Homotopy group

Definition

In the n-sphere ${\displaystyle S^n}$ we choose a base point a. For a space X with base point b, we define ${\displaystyle {\pi }_n(X)}$ to be the set of homotopy classes of maps $${\displaystyle f:S^n\to X\mid f(a)=b}$$ that map the base point a to the base point b. In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, define ${\displaystyle {\pi }_n(X)}$ to be the group of homotopy classes of maps ${\displaystyle g:[0,1{]}^n\to X}$ from the n-cube to X that take the boundary of the n-cube to b.

For ${\displaystyle n\ge 1,}$ the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product ${\displaystyle f\ast g}$ of two loops ${\displaystyle f,g:[0,1]\to X}$ is defined by setting $${\displaystyle (f\ast g)(t)=\{\begin{array}{ll}f(2t)& t\in \left[0,\frac{1}{2}\right]\\ g(2t-1)& t\in \left[\frac{1}{2},1\right]\end{array}}$$

The idea of composition in the fundamental group is that of traveling the first path and the second in succession, or, equivalently, setting their two domains together. The concept of composition that we want for the nth homotopy group is the same, except that now the domains that we stick together are cubes, and we must glue them along a face. We therefore define the sum of maps ${\displaystyle f,g:[0,1{]}^n\to X}$ by the formula $${\displaystyle (f+g)(t_1,t_2,\dots ,t_n)=\{\begin{array}{ll}f(2t_1,t_2,\dots ,t_n)& t_1\in \left[0,\frac{1}{2}\right]\\ g(2t_1-1,t_2,\dots ,t_n)& t_1\in \left[\frac{1}{2},1\right]\end{array}}$$

For the corresponding definition in terms of spheres, define the sum ${\displaystyle f+g}$ of maps ${\displaystyle f,g:S^n\to X}$ to be ${\displaystyle \mathrm{\Psi }}$ composed with h, where ${\displaystyle \mathrm{\Psi }}$ is the map from ${\displaystyle S^n}$ to the wedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to X that is defined to be f on the first sphere and g on the second.

If ${\displaystyle n\ge 2,}$ then ${\displaystyle {\pi }_n}$ is abelian. Further, similar to the fundamental group, for a path-connected space any two choices of basepoint give rise to isomorphic ${\displaystyle {\pi }_n.}$

It is tempting to try to simplify the definition of homotopy groups by omitting the base points, but this does not usually work for spaces that are not simply connected, even for path-connected spaces. The set of homotopy classes of maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of the fundamental group on the homotopy group, and in general has no natural group structure.

A way out of these difficulties has been found by defining higher homotopy groupoids of filtered spaces and of n-cubes of spaces. These are related to relative homotopy groups and to n-adic homotopy groups respectively. A higher homotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on homotopy types. For more background and references, see "Higher dimensional group theory"^{[link removed]} and the references below.

Long exact sequence of a fibration

Let ${\displaystyle p:E\to B}$ be a basepoint-preserving Serre fibration with fiber ${\displaystyle F,}$ that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups $${\displaystyle \cdots \to {\pi }_n(F)\to {\pi }_n(E)\to {\pi }_n(B)\to {\pi }_{n-1}(F)\to \cdots \to {\pi }_0(E)\to 0.}$$

Here the maps involving ${\displaystyle {\pi }_0}$ are not group homomorphisms because the ${\displaystyle {\pi }_0}$ are not groups, but they are exact in the sense that the image equals the kernel.

Example: the Hopf fibration. Let B equal ${\displaystyle S^2}$ and E equal ${\displaystyle S^3.}$ Let p be the Hopf fibration, which has fiber ${\displaystyle S^1.}$ From the long exact sequence $${\displaystyle \cdots \to {\pi }_n(S^1)\to {\pi }_n(S^3)\to {\pi }_n(S^2)\to {\pi }_{n-1}(S^1)\to \cdots }$$

and the fact that ${\displaystyle {\pi }_n(S^1)=0}$ for ${\displaystyle n\ge 2,}$ we find that ${\displaystyle {\pi }_n(S^3)={\pi }_n(S^2)}$ for ${\displaystyle n\ge 3.}$ In particular, ${\displaystyle {\pi }_3(S^2)={\pi }_3(S^3)=\mathbb{Z}.}$

In the case of a cover space, when the fiber is discrete, we have that ${\displaystyle {\pi }_n(E)}$ is isomorphic to ${\displaystyle {\pi }_n(B)}$ for ${\displaystyle n>1,}$ that ${\displaystyle {\pi }_n(E)}$ embeds injectively into ${\displaystyle {\pi }_n(B)}$ for all positive ${\displaystyle n,}$ and that the subgroup of ${\displaystyle {\pi }_1(B)}$ that corresponds to the embedding of ${\displaystyle {\pi }_1(E)}$ has cosets in bijection with the elements of the fiber.

