Homotopy group
A generalization of the fundamental group, proposed by W. Hurewicz [1] in the context of problems on the classification of continuous mappings. Homotopy groups are defined for any . For
the homotopy group is identical with the fundamental group. The definition of homotopy groups is not constructive and for this reason their computation is a difficult task, general methods for which were developed only in the 1950s. Their importance is due to the fact that all problems in homotopy theory can be reduced (cf. Homotopy type), to a greater or lesser extent, to the computation of certain homotopy groups.
Let
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be the dimensional unit cube, let be its face
, and let
be the union of its remaining faces. For any pointed pair
(cf. Pointed object) the symbol
(or simply
) denotes the pointed set of all homotopy classes (cf. Homotopy) of mappings
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The distinguished (zero) element of this set is the constant mapping that maps the whole cube into
. Any continuous mapping
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induces a morphism
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of pointed sets. For any the sets
and the morphisms
constitute a functor
from the category of pointed pairs into the category of pointed sets. This functor is homotopy invariant, i.e.
if
and
are homotopic (as mappings of pointed pairs). Furthermore, it is normalized in the sense that if
, then
.
For it is possible to introduce into the set
an operation of addition, with respect to which it becomes a group (if
even an Abelian group). By definition, if
and
, then
, where
is the mapping
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defined by the formula
![]() | (1) |
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The resulting group is said to be the
-th homotopy group (or the
-dimensional homotopy group) of the pointed pair
; one also speaks of the homotopy group of the pair
at
or of the homotopy group of the space
with respect to the subspace
at
. The mappings
are homomorphisms of these groups. Thus, if
it may be assumed that the function
takes values in the category of groups (if
even in the category of Abelian groups).
For the group
is denoted by
, or simply by
, and is called the absolute homotopy group of the pointed space
(or of the space
at
). Its elements are the homotopy classes of mappings
, where
is the boundary of the cube
. For such mappings formula (1) is meaningful for
as well, and so
is a group. This group coincides with the classical fundamental group. The group operation in
is usually called multiplication. This group is, generally speaking, non-Abelian, while the group
is Abelian. For any
the groups
and the corresponding homomorphisms form a functor from the category of pointed spaces into the category of groups (if
into the category of Abelian groups). This functor is the composition
of the imbedding functor
and the functor
described above.
The functor is extended to include the case
, where
is the pointed set of path-components (cf. Path-connected space) of
; the zero of this set is the component containing
. The set
is not defined for
. In order to simplify the formulations, the sets
and
are usually also called homotopy groups, even though they are not groups in general.
For each element the mapping
represents a mapping
, and thus defines a certain element of the homotopy group
. This element depends only on
and is denoted by the symbol
. The resulting mapping
is a morphism of pointed sets (if
a homomorphism of groups) and is called a boundary homomorphism or a boundary operator. The boundary homomorphism, together with the homomorphisms
and
induced by the imbeddings
and
, makes it possible to write down a sequence of groups and homomorphisms, infinite from the left:
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This is an exact sequence; it is called the exact homotopy sequence of the pair and is usually denoted by
. If
for all
, then the homomorphism
is an isomorphism (also for all
).
The boundary homomorphism is natural, that is, it is a morphism of the functor
into the functor
(more exactly, into the functor
where
). This makes it possible to define
as a functor that takes values in the category of exact sequences of pointed sets which, except for the last six sets, are Abelian groups and, except for the last three sets, are groups.
Let be an arbitrary fibration in the sense of Serre and let
,
,
, and
. The mapping
defines a mapping
of pointed pairs. For any
the induced homomorphism
is an isomorphism. In particular, this is true for
. In the latter case the formula
unambiguously defines a homomorphism
where
is the fibre of
over
. This homomorphisms is called the homotopy transgression. It occurs in the exact sequence
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This sequence is called the homotopy sequence of the fibration . Putting a fibration into correspondence with its homotopy sequence yields a functor on the category of all (pointed) fibrations.
In the particular case when is the standard Serre fibration of paths over a space
, for any
one has the isomorphism
, where
is the loop space of
. This isomorphism is called the Hurewicz isomorphism.
