Obstruction
A concept in homotopy theory: An invariant that equals zero if a (step in a) corresponding problem is solvable and is non-zero otherwise.
Let be a pair of cellular spaces (cf. Cellular space) and let
be a simply-connected (more generally, a homotopy-simple) topological space. Can one extend a given continuous mapping
to a continuous mapping
? The extension can be attempted recursively, over successive skeletons
of
. Suppose one has constructed a mapping
such that
. For any oriented
-dimensional cell
the mapping
gives a mapping
(where
is the
-dimensional unit sphere) and an element
(it is here that one uses that
is homotopy simple, which allows one to ignore the base point). This defines a cochain
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Since for one clearly has
, it follows that
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Clearly if and only if
can be extended to
, i.e.
is an obstruction to extending
to
.
The cochain is a cocycle. The fact that
does not, in general, imply that
cannot be extended to
: It is possible that
cannot be extended to
because of an unsuccessful choice of an extension of
to
. It may turn out that, e.g., the mapping
can be extended to
, i.e. that extension is possible by skipping back one step. It can be shown that the cohomology class
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is an obstruction to this, i.e. if and only if there is a mapping
such that
(in particular,
). The construction of difference chains and cochains is used in the proof of this statement (cf. Difference cochain and chain).
Since the problem of homotopy classification of mappings can be interpreted as an extension problem, obstruction theory is applicable also to the description of the set
of homotopy classes of mappings from
into
. Let
and let
be a subspace of
. Then a pair of mappings
is interpreted as a mapping
,
,
, and the presence of a homotopy between
and
means the presence of a mapping
extending
. If the homotopy
has been constructed on the
-dimensional skeleton of
, then the obstruction to its extension to
is the difference cochain
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As an application one may consider the description of the set ,
, where
is the Eilenberg–MacLane space:
for
;
. Let
be a constant mapping and
an arbitrary continuous mapping. Since
for
, the mappings
and
are homotopic on
and, after having chosen such a homotopy, one can define the difference cochain
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The cohomology class is well-defined, i.e. does not depend on the choice of a homotopy between
and
(since
for
). Further, if two mappings
are such that
, then
, and hence
and
are homotopic on
. The obstructions to extending this homotopy to
lie in the groups
(since
), and hence
and
are homotopic. Thus, the homotopy class of
is completely determined by the element
. Finally, for any
there is a mapping
with
, hence
. Similarly, if
for
and if
, then
.
In studying extension problems one has considered the possibility of extending "by skipping back one step" . A complete solution of the problem requires the analysis of the possibility of skipping back an arbitrary number of steps. Cohomology operations (cf. Cohomology operation) and Postnikov systems (cf. Postnikov system) are used to this end. E.g., in order to describe the set , where
for
,
,
, it is required, in general, to study the possibility of skipping back
steps, for which it is necessary to study the first
levels of the Postnikov system for
, i.e. to use cohomology operations of orders
(in the article Cohomology operation this problem is outlined for
).
The theory of obstructions is also used in the more general situation of extension of sections (cf. Section of a mapping). Let be a fibration with fibre
(where
and
acts trivially on
), let
and let
be a section (i.e. a continuous mapping such that
). Can one extend
to
? The corresponding obstructions lie in the groups
. An extension problem is obtained from this problem if one puts
,
,
,
. Analogously one can also study the classification problem for sections using obstruction theory.
Finally, one can remove the restriction of homotopic simplicity of the space in the extension problem (as well as in the problem on sections); then one must use cohomology with local coefficients.
Obstruction theory was initiated by S. Eilenberg [2]. It was also known to L.S. Pontryagin, who did not formulate it explicitly but used it for the solution of concrete problems, see [1].
A good discussion can be found in [3] and [4].
References
[1] | L.S. Pontryagin, "Classification of continuous transformations of a complex into a sphere" Dokl. Akad. Nauk SSSR , 19 (1938) pp. 361–363 (In Russian) |
[2] | S. Eilenberg, "Cohomology and continuous mappings" Ann. of Math. , 41 (1940) pp. 231–251 |
[3] | S.-T. Hu, "Homotopy theory" , Acad. Press (1959) |
[4] | E. Thomas, "Seminar on fibre spaces" , Springer (1966) |
Comments
The fundamental group acts on the homotopy groups
,
, cf. Homotopy group. The space
is called
-simple if this action (for this
) is trivial;
is called simple or homotopy simple if it is path connected and
-simple for all
. Then
is Abelian and acts trivially on all
. A path-connected
-space is simple.
References
[a1] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) |
[a2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 269–276; 429–432 |
[a3] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) |
[a4] | H.J. Baues, "Obstruction theory" , Springer (1977) |
Obstruction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Obstruction&oldid=11605