Difference cochain and chain

A difference cochain is an obstruction to the extension of a homotopy between mappings. Let be some cellular space, let be a simply-connected topological space and suppose, moreover, that one is given two mappings and a homotopy

(where and is the -dimensional skeleton of ) between these mappings on the -dimensional skeleton. For every oriented -dimensional cell of , the restriction of to gives a mapping ( is the -dimensional sphere) and hence an element of the group . Thus there arises the cochain (the notation would be more precise), which is called the difference cochain; the cochain is an obstruction to the extension of to

The following statements hold: 1) if and only if the homotopy between and can be extended to ; 2) the cochain

is a cocycle; 3) the cohomology class

vanishes if and only if there is a homotopy between and on that coincides with on . Without loss of generality one can assume that and coincide on and that for . Then the following statements hold:

1) , in particular ;

2) ;

3) for any mapping and for any cochain there is a mapping for which and .

Now suppose one is given two mappings , and let and be the obstructions to the extensions of the corresponding mappings. The role of the difference cochain in the theory of obstructions is explained by the following proposition:

Thus, if can be extended to , then and if , then can be extended to .

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228