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 $ \pi ^ {n} ( X) = [ X , S ^ {n} ] $ be the set of homotopy classes of continuous mappings from a pointed space $ X $ to the pointed sphere. The set $ \pi ^ {n} ( X) $ does not always have a natural group structure. (This is the case only for $ n = 1 , 3 $, since $ S ^ {n} $ is then a group.) The group $ \pi ^ {1} ( X) $ is the same as $ H ^ {1} ( X , \mathbf Z ) $.

If $ X $ is a CW-complex of dimension at most $ 2 n - 2 $, then a group structure can be defined on $ \pi ^ {n} ( X) $ in the following way. For $ [ \alpha ] , [ \beta ] \in \pi ^ {n} ( X) $ one considers the mapping

$$ ( \alpha \times \beta ) \circ \Delta : \ X \rightarrow S ^ {n} \times S ^ {n} , $$

where $ \Delta : X \rightarrow X \times X $ is the diagonal mapping and $ \alpha , \beta : X \rightarrow S ^ {n} $ are representatives of the classes $ [ \alpha ] , [ \beta ] $. In view of the restriction on the dimension of $ X $ there is a unique homotopy class of mappings $ f : X \rightarrow S ^ {n} \lor S ^ {n} $( here $ S ^ {n} \lor S ^ {n} $ is a bouquet of pointed spheres) the composite of which with the natural inclusion $ S ^ {n} \lor S ^ {n} \subset S ^ {n} \times S ^ {n} $ is the same as the homotopy class of $ ( \alpha \times \beta ) \circ \Delta $. The homotopy class $ [ \theta \circ f ] \in \pi ^ {n} ( X) $ of $ \theta \circ f : X \rightarrow S ^ {n} $, where $ \theta : S ^ {n} \lor S ^ {n} \rightarrow S ^ {n} $ is the folding mapping, is set equal to $ [ \alpha ] + [ \beta ] \in \pi ^ {n} ( X) $. With respect to this operation the set $ \pi ^ {n} ( X) $ is an Abelian group; therefore, the functor $ \pi ^ {n} $ is often regarded as a functor defined only on the category of CW-complexes of dimension at most $ 2 n - 2 $, with values in the category of Abelian groups. For CW-complexes $ X $ of dimension less than $ n $, $ \pi ^ {n} ( X) = 0 $. Thus, the functor $ \pi ^ {n} $ is of interest in dimensions from $ n $ to $ 2 n - 2 $, that is, in the so-called stable dimensions.

If $ \mathop{\rm dim} X \leq 2 n - 2 $, then $ \pi ^ {n} ( X) \approx \pi ^ {n+} 1 ( S X ) $, where $ S X $ is the suspension of $ X $. This isomorphism is given by the suspension functor: $ [ X , S ^ {n} ] \rightarrow [ S X , S S ^ {n} ] = [ S X , S ^ {n+} 1 ] $. If $ X $ is an arbitrary finite-dimensional CW-complex, then for sufficiently large $ N $ the set $ \pi ^ {n+} N ( S ^ {N} X ) $ has a group structure (for $ N \geq \mathop{\rm dim} X - 2 n + 2 $ one has $ \mathop{\rm dim} ( S ^ {n} X ) = N + \mathop{\rm dim} X \leq 2 ( n + N ) - 2 $). The group $ \pi _ {S} ^ {n} ( X) = \pi ^ {n+} N ( S ^ {N} X ) $ with $ N \geq \mathop{\rm dim} X - 2 n + 2 $ is called the stable cohomotopy group of the CW-complex. The groups $ \pi _ {S} ^ {n} ( X) $ are defined for all integer $ n $( and not merely positive integers). If one chooses for $ X $ two points (one of which is distinguished), then $ \pi _ {S} ^ {n} ( X) = 0 $ for $ n \geq 0 $, $ \pi _ {S} ^ {0} ( x) = \mathbf Z $, and $ \pi _ {S} ^ {n} ( X) = \pi _ {N-} n ( S ^ {N} ) $ are the stable homotopy groups of spheres for $ n < 0 $.

If $ ( X ; A ) $ is a pair of CW-complexes of dimension $ m $, then when $ m \leq 2 n - 2 $, the relative cohomotopy group $ \pi ^ {n} ( X , A ) = \pi ^ {n} ( X / A ) $ is defined. One has the following exact sequence of Abelian groups:

$$ \pi ^ {i} ( X) \rightarrow \ \pi ^ {i} ( A) \rightarrow \ \pi ^ {i+} 1 ( X , A ) \rightarrow \pi ^ {i+} 1 ( X) \rightarrow $$

$$ \rightarrow \ \pi ^ {i+} 1 ( A) \rightarrow \pi ^ {i+} 2 ( X , A ) \rightarrow \dots , $$

extending indefinitely to the right; however, from some term onwards all groups are trivial: $ \pi ^ {i} ( X , A ) = \pi ^ {i} ( X) = \pi ^ {i} ( A) = 0 $ when $ i > m $. This sequence extends to the left only as far as those values of $ i $ for which $ m \leq 2 i - 2 $. In this sequence the homomorphisms $ \pi ^ {i} ( X) \rightarrow \pi ^ {i} ( A) $ and $ \pi ^ {i} ( X / A ) \rightarrow \pi ^ {i} ( X) $ are induced by the natural mappings $ A \subset X $ and $ X \rightarrow X / A $. The homomorphism $ \pi ^ {i} ( A) \rightarrow \pi ^ {i+} 1 ( X / A ) $ is constructed as follows. For a class $ [ f ] \in \pi ^ {i} ( A) = [ A , S ^ {i} ] $ and a representative $ f : A \rightarrow S ^ {i} $ of it, one chooses an extension $ F : X \rightarrow D ^ {i+} 1 $ of $ f $ defined on the subspace $ A \subset X $ with values in $ S ^ {i} \subset D ^ {i+} 1 $. The mapping $ F $ induces a mapping $ X / A \rightarrow D ^ {i+} 1 / S ^ {i} = S ^ {i+} 1 $, the homotopy class of which (an element of $ \pi ^ {i+} 1 ( X , A ) $) is put in correspondence with the class $ [ f ] \in \pi ^ {i} ( A) $.

If $ ( X , A ) $ is a pair of pointed CW-complexes of finite dimension, then there is the exact sequence of stable cohomotopy groups

$$ \dots \rightarrow \pi _ {S} ^ {i} ( X) \rightarrow \pi _ {S} ^ {i} ( A) \rightarrow \pi _ {S} ^ {i+} 1 ( X , A ) \rightarrow \pi _ {S} ^ {i+} 1 ( X) \rightarrow \dots , $$

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 $ X $, let $ \pi _ {S} ^ {i} ( X) = \pi _ {S} ^ {i} ( X \cup x _ {0} , x _ {0} ) $, where $ ( X \cup x _ {0} , x _ {0} ) $ is the pointed CW-complex obtained as the disjoint union of $ X $ with a distinguished point. The functor $ \pi _ {S} ^ {*} $, defined on the category of finite-dimensional CW-complexes, provides a generalized cohomology theory by setting

$$ \pi _ {S} ^ {*} ( X , A ) = \ {\pi _ {S} ^ {*} } tilde ( X / A ) = \ \mathop{\rm Ker} [ \pi _ {S} ^ {*} ( X / A ) \rightarrow \pi _ {S} ^ {*} ( \mathop{\rm pt} ) ] . $$

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)