In

"Higher dimensional group theory"

and the references below.

Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids

EMS Tracts in Mathematics Vol. 15, 703 pages, European Math. Society, Zürich, 2011. * . * * * . * * * {{Topology Homotopy theory cs:Homotopická grupa

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, homotopy groups are used in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

to classify topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s. The first and simplest homotopy group is the fundamental group
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, which records information about loops in a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space.
To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point
In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as ''x''0, that remains unchanged durin ...

) into a given space (with base point) are collected into equivalence class
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

es, called Homotopy class
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

es. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

, called the ''n''-th homotopy group, $\backslash pi\_n(X),$ of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...

), but topological spaces that homeomorphic have the same homotopy groups.
The notion of homotopy of paths was introduced by Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated a ...

.
Introduction

In modern mathematics it is common to study acategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

by 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 associating group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s to topological spaces.
That link between topology and groups lets mathematicians apply insights from group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

to topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

. For example, if two topological objects have different homotopy groups, they can not have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus
In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of revolution does not to ...

is different from the sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

: 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 $T$ is
$$\backslash pi\_1(T)\; =\; \backslash Z^2,$$
because the universal cover
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the torus is the Euclidean plane $\backslash R^2,$ mapping to the torus $T\; \backslash cong\; \backslash R^2/\backslash Z^2.$ Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere $S^2$ satisfies:
$$\backslash pi\_1\backslash left(S^2\backslash right)\; =\; 0,$$
because every loop can be contracted to a constant map (see homotopy groups of spheres
In the mathematical field of algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The ...

for this and more complicated examples of homotopy groups).
Hence the torus is not homeomorphic
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...

to the sphere.
Definition

In the ''n''-sphere $S^n$ we choose a base point ''a''. For a space ''X'' with base point ''b'', we define $\backslash pi\_n(X)$ to be the set of homotopy classes of maps $$f\; :\; S^n\; \backslash to\; X\; \backslash mid\; f(a)\; =\; b$$ 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 $\backslash pi\_n(X)$ to be the group of homotopy classes of maps $g\; :;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$ from the to ''X'' that take theboundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

of the ''n''-cube to ''b''.
For $n\; \backslash ge\; 1,$ the homotopy classes form a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

. To define the group operation, recall that in the fundamental group
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, the product $f\backslash ast\; g$ of two loops $f,\; g:;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$ is defined by setting
$$f\; *\; g\; =\; \backslash begin\; f(2t)\; \&\; t\; \backslash in\; \backslash left;\; href="/html/ALL/s/,\_\backslash tfrac\_\backslash right.html"\; ;"title=",\; \backslash tfrac\; \backslash right">,\; \backslash tfrac\; \backslash right$$
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 ''n''-th 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 $f,\; g\; :;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$ by the formula
$$(f\; +\; g)(t\_1,\; t\_2,\; \backslash ldots,\; t\_n)\; =\; \backslash begin\; f(2t\_1,\; t\_2,\; \backslash ldots,\; t\_n)\; \&\; t\_1\; \backslash in\; \backslash left;\; href="/html/ALL/s/,\_\backslash tfrac\_\backslash right\_.html"\; ;"title=",\; \backslash tfrac\; \backslash right\; ">,\; \backslash tfrac\; \backslash right$$
For the corresponding definition in terms of spheres, define the sum $f\; +\; g$ of maps $f,\; g\; :\; S^n\backslash to\; X$ to be $\backslash Psi$ composed with ''h'', where $\backslash Psi$ is the map from $S^n$ to the wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints ''x''0 and ''y''0) the wedge sum of ''X'' and ''Y'' is th ...

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 $n\; \backslash geq\; 2,$ then $\backslash pi\_n$ is abelian. Further, similar to the fundamental group, for a any two choices of basepoint give rise to isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$\backslash pi\_n.$
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
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

, 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 groupoid
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s 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, se"Higher dimensional group theory"

and the references below.

Long exact sequence of a fibration

Let $p\; :\; E\; \backslash to\; B$ be a basepoint-preserving Serre fibration with fiber $F,$ that is, a map possessing thehomotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from ...

with respect to CW complex
A CW complex is a kind of a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...

es. Suppose that ''B'' is path-connected. Then there is a long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same ...

of homotopy groups
$$\backslash cdots\; \backslash to\; \backslash pi\_n(F)\; \backslash to\; \backslash pi\_n(E)\; \backslash to\; \backslash pi\_n(B)\; \backslash to\; \backslash pi\_(F)\; \backslash to\; \backslash cdots\; \backslash to\; \backslash pi\_0(E)\; \backslash to\; 0.$$
Here the maps involving $\backslash pi\_0$ are not group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s because the $\backslash pi\_0$ are not groups, but they are exact in the sense that the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

equals the kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...

