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Difference cochain and chain

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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
How to Cite This Entry:
Difference cochain and chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Difference_cochain_and_chain&oldid=46652
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article