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A difference cochain is an [[Obstruction|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 } $.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,   "Elements of homotopy theory" , Springer  (1978)  pp. 228</TD></TR></table>