maps $f:X\to Y$ and $g:X\to Y$ such that $g\circ f \simeq 1_X$ and $f\circ g \simeq 1_Y$, where 1X and 1Y denote the identity maps on X and Y. A space is contractible if it is homotopy equivalent to a 1-point space.

Definition

Given a subset $A\subset X$, a relative homotopy fixing A is a homotopy Ft which is the identity on A for all t, i.e. $F_t|_A=1_A$ for all t. Relative homotopy is indicated by writing $F_0\simeq_A F_1$ or $F_0\simeq F_1 \:\mathrm{rel}\: A$. Since relative homotopy is also an equivalence relation, one may also speak of the relative homotopy class of maps that fix A.

Two paths $f:I\to X$ and $g:I\to X$ whose endpoints match are path-homotopic if there is a homotopy between them that fixes the endpoints. The homotopy is then $F_t:I\to X$ with $F_t(0)=x_0$ and $F_t(1)=x_1$ for all t, and one writes $f \simeq g \:\mathrm{rel}\: \{0,1\}$.

A deformation retract is a subset $A\subset X$ with a homotopy $F_t:X\to X$ such that $F_0=1_X$, $F_1|_A=1_A$, and $F_1(X)\subset A$. So the space X is "continuously retracted" onto the subset A. The subset A is a strong deformation retract if it is a deformation retract whose homotopy fixes A, so that $F_t|_A=1_A$ for all t. Both terms imply homotopy equivalence $X\simeq A$.

For example, a mapping cylinder Mf is the space $X\times I$ glued to a second space Y via a map $f:X\times\{0\}\to Y$. The space Mf retracts onto Y by "shrinking" the cylinder down to just $X\times\{0\}$. Because of this, the homotopy type of the mapping cylinder depends only on the properties of f.

The Fundamental Group

It is often the case that the set of relative homotopy classes will have a natural group structure. For this to be possible, one must define an operation on multiple maps $X\to Y$. One can define such an operation if the maps consist of pointed loops, that is paths that start and end at the same point. In such cases, the space of relative homotopy classes (with fixed basepoints) has the structure of a group, called the fundamental group.

Definition

Given a topological space X and a point $x_0\in X$, the fundamental group of X is the group formed out of the relative homotopy classes of loops that fix endpoints of paths at x0, with path composition as the group operation. It is denoted by $\pi_1(X,x_0)$.

To show two paths represent the same group element, one must show they are in the same homotopy class by finding an explicit homotopy between the paths. A key technique for working with these homotopies is reparametrization. If $F_t:I\to X$ is a relative homotopy between F0 and F1 that fixes {0,1}, then any reparametrization $F_{\phi(t)}$ is also a relative homotopy between these two functions. All the reparametrization does is change "how quickly" the deformation occurs. Path composition corresponds to concatenating and reparametrizing two homotopies.

Theorem

The fundamental group is a group, i.e., the relative homotopy classes of pointed loops based at $x_0\in X$ have a group structure under path composition.

Given a different basepoint x1 in the same path component as x0, one can find a group isomorphism between $\pi_1(X,x_0)$ and $\pi_1(X,x_1)$. The isomorphism takes a loop at x0 to one at x1 by adding on a path from x1 to x0 at the beginning of the loop, and the inverse of the path at the end of the loop. Therefore, up to isomorphism, the group $\pi_1(X,x_0)$ depends only on the path component. Because of this, we frequently assume that the space is path-connected and omit the basepoint.

A simply-connected space is a path-connected space with trivial fundamental group. Since homotopy-equivalent spaces have the same fundamental group, all contractible spaces are simply-connected. Indeed, any loop $S^1\to X$ in a space that is homotopically trivial extends to a map from the disk $S^1\hookrightarrow D^2\to X$.

Computing the Fundamental Group

Examples

The Euclidean spaces $\mathbb{R}^n$ are all contractible, so they are all simply-connected. The same is true for disks and balls. Spheres are another matter:

  • In dimension 0, the sphere consists of two disconnected points S0={-1,1}, each of which has trivial fundamental group.
  • In dimension 1, the circle S1 has fundamental group $\mathhb{Z}$. Loosely speaking, paths correspond to the integer n if they "wrap around" the circle n times, and negative numbers correspond to the reverse direction.
  • In larger dimensions, the spheres Sn are simply-connected, even though they are not contractible.

