View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680701.png" /> be a pair of cellular spaces (cf. [[Cellular space|Cellular space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680702.png" /> be a simply-connected (more generally, a homotopy-simple) topological space. Can one extend a given continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680703.png" /> to a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680704.png" />? The extension can be attempted recursively, over successive skeletons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680706.png" />. Suppose one has constructed a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680707.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680708.png" />. For any oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o0680709.png" />-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807010.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807011.png" /> gives a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807012.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807013.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807014.png" />-dimensional unit sphere) and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807015.png" /> (it is here that one uses that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807016.png" /> is homotopy simple, which allows one to ignore the base point). This defines a 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/o/o068/o068070/o06807017.png" /></td> </tr></table>

Since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807018.png" /> one clearly has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807019.png" />, it follows that

<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/o/o068/o068070/o06807020.png" /></td> </tr></table>

Clearly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807021.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807022.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807023.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807024.png" /> is an obstruction to extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807025.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807026.png" />.

The cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807027.png" /> is a cocycle. The fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807028.png" /> does not, in general, imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807029.png" /> cannot be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807030.png" />: It is possible that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807031.png" /> cannot be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807032.png" /> because of an unsuccessful choice of an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807034.png" />. It may turn out that, e.g., the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807035.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807036.png" />, i.e. that extension is possible by skipping back one step. It can be shown that the cohomology class

<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/o/o068/o068070/o06807037.png" /></td> </tr></table>

is an obstruction to this, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807038.png" /> if and only if there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807040.png" /> (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807041.png" />). The construction of difference chains and cochains is used in the proof of this statement (cf. [[Difference cochain and chain|Difference cochain and chain]]).

Since the problem of homotopy classification of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807042.png" /> can be interpreted as an extension problem, obstruction theory is applicable also to the description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807043.png" /> of homotopy classes of mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807044.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807045.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807046.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807047.png" /> be a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807048.png" />. Then a pair of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807049.png" /> is interpreted as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807052.png" />, and the presence of a homotopy between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807054.png" /> means the presence of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807055.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807056.png" />. If the homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807057.png" /> has been constructed on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807058.png" />-dimensional skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807059.png" />, then the obstruction to its extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807060.png" /> is the difference 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/o/o068/o068070/o06807061.png" /></td> </tr></table>

As an application one may consider the description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807064.png" /> is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807065.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807066.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807067.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807068.png" /> be a constant mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807069.png" /> an arbitrary continuous mapping. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807070.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807071.png" />, the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807073.png" /> are homotopic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807074.png" /> and, after having chosen such a homotopy, one can define the difference 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/o/o068/o068070/o06807075.png" /></td> </tr></table>

The cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807076.png" /> is well-defined, i.e. does not depend on the choice of a homotopy between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807078.png" /> (since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807079.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807080.png" />). Further, if two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807081.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807083.png" />, and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807085.png" /> are homotopic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807086.png" />. The obstructions to extending this homotopy to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807087.png" /> lie in the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807088.png" /> (since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807089.png" />), and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807091.png" /> are homotopic. Thus, the homotopy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807092.png" /> is completely determined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807093.png" />. Finally, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807094.png" /> there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807095.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807096.png" />, hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807097.png" />. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o06807099.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070100.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070101.png" />.

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|Cohomology operation]]) and Postnikov systems (cf. [[Postnikov system|Postnikov system]]) are used to this end. E.g., in order to describe the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070103.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070106.png" />, it is required, in general, to study the possibility of skipping back <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070107.png" /> steps, for which it is necessary to study the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070108.png" /> levels of the Postnikov system for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070109.png" />, i.e. to use cohomology operations of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070110.png" /> (in the article [[Cohomology operation|Cohomology operation]] this problem is outlined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070111.png" />).

The theory of obstructions is also used in the more general situation of extension of sections (cf. [[Section of a mapping|Section of a mapping]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070112.png" /> be a fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070113.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070115.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070116.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070117.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070118.png" /> be a section (i.e. a continuous mapping such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070119.png" />). Can one extend <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070120.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070121.png" />? The corresponding obstructions lie in the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070122.png" />. An extension problem is obtained from this problem if one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070126.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070127.png" /> 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 [[#References|[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 [[#References|[1]]].

A good discussion can be found in [[#References|[3]]] and [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Classification of continuous transformations of a complex into a sphere"  ''Dokl. Akad. Nauk SSSR'' , '''19'''  (1938)  pp. 361–363  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Eilenberg,   "Cohomology and continuous mappings"  ''Ann. of Math.'' , '''41'''  (1940)  pp. 231–251</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.-T. Hu,   "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Thomas,   "Seminar on fibre spaces" , Springer  (1966)</TD></TR></table>



====Comments====
The fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070128.png" /> acts on the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070130.png" />, cf. [[Homotopy group|Homotopy group]]. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070131.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070133.png" />-simple if this action (for this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070134.png" />) is trivial; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070135.png" /> is called simple or homotopy simple if it is path connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070136.png" />-simple for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070137.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070138.png" /> is Abelian and acts trivially on all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070139.png" />. A path-connected [[H-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068070/o068070140.png" />-space]] is simple.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)  pp. 269–276; 429–432</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Whitehead,   "Elements of homotopy theory" , Springer  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.J. Baues,   "Obstruction theory" , Springer  (1977)</TD></TR></table>