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A difference cochain is an [[Obstruction|obstruction]] to the extension of a homotopy between mappings. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316601.png" /> be some cellular space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316602.png" /> be a simply-connected topological space and suppose, moreover, that one is given two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316603.png" /> and a homotopy
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316604.png" /></td> </tr></table>
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(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316606.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316607.png" />-dimensional skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316608.png" />) between these mappings on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d0316609.png" />-dimensional skeleton. For every oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166010.png" />-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166012.png" />, the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166014.png" /> gives a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166016.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166017.png" />-dimensional sphere) and hence an element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166018.png" />. Thus there arises the cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166019.png" /> (the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166020.png" /> would be more precise), which is called the difference cochain; the cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166021.png" /> is an obstruction to the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166022.png" /> to
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166023.png" /></td> </tr></table>
+
$$
 +
F \  \mathop{\rm on}  ( X \times 0) \cup
 +
( X ^ {n - 1 } \times I ) \cup ( X \times 1)
 +
$$
  
The following statements hold: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166024.png" /> if and only if the homotopy between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166026.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166027.png" />; 2) the cochain
+
(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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166028.png" /></td> </tr></table>
+
$$
 +
( X \times 0 ) \cup ( X  ^ {n} \times I ) \cup ( X \times 1)  = \
 +
( X \times I) ^ {n - 1 } \cup ( X \times \{ 0, 1 \} ) .
 +
$$
  
is a cocycle; 3) the cohomology class
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166029.png" /></td> </tr></table>
+
$$
 +
d  ^ {n} ( f, g)  \in \
 +
C  ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y))
 +
$$
  
vanishes if and only if there is a homotopy between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166032.png" /> that coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166034.png" />. Without loss of generality one can assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166036.png" /> coincide on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166037.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166039.png" />. Then the following statements hold:
+
is a cocycle; 3) the cohomology class
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166040.png" />, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166041.png" />;
+
$$
 +
[ d ^ {n} ( f, g)]  \in \
 +
H  ^ {n} ( X \times I, X \times \{ 0, 1 \} ; \pi _ {n} ( Y) )
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166042.png" />;
+
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:
  
3) for any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166043.png" /> and for any cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166044.png" /> there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166045.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166047.png" />.
+
1) d ^ {n} ( f, g) = - d ^ {n} ( g, f  ) $,
 +
in particular  $  d ^ {n} ( f, f  ) = 0 $;
  
Now suppose one is given two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166049.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166051.png" /> 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:
+
2)  $  d ^ {n} ( f, g) + d ^ {n} ( g, h) = d ^ {n} ( f, h) $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166052.png" /></td> </tr></table>
+
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 $.
  
Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166053.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166055.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166057.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031660/d03166058.png" />.
+
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
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