When the fibration is the mapping fibre, or dually, the cofibration is the mapping cone, then the resulting exact (or dually, coexact) sequence is given by the Puppe sequence.

Homogeneous spaces and spheres

There are many realizations of spheres as homogeneous spaces, which provide good tools for computing homotopy groups of Lie groups, and the classification of principal bundles on spaces made out of spheres.

Special orthogonal group

There is a fibration $${\displaystyle \mathrm{S}\mathrm{O}(n-1)\to \mathrm{S}\mathrm{O}(n)\to \mathrm{S}\mathrm{O}(n)/\mathrm{S}\mathrm{O}(n-1)\cong S^{n-1}}$$ giving the long exact sequence $${\displaystyle \cdots \to {\pi }_i(\mathrm{S}\mathrm{O}(n-1))\to {\pi }_i(\mathrm{S}\mathrm{O}(n))\to {\pi }_i\left(S^{n-1}\right)\to {\pi }_{i-1}(\mathrm{S}\mathrm{O}(n-1))\to \cdots }$$ which computes the low order homotopy groups of ${\displaystyle {\pi }_i(\mathrm{S}\mathrm{O}(n-1))\cong {\pi }_i(\mathrm{S}\mathrm{O}(n))}$ for ${\displaystyle i<n-1,}$ since ${\displaystyle S^{n-1}}$ is ${\displaystyle (n-2)}$-connected. In particular, there is a fibration

$${\displaystyle \mathrm{S}\mathrm{O}(3)\to \mathrm{S}\mathrm{O}(4)\to S^3}$$ whose lower homotopy groups can be computed explicitly. Since ${\displaystyle \mathrm{S}\mathrm{O}(3)\cong {\mathbb{R}\mathbb{P}}^3,}$ and there is the fibration $${\displaystyle \mathbb{Z}/2\to S^n\to {\mathbb{R}\mathbb{P}}^n}$$ we have ${\displaystyle {\pi }_i(\mathrm{S}\mathrm{O}(3))\cong {\pi }_i(S^3)}$ for ${\displaystyle i>1.}$ Using this, and the fact that ${\displaystyle {\pi }_4\left(S^3\right)=\mathbb{Z}/2,}$ which can be computed using the Postnikov system, we have the long exact sequence $${\displaystyle \begin{array}{cc}\cdots \to & {\pi }_4(\mathrm{S}\mathrm{O}(3))\to {\pi }_4(\mathrm{S}\mathrm{O}(4))\to {\pi }_4(S^3)\to \\ \to & {\pi }_3(\mathrm{S}\mathrm{O}(3))\to {\pi }_3(\mathrm{S}\mathrm{O}(4))\to {\pi }_3(S^3)\to \\ \to & {\pi }_2(\mathrm{S}\mathrm{O}(3))\to {\pi }_2(\mathrm{S}\mathrm{O}(4))\to {\pi }_2(S^3)\to \cdots \end{array}}$$

Since ${\displaystyle {\pi }_2\left(S^3\right)=0}$ we have ${\displaystyle {\pi }_2(\mathrm{S}\mathrm{O}(4))=0.}$ Also, the middle row gives ${\displaystyle {\pi }_3(\mathrm{S}\mathrm{O}(4))\cong \mathbb{Z}\oplus \mathbb{Z}}$ since the connecting map ${\displaystyle {\pi }_4\left(S^3\right)=\mathbb{Z}/2\to \mathbb{Z}={\pi }_3\left({\mathbb{R}\mathbb{P}}^3\right)}$ is trivial. Also, we can know ${\displaystyle {\pi }_4(\mathrm{S}\mathrm{O}(4))}$ has two-torsion.