.
Example: the Hopf fibration
In the mathematical field of differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

. Let ''B'' equal $S^2$ and ''E'' equal $S^3.$ Let ''p'' be the Hopf fibration
In the mathematical field of differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, which has fiber $S^1.$ From the long exact sequence
$$\backslash cdots\; \backslash to\; \backslash pi\_n\backslash left(S^1\backslash right)\; \backslash to\; \backslash pi\_n\backslash left(S^3\backslash right)\; \backslash to\; \backslash pi\_n\backslash left(S^2\backslash right)\; \backslash to\; \backslash pi\_\backslash left(S^1\backslash right)\; \backslash to\; \backslash cdots$$
and the fact that $\backslash pi\_n\backslash left(S^1\backslash right)\; =\; 0$ for $n\; \backslash geq\; 2,$ we find that $\backslash pi\_n\backslash left(S^3\backslash right)\; =\; \backslash pi\_n\backslash left(S^2\backslash right)$ for $n\; \backslash geq\; 3.$ In particular, $\backslash pi\_3\backslash left(S^2\backslash right)\; =\; \backslash pi\_3\backslash left(S^3\backslash right)\; =\; \backslash Z.$
In the case of a cover space, when the fiber is discrete, we have that $\backslash pi\_n(E)$ is isomorphic to $\backslash pi\_n(B)$ for $n\; >\; 1,$ that $\backslash pi\_n(E)$ embeds injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

ly into $\backslash pi\_n(B)$ for all positive $n,$ and that the subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

of $\backslash pi\_1(B)$ that corresponds to the embedding of $\backslash pi\_1(E)$ has cosets in bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 ashomogeneous space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, 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 $$SO(n-1)\; \backslash to\; SO(n)\; \backslash to\; SO(n)/SO(n-1)\; \backslash cong\; S^$$ giving the long exact sequence $$\backslash cdots\; \backslash to\; \backslash pi\_i(SO(n-1))\; \backslash to\; \backslash pi\_i(SO(n))\; \backslash to\; \backslash pi\_i\backslash left(S^\backslash right)\; \backslash to\; \backslash pi\_(SO(n-1))\; \backslash to\; \backslash cdots$$ which computes the low order homotopy groups of $\backslash pi\_i(SO(n-1))\; \backslash cong\; \backslash pi\_i(SO(n))$ for $i\; <\; n-1,$ since $S^$ is $(n-2)$-connected. In particular, there is a fibration $$SO(3)\; \backslash to\; SO(4)\; \backslash to\; S^3$$ whose lower homotopy groups can be computed explicitly. Since $SO(3)\; \backslash cong\; \backslash mathbb^3,$ and there is the fibration $$\backslash Z/2\; \backslash to\; S^n\; \backslash to\; \backslash mathbb^n$$ we have $\backslash pi\_i(SO(3))\; \backslash cong\; \backslash pi\_i\backslash left(S^3\backslash right)$ for $i\; >\; 1.$ Using this, and the fact that $\backslash pi\_4\backslash left(S^3\backslash right)\; =\; \backslash Z/2,$ which can be computed using thePostnikov systemIn homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an Ind-completion, inverse system of topological spaces whose Homotopy, homotopy type at d ...

, we have the long exact sequence
$$\backslash begin\; \backslash cdots\; \backslash to\; \&\backslash pi\_4(SO(3))\; \backslash to\; \backslash pi\_4(SO(4))\; \backslash to\; \backslash pi\_4\backslash left(S^3\backslash right)\; \backslash to\; \backslash \backslash \; \backslash to\; \&\backslash pi\_3(SO(3))\; \backslash to\; \backslash pi\_3(SO(4))\; \backslash to\; \backslash pi\_3\backslash left(S^3\backslash right)\; \backslash to\; \backslash \backslash \; \backslash to\; \&\backslash pi\_2(SO(3))\; \backslash to\; \backslash pi\_2(SO(4))\; \backslash to\; \backslash pi\_2\backslash left(S^3\backslash right)\; \backslash to\; \backslash cdots\; \backslash \backslash \; \backslash end$$
Since $\backslash pi\_2\backslash left(S^3\backslash right)\; =\; 0$ we have $\backslash pi\_2(SO(4))\; =\; 0.$ Also, the middle row gives $\backslash pi\_3(SO(4))\; \backslash cong\; \backslash Z\backslash oplus\backslash Z$ since the connecting map $\backslash pi\_4\backslash left(S^3\backslash right)\; =\; \backslash Z/2\; \backslash to\; \backslash Z\; =\; \backslash pi\_3\backslash left(\backslash mathbb^3\backslash right)$ is trivial. Also, we can know $\backslash pi\_4(SO(4))$ has two-torsion.
= Application to sphere bundles

= Milnor used the fact $\backslash pi\_3(SO(4))\; =\; \backslash Z\backslash oplus\backslash Z$ to classify 3-sphere bundles over $S^4,$ in particular, he was able to findExotic sphere
In an area of mathematics called differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

s which are smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

s called Milnor's spheres only homeomorphic to $S^7,$ not diffeomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. Note that any sphere bundle can be constructed from a $4$-vector bundle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

, which have structure group $SO(4)$ since $S^3$ can have the structure of an oriented
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...

