Difference cochain and chain

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

$$F \ \mathop{\rm on} ( X \times 0) \cup ( X ^ {n - 1 } \times I ) \cup ( X \times 1)$$

(where $I = [ 0, 1]$ and $X ^ {n}$ is the $n$- dimensional skeleton of $X$) between these mappings on the $( n - 1)$- dimensional skeleton. For every oriented $n$- dimensional cell $e ^ {n}$ of $X$, the restriction of $F$ to $\partial ( \overline{e}\; \times I)$ gives a mapping $S ^ {n} \rightarrow Y$( $S ^ {n}$ is the $n$- dimensional sphere) and hence an element of the group $\pi _ {n} ( Y)$. Thus there arises the cochain $d ^ {n} ( f, g) \in C ^ {n} ( X; \pi _ {n} ( Y))$( the notation $d _ {F} ^ {n} ( f, g)$ would be more precise), which is called the difference cochain; the cochain $d ^ {n} ( f, g)$ is an obstruction to the extension of $F$ to

$$( X \times 0 ) \cup ( X ^ {n} \times I ) \cup ( X \times 1) = \ ( X \times I) ^ {n - 1 } \cup ( X \times \{ 0, 1 \} ) .$$

The following statements hold: 1) $d ^ {n} ( f, g) = 0$ if and only if the homotopy between $f$ and $g$ can be extended to $X ^ {n}$; 2) the cochain

$$d ^ {n} ( f, g) \in \ C ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y))$$

is a cocycle; 3) the cohomology class

$$[ d ^ {n} ( f, g)] \in \ H ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y) )$$

vanishes if and only if there is a homotopy between $f$ and $g$ on $X ^ {n}$ that coincides with $F$ on $X ^ {n - 2 }$. Without loss of generality one can assume that $f$ and $g$ coincide on $X ^ {n - 1 }$ and that $F ( x, t) = f ( x) = g ( x)$ for $x \in X ^ {n - 2 }$. Then the following statements hold:

1) $d ^ {n} ( f, g) = - d ^ {n} ( g, f )$, in particular $d ^ {n} ( f, f ) = 0$;

2) $d ^ {n} ( f, g) + d ^ {n} ( g, h) = d ^ {n} ( f, h)$;

3) for any mapping $f: X \rightarrow Y$ and for any cochain $d \in C ^ {n} ( X; \pi _ {n} ( Y))$ there is a mapping $g$ for which $f \mid _ {X ^ {n - 1 } } = g \mid _ {X ^ {n - 1 } }$ and $d ^ {n} ( f, g) = d$.

Now suppose one is given two mappings $f, g: X ^ {n} \rightarrow Y$, $f \mid _ {X ^ {n - 1 } } = g \mid _ {X ^ {n - 1 } }$ and let $c _ {f} ^ {n + 1 }$ and $c _ {g} ^ {n + 1 }$ 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:

$$c _ {f} ^ {n + 1 } - c _ {g} ^ {n + 1 } = \ \delta d ^ {n} ( f, g).$$

Thus, if $g$ can be extended to $X ^ {n + 1 }$, then $[ c _ {f} ^ {n + 1 } ] = 0$ and if $[ c _ {f} ^ {n + 1 } ] = 0$, then $f \mid _ {X ^ {n - 1 } }$ can be extended to $X ^ {n + 1 }$.

References

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