Difference cochain and chain
A difference cochain is an obstruction to the extension of a homotopy between mappings. Let be some cellular space, let
be a simply-connected topological space and suppose, moreover, that one is given two mappings
and a homotopy
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(where and
is the
-dimensional skeleton of
) between these mappings on the
-dimensional skeleton. For every oriented
-dimensional cell
of
, the restriction of
to
gives a mapping
(
is the
-dimensional sphere) and hence an element of the group
. Thus there arises the cochain
(the notation
would be more precise), which is called the difference cochain; the cochain
is an obstruction to the extension of
to
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The following statements hold: 1) if and only if the homotopy between
and
can be extended to
; 2) the cochain
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is a cocycle; 3) the cohomology class
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vanishes if and only if there is a homotopy between and
on
that coincides with
on
. Without loss of generality one can assume that
and
coincide on
and that
for
. Then the following statements hold:
1) , in particular
;
2) ;
3) for any mapping and for any cochain
there is a mapping
for which
and
.
Now suppose one is given two mappings ,
and let
and
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:
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Thus, if can be extended to
, then
and if
, then
can be extended to
.
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