Application to sphere bundles

Milnor used the fact ${\displaystyle {\pi }_3(\mathrm{S}\mathrm{O}(4))=\mathbb{Z}\oplus \mathbb{Z}}$ to classify 3-sphere bundles over ${\displaystyle S^4,}$ in particular, he was able to find exotic spheres which are smooth manifolds called Milnor's spheres only homeomorphic to ${\displaystyle S^7,}$ not diffeomorphic. Note that any sphere bundle can be constructed from a ${\displaystyle 4}$-vector bundle, which have structure group ${\displaystyle \mathrm{S}\mathrm{O}(4)}$ since ${\displaystyle S^3}$ can have the structure of an oriented Riemannian manifold.

Complex projective space

There is a fibration $${\displaystyle S^1\to S^{2n+1}\to {\mathbb{C}\mathbb{P}}^n}$$ where ${\displaystyle S^{2n+1}}$ is the unit sphere in ${\displaystyle {\mathbb{C}}^{n+1}.}$ This sequence can be used to show the simple-connectedness of ${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ for all ${\displaystyle n.}$

Methods of calculation

Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the excision theorem for singular homology and cohomology, there is no simple known way to calculate the homotopy groups of a space by breaking it up into smaller spaces. However, methods developed in the 1980s involving a van Kampen type theorem for higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. See for a sample result the 2010 paper by Ellis and Mikhailov.

For some spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called aspherical spaces. However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of ${\displaystyle S^2}$ one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.

Certain homotopy groups of n-connected spaces can be calculated by comparison with homology groups via the Hurewicz theorem.

A list of methods for calculating homotopy groups

• The long exact sequence of homotopy groups of a fibration.
• Hurewicz theorem, which has several versions.
• Blakers–Massey theorem, also known as excision for homotopy groups.
• Freudenthal suspension theorem, a corollary of excision for homotopy groups.

Relative homotopy groups

There is also a useful generalization of homotopy groups, ${\displaystyle {\pi }_n(X),}$ called relative homotopy groups ${\displaystyle {\pi }_n(X,A)}$ for a pair ${\displaystyle (X,A),}$ where A is a subspace of ${\displaystyle X.}$

The construction is motivated by the observation that for an inclusion ${\displaystyle i:(A,x_0)\hookrightarrow (X,x_0),}$ there is an induced map on each homotopy group ${\displaystyle i_{\ast }:{\pi }_n(A)\to {\pi }_n(X)}$ which is not in general an injection. Indeed, elements of the kernel are known by considering a representative ${\displaystyle f:I^n\to X}$ and taking a based homotopy ${\displaystyle F:I^n\times I\to X}$ to the constant map ${\displaystyle x_0,}$ or in other words ${\displaystyle H_{I^n\times 1}=f,}$ while the restriction to any other boundary component of ${\displaystyle I^{n+1}}$ is trivial. Hence, we have the following construction:

The elements of such a group are homotopy classes of based maps ${\displaystyle D^n\to X}$ which carry the boundary ${\displaystyle S^{n-1}}$ into A. Two maps ${\displaystyle f,g}$ are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy ${\displaystyle F:D_n\times [0,1]\to X}$ such that, for each p in ${\displaystyle S^{n-1}}$ and t in ⁠${\displaystyle [0,1]}$⁠, the element ${\displaystyle F(p,t)}$ is in A. Note that ordinary homotopy groups are recovered for the special case in which ${\displaystyle A=\{x_0\}}$ is the singleton containing the base point.

These groups are abelian for ${\displaystyle n\ge 3}$ but for ${\displaystyle n=2}$ form the top group of a crossed module with bottom group ${\displaystyle {\pi }_1(A).}$

There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence:

${\displaystyle \cdots \to {\pi }_n(A)\to {\pi }_n(X)\to {\pi }_n(X,A)\to {\pi }_{n-1}(A)\to \cdots }$

Related notions

The homotopy groups are fundamental to homotopy theory, which in turn stimulated the development of model categories. It is possible to define abstract homotopy groups for simplicial sets.

Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are often very complex and hard to compute. In contrast, homology groups are commutative (as are the higher homotopy groups). Given a topological space ${\displaystyle X,}$ its nth homotopy group is denoted by ${\displaystyle {\pi }_n(X),}$ and its nth homology group is denoted by ${\displaystyle H_n(X)}$ or ${\displaystyle H_n(X;\mathbb{Z}).}$