Cohomotopy group
One of the generalizations of the one-dimensional cohomology group; a concept which is, in a certain sense, dual to that of homotopy group.
Let be the set of homotopy classes of continuous mappings from a pointed space
to the pointed sphere. The set
does not always have a natural group structure. (This is the case only for
, since
is then a group.) The group
is the same as
.
If is a CW-complex of dimension at most
, then a group structure can be defined on
in the following way. For
one considers the mapping
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where is the diagonal mapping and
are representatives of the classes
. In view of the restriction on the dimension of
there is a unique homotopy class of mappings
(here
is a bouquet of pointed spheres) the composite of which with the natural inclusion
is the same as the homotopy class of
. The homotopy class
of
, where
is the folding mapping, is set equal to
. With respect to this operation the set
is an Abelian group; therefore, the functor
is often regarded as a functor defined only on the category of CW-complexes of dimension at most
, with values in the category of Abelian groups. For CW-complexes
of dimension less than
,
. Thus, the functor
is of interest in dimensions from
to
, that is, in the so-called stable dimensions.
If , then
, where
is the suspension of
. This isomorphism is given by the suspension functor:
. If
is an arbitrary finite-dimensional CW-complex, then for sufficiently large
the set
has a group structure (for
one has
). The group
with
is called the stable cohomotopy group of the CW-complex. The groups
are defined for all integer
(and not merely positive integers). If one chooses for
two points (one of which is distinguished), then
for
,
, and
are the stable homotopy groups of spheres for
.
If is a pair of CW-complexes of dimension
, then when
, the relative cohomotopy group
is defined. One has the following exact sequence of Abelian groups:
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extending indefinitely to the right; however, from some term onwards all groups are trivial: when
. This sequence extends to the left only as far as those values of
for which
. In this sequence the homomorphisms
and
are induced by the natural mappings
and
. The homomorphism
is constructed as follows. For a class
and a representative
of it, one chooses an extension
of
defined on the subspace
with values in
. The mapping
induces a mapping
, the homotopy class of which (an element of
) is put in correspondence with the class
.
If is a pair of pointed CW-complexes of finite dimension, then there is the exact sequence of stable cohomotopy groups
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extending indefinitely in both directions. This circumstance enables one to convert the stable cohomotopy groups into a generalized cohomology theory. For an arbitrary (non-pointed) finite-dimensional CW-complex , let
, where
is the pointed CW-complex obtained as the disjoint union of
with a distinguished point. The functor
, defined on the category of finite-dimensional CW-complexes, provides a generalized cohomology theory by setting
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The value at a point of this theory is the same as the stable homotopy groups of spheres.
As for homotopy groups, the cohomotopy groups cannot be explicitly calculated even in the simplest cases, and this severely restricts the possibility of practical application of the above functors.
References
[1] | S.-T. Hu, "Homotopy theory" , Acad. Press (1959) |
[2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
Comments
References
[a1] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) |
Cohomotopy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomotopy_group&oldid=46395