The fundamental group behaves nicely under products, in the sense that $\pi_1(X\times Y,(x_0,y_0))=\pi_1(X,x_0)\times\pi_1(Y,y_0)$. Thus, the fundamental group of the n-torus $T^n=S^1\times\cdots\times S^1$ is $\mathbb{Z}\times\cdots\times\mathbb{Z}$.

Since deformation retracts preserve homotopy type, any space may be crossed with an interval I without changing the fundamental group. Thus, the fundamental group of the annulus S1xI is also $\mathbb{Z}$.

If X and Y are path-connected spaces, their one-point union is formed by identifying a single point in X with one in Y, sometimes denoted X*Y. The fundamental group of this space is a free product of the individual fundamental groups: $\pi_1(X*Y,(x_0,y_0))=\pi_1(X,x_0)*\pi_1(Y,y_0)$. Thus, the n-petal "rose" $S^1*\cdots*S^1$, which is comprised of n circles glued together at a common point, has fundamental group $\mathbb{Z}*\cdots*\mathbb{Z}$, the free group on n elements.

The Seifert-van Kampen Theorem

The preceding discussion of one-point unions is a special case of the Seifert-van Kampen Theorem, a key tool used for computing the fundamental group, as it can express the fundamental group of a space in terms of the fundamental group of simpler components.

Theorem (Seifert-Van Kampen Theorem)

Suppose that $X=U\cup V$ for nonempty connected subsets U and V that overlap, and let $x_0\in U\cap V$. Then $\pi_1(X,x_0)$ is the free product of the fundamental groups of U and V, with relations stating that any loop common to both U and V (and therefore in $U\cap V$) is considered the same. In formal terms,$\pi_1(X,x_0) \cong \pi_1(U)*_{\pi_1(U\cap V,x_0)} \pi_1(V,x_0)$, the amalgamated free product of $\pi_1(U,x_0)$ and $\pi_1(V,x_0)$.

Here are two sample calculations that use this theorem:

  • If $U\cap V$ is simply connected, then $\pi_1(X,x_0) \cong \pi_1(U,x_0) * \pi_1(V,x_0)$.
  • If V is simply connected, then $\pi_1(X,x_0) \cong \pi_1(U,x_0)/N$, where N is the subgroup of $\pi_1(U,x_0)$ generated by the image of $\pi_1(U\cap V,x_0)$.

Surfaces

// needs completion… //

Covering Spaces

Covering Maps

Definition

Given two path-connected, locally path-connected Hausdorff spaces W and X, a map $p:W\to X$ is a covering map if each point $x\in X$ has a neighborhood U such that the inverse image p-1(U) is comprised of disjoint sets $U_\alpha$, each of which is homeomorphic to U with homeomorphism $p|_{U_\alpha}$. The space W is called a covering space of X.


The number of points in the inverse image of a point is constant, called the number of sheets of the covering.

One of the simplest examples of a covering space is $\mathbb{R}\to S^1$ with $t\mapsto e^{2\pi it}$ (an infinite sheeted covering). Similarly, the map $S^2 \to \mathbb{R}P^2$ from the sphere to the projective plane is a double covering.

In the context of a covering map $p:W\to X$, a lift of a map $f:Z\to X$ is a map $g:Z\to W$ such that $f=g\circ p$. The path-lifting property says that a path $f:I\to X$ can be uniquely lifted to a path $g:I\to W$, and the homotopy-lifting theorem says that a homotopy $F:Z\times I\to X$ with partial lift $f:Z\times\{0\}\to W$ can be lifted uniquely to a homotopy $G:Z\times I\to W$. In general, a unique lift exists if and only if $\mathrm{image}\: f_\#\subset \mathrm{image}\: p_\#$, where $f_\#:\pi_1(Z)\to\pi_1(X)$ is the homomorphism induced on the fundamental group by the map f and $p_\#$ is defined similarly. It can be shown that $p_\#:\pi_1(W)\to\pi_1(X)$ is one-to-one. Consequently, a space is simply connected if and only if it has no nontrivial covers.

For most spaces, one may define a special cover that is unique up to homeomorphism:

Definition

A universal cover of X is a simply-connected cover of X.


It can also be shown that the universal cover is also a covering space for any other cover of X.

Deck Transformations

Given a covering map $p:W\to X$, the group $G=\pi_1(X,x_0)$ acts on the fiber $p^{-1}(x_0)$ as a group of permutations. The action is given by looking at how a loop in G lifts (under the path-lifting property) to a path starting at some $w_0\in p^{-1}(x_0)$. The endpoint of this path is the result of the action on w0.

