Inmathematics,homotopy groups are used inalgebraic topology to classifytopological spaces. The first and simplest homotopy group is thefundamental group, denoted${\displaystyle {\pi }_1(X),}$ which records information aboutloops in aspace. Intuitively, homotopy groups record information about the basic shape, orholes, of a topological space.

To define thenth homotopy group, the base-point-preserving maps from ann-dimensional sphere (withbase point) into a given space (with base point) are collected intoequivalence classes, calledhomotopy classes. Two mappings arehomotopic if one can be continuously deformed into the other. These homotopy classes form agroup, called thenth homotopy group,${\displaystyle {\pi }_n(X),}$ of the given spaceX with base point. Topological spaces with differing homotopy groups are neverhomeomorphic, but topological spaces thatare not homeomorphiccan have the same homotopy groups.

The notion of homotopy ofpaths was introduced byCamille Jordan.^[1]

In modern mathematics it is common to study acategory byassociating to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associatinggroups to topological spaces.

That link between topology and groups lets mathematicians apply insights fromgroup theory totopology. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. For example, thetorus is different from thesphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.

As for the example: the first homotopy group of the torus${\displaystyle T}$ is$${\displaystyle {\pi }_1(T)={\mathbb{Z}}^2,}$$because theuniversal cover of the torus is the Euclidean plane${\displaystyle {\mathbb{R}}^2,}$ mapping to the torus${\displaystyle T\cong {\mathbb{R}}^2/{\mathbb{Z}}^2.}$ Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere${\displaystyle S^2}$ satisfies:$${\displaystyle {\pi }_1\left(S^2\right)=0,}$$because every loop can be contracted to a constant map (seehomotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is nothomeomorphic to the sphere.

In then-sphere${\displaystyle S^n}$ we choose a base pointa. For a spaceX with base pointb, 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 pointa to the base pointb. 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 then-cube toX that take theboundary of then-cube tob.

For${\displaystyle n\ge 1,}$ the homotopy classes form agroup. To define the group operation, recall that in thefundamental group, the product${\displaystyle f\ast g}$ of two loops${\displaystyle f,g:[0,1]\to X}$ is defined by setting

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 thenth 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

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 withh, where${\displaystyle \mathrm{\Psi }}$ is the map from${\displaystyle S^n}$ to thewedge sum of twon-spheres that collapses the equator andh is the map from the wedge sum of twon-spheres toX that is defined to bef on the first sphere andg on the second.

If${\displaystyle n\ge 2,}$ then${\displaystyle {\pi }_n}$ isabelian.^[2] Further, similar to the fundamental group, for apath-connected space any two choices of basepoint give rise toisomorphic${\displaystyle {\pi }_n.}$^[3]

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 notsimply 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 homotopygroupoids of filtered spaces and ofn-cubes of spaces. These are related to relative homotopy groups and ton-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" and the references below.

A topological space has ahole with ad-dimensional boundary if-and-only-if it contains ad-dimensional sphere that cannot be shrunk continuously to a single point. This holds if-and-only-if there is a mapping$S^d\to X$ that is not homotopic to aconstant function. This holds if-and-only-if thedth homotopy group ofX is not trivial. In short,X has a hole with ad-dimensional boundary, if-and-only-if${\displaystyle {\pi }_d(X)\ncong 0}$.

Let${\displaystyle p:E\to B}$ be a basepoint-preservingSerre fibration with fiber${\displaystyle F,}$ that is, a map possessing thehomotopy lifting property with respect toCW complexes. Suppose thatB is path-connected. Then there is a longexact 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 notgroup homomorphisms because the${\displaystyle {\pi }_0}$ are not groups, but they are exact in the sense that theimage equals thekernel.

Example: theHopf fibration. LetB equal${\displaystyle S^2}$ andE equal${\displaystyle S^3.}$ Letp be theHopf 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${\displaystyle {\pi }_n(E)}$ embedsinjectively into${\displaystyle {\pi }_n(B)}$ for all positive${\displaystyle n,}$ and that thesubgroup of${\displaystyle {\pi }_1(B)}$ that corresponds to the embedding of${\displaystyle {\pi }_1(E)}$ has cosets inbijection with the elements of the fiber.

