Difference between revisions of "Difference cochain and chain"
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− | + | A difference cochain is an [[Obstruction|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 | |
− | + | $$ | |
+ | [ d ^ {n} ( f, g)] \in \ | ||
+ | H ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y) ) | ||
+ | $$ | ||
− | 2) | + | 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==== | ====Comments==== | ||
− | |||
====References==== | ====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> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228</TD></TR></table> |
Latest revision as of 17:33, 5 June 2020
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 } .
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=15769