The isotropy subgroup is the set of elements in G that fix w0: $G_{w_0}=\{\alpha \in G: w_0 \cdot \alpha = w_0\}$. Equivalently, it consists of loops in X that lift to loops based at w0, and it may also be identified with the image of p# in $\pi_1(W,w_0)$. There is a one-to-one correspondence between the right cosets $\left(p_\#\pi_1(W,w_0)\right)\pi_1(X,x_0)$ and the fiber $p^{-1}(x_0)$. This in turn implies that the number of sheets of the covering map is precisely the index of $p_\#(\pi_1(W,w_0))$ in $\pi_1(X,x_0)$. In the specific case of the universal cover, the number of sheets is just the order of $\pi_1(X,x_0)$.

Given a covering map $p:W\to X$, a deck transformation is a homeomorphism $D:W\to W$ of the cover. Deck transformations form a group $\Delta=\Delta_p$ under map composition. One might expect that the quotient $W/\Delta$ could be identified with Y. This particular case can be assured if the subgroup $p_\#\pi_1(W,w_0)$ is normal in $\pi_1(X,x_0)$, or alternately if $\Delta$ acts transitively on $p^{-1}(x_0)$, in which case the covering map is said to be regular and $\Delta \cong \pi_1(X,x_0)/p_\#\pi_1(W,w_0)$. In particular, if W is simply connected (perhaps the universal cover), then $\Delta \cong \pi_1(X,x_0)$.

Going Further

Higher Homotopy Groups

Denote the set of homotopy classes of maps $X\to Y$ by $[X;Y]$, and of maps $(X,A)\to (Y,B)$ by $[X,A;Y,B]$.

A pointed space is a space with a specified base point, such as $(X,x_0)$.

Definition

A homotopy group is constructed as follows. Maps preserving base points form the group $[X;Y]_*$ so that $[SX;Y]_* \cong [X\times I,A;Y,\{y_0\}]$, where $SX=(X\times I)/(\{x_0\}\times I\cup X\times\del I)$ is the reduced suspension. $[SX;Y]$ forms a group with operation being the composition of maps, called a homotopy group.

The nth homotopy group is defined by $\pi_n(Y,\{y_0\})=[S^n;Y]_*$, where $S^n$ is the $n$-sphere, and can be thought of as the n-fold reduced suspension of $S^0=\{0,1\}$ with base point 0. An alternate definition would be $\pi_n(Y,\{y_0\}=[I^n,\del I^n;Y,\{y_0\}]$, since $S^n$ is formed from $I^n$ by collapsing the boundary to a point.

The homotopy of spheres is most easily calculated. $\pi_n(S^n)\cong\mathbb{Z}$, and $\pi_n(S^k)=0$ for $n<k$. $\pi_n(S^1)=0$ for $n>1$, but $\pi_3(S^2)\cong\mathbb{Z}$ and $\pi_{n+1}(S^n)\cong\mathbb{Z}_2$ for $n>2$. In general, the homotopy groups are stable, in the sense that $\pi_{n+k}(S^n)$ is independent of $n$ for large $n$.

The homotopy group is functorial. This means that a map $\phi \in [Y,W]_*$ induces a group homomorphism $\phi_\#:\pi_1(Y,\{y_0\})\to\pi_1(W,\{w_0\})$, with $\psi_\#\circ\phi_\#=(\psi\circ\phi)_\#$ and $\mathsf{Id}_\#=\mathsf{Id}$. Moreover, if $\psi$ and $\phi$ are homotopic, then $\psi_\#=\phi_\#$.

The Road Ahead

The fundamental group is just the introduction to the vast subject of algebraic topology. It is a rather intuitive concept, but can be very difficult to calculate. There is another topological invariant, called homology, which turns out to be easier. It is less intuitive, calculated in terms of boundaries and pieces of a space rather than something concrete. However, the groups one obtains are always abelian; in fact, the first homology group is the abelianization of the fundamental group.

Another vital aspect of modern algebraic topology is cohomology theory, which deals with maps from bits and pieces of a space into some nice group like $\mathbb{Z}$. It pairs up nicely with homology, meaning there is a natural correspondence (called duality) between the two.

A third piece of algebraic topology is higher homotopy theory, which generalizes the fundamental group to maps from spheres $S^n$ into a space. It retains the calculational complexity of the fundamental group, and working with this complexity requires a very deep theory best approached after working through homology and cohomology theory.

References

  • Allen Hatcher, Algebraic Topology
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License