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

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

There is a fibration^[4]$${\displaystyle \mathrm{S}\mathrm{O}(n-1)\to \mathrm{S}\mathrm{O}(n)\to \mathrm{S}\mathrm{O}(n)/\mathrm{S}\mathrm{O}(n-1)\cong S^{n-1}}$$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 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 thePostnikov system, we have the long exact sequence

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.

Milnor^[5] 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 findexotic spheres which aresmooth manifolds calledMilnor's spheres only homeomorphic to${\displaystyle S^7,}$ notdiffeomorphic. 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 anorientedRiemannian manifold.

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${\displaystyle n.}$

Calculation of homotopy groups is in general much more difficult than some of the other homotopyinvariants learned in algebraic topology. Unlike theSeifert–van Kampen theorem for the fundamental group and theexcision theorem forsingular homology andcohomology, 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.^[6]

For some spaces, such astori, all higher homotopy groups (that is, second and higher homotopy groups) aretrivial. These are the so-calledaspherical 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 theSerre spectral sequence was constructed for just this purpose.

Certain homotopy groups ofn-connected spaces can be calculated by comparison withhomology groups via theHurewicz theorem.

• The long exact sequence of homotopy groups of a fibration.
• Hurewicz theorem, which has several versions.
• Blakers–Massey theorem, also known asexcision for homotopy groups.
• Freudenthal suspension theorem, a corollary of excision for 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 apair${\displaystyle (X,A),}$ whereA is asubspace 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${\displaystyle S^{n-1}}$ intoA. Two maps${\displaystyle f,g}$ are called homotopicrelative toA if they are homotopic by a basepoint-preserving homotopy${\displaystyle F:D_n\times [0,1]\to X}$ such that, for eachp in${\displaystyle S^{n-1}}$ andt in⁠${\displaystyle [0,1]}$⁠, the element${\displaystyle F(p,t)}$ is inA. 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 acrossed module with bottom group${\displaystyle {\pi }_1(A).}$

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

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

The homotopy groups are fundamental tohomotopy theory, which in turn stimulated the development ofmodel categories. It is possible to define abstract homotopy groups forsimplicial 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${\displaystyle X,}$ itsnth homotopy group is denoted by${\displaystyle {\pi }_n(X),}$ and itsnth homology group is denoted by${\displaystyle H_n(X)}$ or${\displaystyle H_n(X;\mathbb{Z}).}$

• Fibration
• Hopf fibration
• Hopf invariant
• Knot theory
• Homotopy class
• Homotopy groups of spheres
• Topological invariant
• Homotopy group with coefficients
• Pointed set

• Ronald Brown, `Groupoids and crossed objects in algebraic topology',Homology, Homotopy and Applications, 1 (1999) 1–78.
• Ronald Brown, Philip J. Higgins, Rafael Sivera,Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages, European Math. Society, Zürich, 2011.doi:10.4171/083MR2841564
• Čech, Eduard (1932), "Höherdimensionale Homotopiegruppen",Verhandlungen des Internationalen Mathematikerkongress, Zürich.
• Hatcher, Allen (2002),Algebraic topology,Cambridge University Press,ISBN 978-0-521-79540-1
• "Homotopy group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Hopf, Heinz (1931),"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche",Mathematische Annalen,104 (1):637–665,doi:10.1007/BF01457962.
• Kamps, Klaus H.; Porter, Timothy (1997).Abstract homotopy and simple homotopy theory. River Edge, NJ: World Scientific Publishing.doi:10.1142/9789812831989.ISBN 981-02-1602-5.MR 1464944.
• Toda, Hiroshi (1962).Composition methods in homotopy groups of spheres. Annals of Mathematics Studies. Vol. 49. Princeton, N.J.: Princeton University Press.ISBN 0-691-09586-8.MR 0143217.{{cite book}}:ISBN / Date incompatibility (help)
• Whitehead, George William (1978).Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744.ISBN 978-0-387-90336-1.MR 0516508.

• Homotopy theory

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