The above properties actually unambiguously define the homotopy groups , i.e. may be taken as axioms which describe these groups. In fact, let
be an arbitrary sequence of homotopy-invariant normalized functors, defined on the category of pointed spaces, taking values in the category of pointed sets, and having the following property: For any fibration in the sense of Serre
, any subset
and any point
, the induced homomorphism
is an isomorphism. Such a sequence is called a homotopy system if for any
there is defined a morphism
of the functor
into the functor
(if
, into
) that is an isomorphism for any pointed pair
for which
for all
. Any homotopy system is isomorphic to the homotopy system constructed above, which consists of homotopy groups. Furthermore, if
, a group structure can be uniquely introduced into the pointed sets
(and also into the sets
) so that all morphisms
are homomorphism (this structure accordingly corresponds to that described by formula (1)). On the other hand, the sets
if
and
carry only the inverse group operation. All this means that the above properties unambiguously define the homotopy groups (up to the order of multiplication in non-commutative groups).
For any mapping and any path
connecting two points
and
, the formula
, defines a homotopy of
. By the homotopy extension axiom (cf. Cofibration) this homotopy can be extended to a homotopy
for which
. The final mapping
of this homotopy maps
into
, i.e. represents a mapping
. The corresponding element of the homotopy group depends only on the class
of
and the homotopy class
of
, and is denoted by the symbol
(if
, by the symbol
). The family
is thus defined as a local family on the space
, i.e. on the fundamental groupoid of this space. In particular, for any point
the group
operates on
. If
these operators act as inner automorphisms:
, and if
they make the group
into a
-module. For any continuous mapping
the induced homomorphisms
are operator homomorphisms (homomorphisms of modules):
.
In a similar way, the groups ,
, constitute a local family of homotopy groups on the subspace
. In particular, the group
operates on the homotopy group
so that if
the group
is a
-module. The group
is said to be a crossed
-module (cf. Crossed modules), where
is the boundary homomorphism.
The group acts as a group of operators not only on the groups
but also on the groups
, and also, by virtue of the natural homomorphism
, on the groups
. With respect to these actions of
all homomorphisms of the exact sequence
are operator homomorphisms, so that
can be regarded as a group of operators on the sequence
. This is equivalent to saying that the sequences
,
, constitute a local family of exact sequences of the subspace
.
If the complement is represented as a union of disjoint open
-dimensional cells, then the
-module
is a free module (if
, a free crossed module) and has a system of free generators — a basis in bijective (not necessarily natural) correspondence with the cells of
(Whitehead's theorem).
The mappings are in bijective correspondence with the mappings
, where
is an
-dimensional sphere and
is some point on it. For this reason the elements of
can be regarded as the homotopy classes of mappings
. This is also true if
. The above identification depends on the selection of some relative homeomorphism
. It is common to select and fix the sphere
and the homeomorphism
once and for all. In the original definition of Hurewicz, which is not frequently used nowadays,
was not fixed, while
was given up to a homotopy. Such a specification of
is equivalent to specifying an orientation on
. Thus, according to Hurewicz, the elements of
are pointed homotopy classes of mappings of an oriented
-dimensional sphere into
. The set
of non-pointed homotopy classes of mappings
is in bijective correspondence with the orbits of the action of
on
(cf. Orbit). If
(or, more generally, if
acts trivially on
), then
is said to be homotopically
-simple. In this case
is independent of
(so that the notation
is fully justified). This group is naturally identified with the set
, which, as a consequence, has a group structure. A space that is homotopically
-simple for all
is said to be Abelian.
Let be the orientation class of the sphere
and let
,
. This defines a homomorphism
, the so-called Hurewicz homomorphism. Its kernel contains all elements of the form
,
,
(if
, all elements of the form
, i.e. it contains the commutator
of
). Poincaré's classical theorem states that for
the kernel of
coincides with the commutator
, so that the group
is isomorphic to the Abelianization of the fundamental group
. Hurewicz's theorem, which is a generalization of Poincaré's theorem to the case
, states that if
for
, then the homomorphism
is an isomorphism (and the homomorphism
is an epimorphism).
In a similar way, the elements of can be regarded as (pointed) homotopy classes of mappings
, where
is an (oriented)
-dimensional ball and
is its boundary. If the pair
is homotopically
-simple (i.e. if
acts trivially on
), then the requirement of pointedness may be dropped in this definition. The formula
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where is the orientation class of the pair
and
defines the Hurewicz homomorphism
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If and
for
, this homomorphism is an isomorphism (Hurewicz's theorem for relative groups).