.
Complex projective space

There is a fibration $$S^1\; \backslash to\; S^\; \backslash to\; \backslash mathbb^n$$ where $S^$ is the unit sphere in $\backslash Complex^n.$ This sequence can be used to show the simple-connectedness of $\backslash mathbb^n$ for all $n.$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 theSeifert–van Kampen theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

for the fundamental group and the excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem ...

for singular homology
In algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify ...

and cohomology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, 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
Tori may refer to:
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* Tori, Järva County, Estonia
* Tori, ...

, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called aspherical spaceIn topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups \pi_n(X) equal to 0 when n>1.
If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whos ...

s. 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 $S^2$ one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequenceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

was constructed for just this purpose.
Certain homotopy groups of ''n''-connected spaces can be calculated by comparison with homology group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s via the Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results o ...

.
A list of methods for calculating homotopy groups

* The long exact sequence of homotopy groups of a fibration. *Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results o ...

, which has several versions.
* Blakers–Massey theorem, also known as excision for homotopy groups.
* Freudenthal suspension theorem
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the beh ...

, a corollary of excision for homotopy groups.
Relative homotopy groups

There is also a useful generalization of homotopy groups, $\backslash pi\_n(X),$ called relative homotopy groups $\backslash pi\_n(X,\; A)$ for a pair $(X,\; A),$ where ''A'' is a of $X.$ The construction is motivated by the observation that for an inclusion $i\; :\; (A,x\_0)\; \backslash hookrightarrow\; (X,x\_0),$ there is an induced map on each homotopy group $i\_*\; :\; \backslash pi\_n(A)\; \backslash to\; \backslash pi\_n(X)$ which is not in general an injection. Indeed, elements of the kernel are known by considering a representative $f\; :\; I^n\; \backslash to\; X$ and taking a based homotopy $F\; :\; I^n\; \backslash times\; I\; \backslash to\; X$ to the constant map $x\_0,$ or in other words $H\_\; =\; f,$ while the restriction to any other boundary component of $I^$ is trivial. Hence, we have the following construction: The elements of such a group are homotopy classes of based maps $D^n\; \backslash to\; X$ which carry the boundary $S^$ into ''A''. Two maps $f,\; g$ are called homotopic relative to ''A'' if they are homotopic by a basepoint-preserving homotopy $F\; :\; D\_n\; \backslash times$, 1
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

\to X such that, for each ''p'' in $S^$ and ''t'' in $$, 1
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

the element $F(p,\; t)$ is in ''A''. Note that ordinary homotopy groups are recovered for the special case in which $A\; =\; \backslash $ is the singleton containing the base point.
These groups are abelian for $n\; \backslash geq\; 3(E)$ but for $n\; =\; 2$ form the top group of a crossed moduleIn mathematics, and especially in homotopy theory, a crossed module consists of group (mathematics), groups ''G'' and ''H'', where ''G'' Group action (mathematics), acts on ''H'' by automorphisms (which we will write on the left, (g,h) \mapsto g \cd ...

with bottom group $\backslash pi\_1(A).$
There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence:
:$\backslash cdots\; \backslash to\; \backslash pi\_n(A)\; \backslash to\; \backslash pi\_n(X)\; \backslash to\; \backslash pi\_n(X,A)\; \backslash to\; \backslash pi\_(A)\backslash to\; \backslash cdots$
Related notions

The homotopy groups are fundamental tohomotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps come with homotopy, homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Be ...

, which in turn stimulated the development of model categories. It is possible to define abstract homotopy groups for simplicial set In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and Category (mathematics), categories. Formally, a simplicial ...

s.
Homology group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are usually not commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, and often very complex and hard to compute. In contrast, homology groups are commutative (as are the higher homotopy groups). Hence, it is sometimes said that "homology is a commutative alternative to homotopy". Given a topological space $X,$ its ''n''-th homotopy group is usually denoted by $\backslash pi\_n(X),$ and its ''n''-th homology group is usually denoted by $H\_n(X).$
See also

*Fibration
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

*Hopf fibration
In the mathematical field of differential topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

*Hopf invariant In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

*Knot theory
In the mathematical field of topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...

*Homotopy class
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

*Homotopy groups of spheres
In the mathematical field of algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The ...

*Topological invariantIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

* Homotopy group with coefficients
*Pointed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

Notes

References

* Ronald Brown, `Groupoids and crossed objects in algebraic topology',Homology, Homotopy and Applications
''Homology, Homotopy and Applications'' is a peer review, peer-reviewed delayed open access mathematics journal published by International Press. It was established in 1999 and covers research on algebraic topology. The journal "Homology, Homotopy a ...

, 1 (1999) 1–78.
* Ronald Brown, Philip J. Higgins, Rafael SiveraNonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids

EMS Tracts in Mathematics Vol. 15, 703 pages, European Math. Society, Zürich, 2011. * . * * * . * * * {{Topology Homotopy theory cs:Homotopická grupa