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
 |
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:
 |
 |
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
 |
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
 |
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