Two principal methods are known for the computation of the homotopy groups of specific spaces: the method of killing spaces (cf. Killing space) and the method of homotopy resolutions (cf. Homotopy type; Postnikov system). The first method is based on the isomorphism , which follows from Hurewicz's theorem and the definition of the killing space
. This isomorphism reduces the computation of
to the problem of computing the homology groups
. The space
fibres over the space
with fibre
, and the homology groups of the space
are known. Therefore one may try to find the lower homology groups of killing spaces by induction. The problem of computing the homology groups of a fibre space from the homology groups of its base and fibre is still not completely solved in its general formulation (and, obviously, a general satisfactory solution does not exist). However, extensive information on the homology groups of the spaces
can be extracted from the corresponding Serre spectral sequence. In many cases this information is sufficient for the computation of
, at least for some
. An essential technical simplification of the problem is obtained on the basis of the Serre's theory of classes of Abelian groups and the
-approximation derived from it. With this theory it is possible to compute entirely in the cohomology and only for the coefficient groups
. The geometric principles on which this technique is based were first clarified by J.F. Adams and D. Sullivan on the basis of the concept of localization of topological spaces at a given prime number
.
The second (also inductive) method of computing homotopy groups consists of a stepwise construction of the homotopy resolution of the space . Suppose the
-th term of this resolution is known (e.g. if
, then
). The next term must be the fibre space over
with fibre
; moreover, the group
must be isomorphic to the known group
. This gives (on the basis of the corresponding spectral sequence) definite information on the group
, which, in many cases, makes it possible to compute it completely. For example, for
by this method all groups
,
, can be found. In its modern form, this method is also based on the concept of localization.
The method of homology resolutions was extended (cf. [4]) to an algorithm that is applicable to any simply-connected finite -complex and that gives all its homotopy groups. However, for practical use this algorithm is too complicated.
Since the homotopy theory is completely equivalent to the homotopy theory of simplicial sets, the definition of a homotopy group may be transferred to any (complete) simplicial set. The "combinatorial" definition obtained (due to D. Kan) can easily be extended to an algorithm. However, this algorithm is also too complicated for practical use.
From any of the above methods it is easy to establish that the homotopy groups of a simply-connected space having finitely-generated homology groups, are also finitely generated. The analogous statement for non-simply connected spaces (i.e. its homology groups should be finitely generated as -modules) is, in general, not true.
Let be the (reduced) suspension functor, and let
be the loop functor. Since these functors are adjoint, the identity mapping
defines an imbedding
, for any
. Since
, this imbedding defines a homomorphism
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which is known as the suspension homomorphism. It coincides with the homomorphism obtained by assigning to an arbitrary (pointed) mapping its suspension
. This homomorphism occurs in an exact sequence:
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This sequence is called the suspension sequence of the space . The homomorphism
in it is a generalization of the classical Hopf invariant.
If is a countable CW-complex with one vertex, the space
may be replaced by the infinite reduced product
of the complex
. This shows that if
for
, then
is an isomorphism for all
and an epimorphism if
. This theorem is known as Freudenthal's suspension theorem (H. Freudenthal first published the proof for the case
, although the theorem was known much earlier.)
Freudenthal's theorem shows that for the group
is independent of
. It is called the
-th stable homotopy group of the sphere (cf. also Stable homotopy group). Similar stabilization phenomena occur for the homotopy groups of the orthogonal groups, of the Thom spaces
(cf. Thom space) and in many other cases. The general study of these phenomena is most conveniently done within the framework of the so-called theory of spectra. In this theory stable homotopy groups arise as the homotopy groups of spectra. These groups have an essentially simpler structure than the homotopy groups of a space and their study (and computation) is an easier task. For example, for the computation of these groups one has a special device: the Adams spectral sequence.
Homotopy groups have been generalized in various directions. For example, an attempt was made to replace the spheres by other spaces. Here one may note toroidal homotopy groups, obtained by interpreting the Whitehead product as a commutator. It was also shown that the set of homotopy classes of mappings admits a group operation which is natural with respect to
if and only if
is a co-
-space. Homotopy groups with coefficients were obtained by replacing the spheres
by the Moore spaces
(cf. Moore space). This definition of homotopy groups with coefficients was not very successful. A more satisfactory definition (compatible with the general Eckmann–Hilton duality principle) was obtained by replacing the Moore
-spaces by co-
-spaces. However, these homotopy groups were not defined for all
(e.g. for
the additive group of real numbers, these groups are not defined).
