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)

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978)