In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, homotopy groups are used 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 topological spaces up t ...
to classify
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s. The first and simplest homotopy group is the
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 loops contained in the space. It records information about the basic shape, or holes, of ...
, denoted
which records information about
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s in a
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
. Intuitively, homotopy groups record information about the basic shape, or ''
holes
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
'', 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 or based 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 ...
) into a given space (with base point) are collected into
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es, called
homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a
group, called the ''n''-th homotopy group,
of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent (
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
), but topological spaces that homeomorphic have the same homotopy groups.
The notion of homotopy of
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
s was introduced by
Camille Jordan.
Introduction
In modern mathematics it is common to study a
category by
associating 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
groups to topological spaces.
That link between topology and groups lets mathematicians apply insights from
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
to
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. 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 or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
is different from the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
: 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
is
because the
universal cover of the torus is the Euclidean plane
mapping to the torus
Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere
satisfies:
because every loop can be contracted to a constant map (see
homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the sphere.
Definition
In the
''n''-sphere we choose a base point ''a''. For a space ''X'' with base point ''b'', we define
to be the set of homotopy classes of maps
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
to be the group of homotopy classes of maps
from the
''n''-cube to ''X'' that take the
boundary of the ''n''-cube to ''b''.
For
the homotopy classes form a
group. To define the group operation, recall that in the
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 loops contained in the space. It records information about the basic shape, or holes, of ...
, the product
of two loops
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 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
by the formula
For the corresponding definition in terms of spheres, define the sum
of maps
to be
composed with ''h'', where
is the map from
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 the qu ...
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
then
is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
. Further, similar to the fundamental group, for a
path-connected space any two choices of basepoint give rise to
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
, 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, se
"Higher dimensional group theory"and the references below.
Homotopy groups and holes
A topological space has a
''hole'' with a ''d''-dimensional boundary if-and-only-if it contains a ''d''-dimensional sphere that cannot be shrunk continuously to a single point. This holds if-and-only-if there is a mapping
that is not homotopic to a
constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties ...
. This holds if-and-only-if the ''d''-th homotopy group of ''X'' is not trivial. In short, X has a hole with a ''d''-dimensional boundary, if-and-only-if
.
Long exact sequence of a fibration
Let
be a basepoint-preserving
Serre fibration with fiber
that is, a map possessing the
homotopy lifting property with respect to
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
es. Suppose that ''B'' is path-connected. Then there is a long
exact sequence of homotopy groups
Here the maps involving
are not
group homomorphisms because the
are not groups, but they are exact in the sense that the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
equals the
kernel.
Example: the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
. Let ''B'' equal
and ''E'' equal
Let ''p'' be the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
, which has fiber
From the long exact sequence
and the fact that
for
we find that
for
In particular,
In the case of a cover space, when the fiber is discrete, we have that
is isomorphic to
for
that
embeds
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
ly into
for all positive
and that the
subgroup of
that corresponds to the embedding of
has cosets in
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the elements of the fiber.
When the fibration is the
mapping fibre, or dually, the cofibration is the
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexes ...
, then the resulting exact (or dually, coexact) sequence is given by the
Puppe sequence In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone ...
.
Homogeneous spaces and spheres
There are many realizations of spheres as
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
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
giving the long exact sequence
which computes the low order homotopy groups of
for
since
is
-connected. In particular, there is a fibration
whose lower homotopy groups can be computed explicitly. Since
and there is the fibration
we have
for
Using this, and the fact that
which can be computed using the
Postnikov system
In 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 inverse system of topological spaces whose homotopy type at degree k agrees with the ...
, we have the long exact sequence
Since
we have
Also, the middle row gives
since the connecting map
is trivial. Also, we can know
has two-torsion.
= Application to sphere bundles
=
Milnor used the fact
to classify 3-sphere bundles over
in particular, he was able to find
exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...
s which are
smooth manifolds called
Milnor's spheres only homeomorphic to
not
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
. Note that any sphere bundle can be constructed from a
-
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
, which have structure group
since
can have the structure of an
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
.
Complex projective space
There is a fibration
where
is the unit sphere in
This sequence can be used to show the simple-connectedness of
for all
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
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in te ...
for the fundamental group and the
excision theorem for
singular homology and
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, 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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
. These are the so-called
aspherical space In 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 who ...
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
one needs much more advanced techniques than the definitions might suggest. In particular the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
was constructed for just this purpose.
Certain homotopy groups of
''n''-connected spaces can be calculated by comparison with
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
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 ...
.
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 ...
, 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 b ...
, a corollary of excision for homotopy groups.
Relative homotopy groups
There is also a useful generalization of homotopy groups,
called relative homotopy groups
for a pair
where ''A'' is a
subspace of
The construction is motivated by the observation that for an inclusion
there is an induced map on each homotopy group
which is not in general an injection. Indeed, elements of the kernel are known by considering a representative
and taking a based homotopy
to the constant map
or in other words
while the restriction to any other boundary component of
is trivial. Hence, we have the following construction:
The elements of such a group are homotopy classes of based maps
which carry the boundary
into ''A''. Two maps
are called homotopic relative to ''A'' if they are homotopic by a basepoint-preserving homotopy
such that, for each ''p'' in
and ''t'' in
the element
is in ''A''. Note that ordinary homotopy groups are recovered for the special case in which
is the singleton containing the base point.
These groups are abelian for
but for
form the top group of a
crossed module In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups
: d\colon H \longrightarro ...
with bottom group
There is also a long exact sequence of relative homotopy groups that can be obtained via the
Puppe sequence In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone ...
:
:
Related notions
The homotopy groups are fundamental to
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
, which in turn stimulated the development of
model categories
In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
. It is possible to define abstract homotopy groups for
simplicial sets.
Homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s 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 homology groups). Hence, it is sometimes said that "homology is a commutative alternative to homotopy".
Given a topological space
its ''n''-th homotopy group is usually denoted by
and its ''n''-th homology group is usually denoted by
See also
*
Fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
*
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
*
Hopf invariant
*
Knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
*
Homotopy class
*
Homotopy groups of spheres
*
Topological invariant
*
Homotopy group with coefficients In topology, a branch of mathematics, for i \ge 2, the ''i''-th homotopy group with coefficients in an abelian group ''G'' of a based space ''X'' is the pointed set of homotopy classes of based maps from the Moore space of type (G, i) to ''X'', an ...
*
Pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.
Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...
Notes
References
*
Ronald Brown, `Groupoids and crossed objects in algebraic topology',
Homology, Homotopy and Applications
''Homology, Homotopy and Applications'' is a 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 and Applicat ...
, 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.
* .
*
*
* .
*
*
*
{{Topology
Homotopy theory
cs:Homotopická grupa