The question of the construction of homotopy groups in categories other than the category of pointed pairs has been studied in detail. First of all one has to mention the homotopy groups of a triad (cf. Triads, see, e.g., [3]), which were very useful in the study of the homomorphism . A very general construction of homotopy groups was proposed in connection with studies on duality. On the basis of the concept of a standard construction (see [6]) the construction of homotopy groups was transferred to arbitrary categories. A fundamental role in this construction is played by the homotopy groups of simplicial sets mentioned earlier.
References
[1] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) |
[2] | V.G. Boltyanskii, "The homotopy theory of continuous mapping and vector fields" , Moscow (1955) (In Russian) |
[3] | S.-T. Hu, "Homotopy theory" , Acad. Press (1959) |
[4] | E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20 |
[5] | D. Kan, "A combinatorial definition of homotopy groups" Ann. of Math. (2) , 67 (1958) pp. 282–313 |
[6] | J. Stallings, "A finitely presented group whose 3-dimensional integral homology is not finitely generated" Amer. J. Math. , 85 (1963) pp. 541–543 |
[7a] | B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. , 246 (1958) pp. 2444–2447 |
[7b] | B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. , 246 (1958) pp. 2555–2558 |
[7c] | B. Eckmann, P. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. , 246 (1958) pp. 2991–2993 |
[7d] | B. Eckmann, P. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. , 247 (1958) pp. 620–623 |
[7e] | B. Eckmann, P. Hilton, "Décomposition homologique d'une polyèdre simplement connexe" C.R. Acad. Sci. , 248 (1959) pp. 2054–2056 |
[8] | D. Sullivan, "Geometric topology" , M.I.T. (1971) (Notes) |
Comments
Although W. Hurewicz
was the first to study the higher homotopy groups in detail, the definition was in fact suggested a few years earlier by E. Čech [a2]. The action of the fundamental group on the higher homotopy groups was first studied by S. Eilenberg [a3]. A good general reference for homotopy groups is [a4].
The stable homotopy groups form a generalized homology theory, i.e. a theory which satisfies all the Eilenberg–Steenrod axioms except possibly the dimension axiom. This theory is in fact defined by the spectrum of spheres , cf. Spectrum of spaces. The corresponding generalized cohomology theory (cf. Generalized cohomology theories) defined by this spectrum consists of the cohomotopy group. Cf., e.g., [a4] and [a11] for more details. Powerful tools for computing the stable homotopy groups of the spheres (besides the (classical) Adams spectral sequence) involve the Adams–Novikov spectral sequence, the so-called chromatic spectral sequence and complex cobordism, cf. [a12].
References
[a1a] | W. Hurewicz, "Beiträge zur Topologie der Deformationen I-II" Proc. Ned. Akad. Weten. Ser. A , 38 (1935) pp. 112–119; 521–528 |
[a1b] | W. Hurewicz, "Beiträge zur Topologie der Deformationen III-IV" Proc. Ned. Akad. Weten. Ser. A , 39 (1936) pp. 117–126; 215–224 |
[a2] | E. Čech, "Höherdimensionale Homotopiegruppen" , Verh. Intern. Mathematikerkongress Zürich, 1932 , O. Füssli (1932) pp. 203 |
[a3] | S. Eilenberg, "On the relation between the fundamental group of a space and the higher homotopy groups" Fund. Math. , 32 (1939) pp. 167–175 |
[a4] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 23; 415–455 |
[a5] | B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. §12 |
[a6] | P.J. Hilton, "An introduction to homotopy theory" , Cambridge Univ. Press (1953) |
[a7] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1960) |
[a8] | D. Sullivan, "Genetics of homotopy theory and the Adams conjecture" Ann. of Math. , 100 (1974) pp. 1–79 |
[a9] | D.G. Quillen, "Homotopical algebra" , Springer (1967) |
[a10] | B. Eckmann, "Homotopie et dualité" , Coll. Topol. Algébrique Louvain, 1956 , Masson (1957) pp. 41–53 |
[a11] | R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) |
[a12] | D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986) |
Homotopy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy_group&oldid=14008