Difference cochain and chain
A difference cochain is an obstruction to the extension of a homotopy between mappings. Let
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
[a1] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228 |
Difference cochain and chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Difference_cochain_and_chain&oldid=46652