, homotopy groups are used in algebraic topology
to classify topological space
s. The first and simplest homotopy group is the fundamental group
, which records information about loop
s in a space
. 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
) into a given space (with base point) are collected into equivalence class
es, called homotopy class
es. 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
), but topological spaces that ''are not'' homeomorphic ''can'' have the same homotopy groups.
The notion of homotopy of path
s was introduced by Camille Jordan
In modern mathematics it is common to study a category
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
s to topological spaces.
That link between topology and groups lets mathematicians apply insights from group theory
. For example, if two topological objects have different homotopy groups, they can't have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus
is different from the sphere
: 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
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
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
to the sphere.
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, we can define π''n''
(''X'') 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''.
240px|Composition in the fundamental group
, the homotopy classes form a group
. To define the group operation, recall that in the fundamental group
, 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
composed with ''h'', where
is the map from
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.
. Further, similar to the fundamental group, for a path-connected space
any two choices of basepoint give rise to isomorphic
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 groupoid
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'' → ''B'' be a basepoint-preserving Serre fibration
with fiber ''F'', that is, a map possessing the homotopy lifting property
with respect to CW complex
es. Suppose that ''B'' is path-connected. Then there is a long exact sequence
of homotopy groups
Here the maps involving π0
are not group homomorphism
s because the π0
are not groups, but they are exact in the sense that the image
equals the kernel
Example: the Hopf fibration
. Let ''B'' equal ''S''2
and ''E'' equal ''S''3
. Let ''p'' be the Hopf fibration
, which has fiber ''S''1
. From the long exact sequence
and the fact that π''n''
) = 0 for ''n'' ≥ 2, we find that π''n''
) = π''n''
) for ''n'' ≥ 3. In particular,
In the case of a cover space, when the fiber is discrete, we have that π''n''
(''E'') is isomorphic to π''n''
(''B'') for ''n'' > 1, that π''n''
(''E'') embeds injective
ly into π''n''
(''B'') for all positive ''n'', and that the subgroup
(''B'') that corresponds to the embedding of π1
(''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 space
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
-connected. In particular, there is a fibration
whose lower homotopy groups can be computed explicitly. Since
, and there is the fibration
. Using this, and the fact that
, which can be computed using the Postnikov system
, we have the long exact sequence
. Also, the middle row gives
since the connecting map
is trivial. Also, we can know
= Application to sphere bundles
Milnor used the fact
to classify 3-sphere bundles over
, in particular, he was able to find Exotic sphere
s which are smooth manifold
s called Milnor's spheres
only homeomorphic to
, not diffeomorphic
. Note that any sphere bundle can be constructed from a
, which have structure group
can have the structure of an oriented Riemannian manifold
Complex projective space
There is a fibration
is the unit sphere in
. This sequence can be used to show the simple-connectedness of
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
, 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 space
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 sequence
was constructed for just this purpose.
Certain homotopy groups of ''n''-connected
spaces can be calculated by comparison with homology group
s 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,
, called relative homotopy groups
for a pair
, where ''A'' is a subspace
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 ''f, g'' are called homotopic relative to ''A'' if they are homotopic by a basepoint-preserving homotopy ''F'' : ''Dn
'' × , 1
→ ''X'' such that, for each ''p'' in ''S''''n''−1
and ''t'' in , 1
the element ''F''(''p'', ''t'') is in ''A''. Note that ordinary homotopy groups are recovered for the special case in which
is the base point.
These groups are abelian for ''n'' ≥ 3 but for ''n'' = 2 form the top group of a crossed module
with bottom group π1
There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence
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 set
s are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are usually not commutative
, 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
, and its ''n''-th homology group is usually denoted by
*Homotopy groups of spheres
*Homotopy group with coefficients
* Ronald Brown
, `Groupoids and crossed objects in algebraic topology', Homology, Homotopy and Applications
, 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.