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''bordantism''
 
''bordantism''
  
 
A term used by itself or as a part of standard expressions in a number of similar meanings. The older term, cobordism, is still employed.
 
A term used by itself or as a part of standard expressions in a number of similar meanings. The older term, cobordism, is still employed.
  
The simplest variant is the following. Two smooth closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170301.png" />-dimensional manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170303.png" /> are bordant (cobounding or internally homologous) if there exists a smooth compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170304.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170305.png" /> (a "membrane" ) the boundary of which consists of two manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170307.png" /> which are diffeomorphic, respectively, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b0170309.png" /> under certain diffeomorphisms (cf. [[Diffeomorphism|Diffeomorphism]])
+
The simplest variant is the following. Two smooth closed $  n $-dimensional manifolds $  M _ {0} $
 
+
and $  M $
<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/b/b017/b017030/b01703010.png" /></td> </tr></table>
+
are bordant (cobounding or internally homologous) if there exists a smooth compact $  (n + 1) $-dimensional manifold $  W $ (a "membrane" ) the boundary of which consists of two manifolds $  N _ {0} $
 +
and $  N $
 +
which are diffeomorphic, respectively, to $  M _ {0} $
 +
and $  M $
 +
under certain diffeomorphisms (cf. [[Diffeomorphism|Diffeomorphism]])
  
A set of mutually bordant manifolds is called a bordism class, while the triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703011.png" /> is sometimes called a bordism (it would be more accurate to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703012.png" />). The set of bordism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703013.png" />-dimensional manifolds forms an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703014.png" /> with respect to unconnected union. The zero element of the group is the bordism class consisting of the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703015.png" /> which constitute the boundary of a certain manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703016.png" /> (one formally speaks of the triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703017.png" /> with empty <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703018.png" />; other appellations are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703019.png" /> is a "bounding manifold" , or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703020.png" /> is "internally homologous to zero" or is "bordantly zero" ). The element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703021.png" /> inverse to a given bordism class is this class itself (since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703022.png" /> is diffeomorphic to the boundary of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703023.png" />). The direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703024.png" /> of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703025.png" /> is a commutative graded ring in which the multiplication is induced by the direct product of manifolds with unit element given by the bordism class of a point.
+
$$
 +
g _ {0} : N _ {0}  \rightarrow \
 +
M _ {0} \ \textrm{ and } \ \
 +
gN \rightarrow M.
 +
$$
  
More complex variants comprise the bordism of smooth closed manifolds with a supplementary structure. For instance, two oriented manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703027.png" /> are orientedly bordant if they are bordant in the sense explained above, if the "membrane" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703028.png" /> is oriented and if, using the above notation, the orientation induced by the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703031.png" /> (which are parts of the boundary) goes over under the diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703033.png" />, respectively, to the original orientation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703034.png" /> and the opposite orientation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703035.png" />. One then speaks of oriented bordism; if it is desired to stress the difference between such bordism and bordism in the sense given above, then one speaks of the latter as non-oriented bordism. In analogy with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703037.png" /> one introduces the groups of oriented bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703038.png" /> and the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703039.png" />.
+
A set of mutually bordant manifolds is called a bordism class, while the triplet  $  (W, M _ {0} , M) $
 +
is sometimes called a bordism (it would be more accurate to take $ (W, M _ {0} , M, g _ {0} , g) $).  
 +
The set of bordism classes of  $  n $-dimensional manifolds forms an Abelian group  $  \mathfrak N _ {n} $
 +
with respect to unconnected union. The zero element of the group is the bordism class consisting of the manifolds  $  M $
 +
which constitute the boundary of a certain manifold  $  W $ (one formally speaks of the triplet  $  (W, M _ {0} , M) $
 +
with empty  $  M _ {0} $;
 +
other appellations are: $  M $
 +
is a "bounding manifold" , or  $  M $
 +
is "internally homologous to zero" or is "bordantly zero" ). The element of $  \mathfrak N _ {n} $
 +
inverse to a given bordism class is this class itself (since  $  M \cup M $
 +
is diffeomorphic to the boundary of the direct product  $  M \times [0, 1] $).  
 +
The direct sum  $  \mathfrak N _ {*} $
 +
of the groups $  \mathfrak N _ {n} $
 +
is a commutative graded ring in which the multiplication is induced by the direct product of manifolds with unit element given by the bordism class of a point.
  
Historically, the first example was that of the bordism of nested manifolds, introduced in 1938 by L.S. Pontryagin, who showed that the classification of such bordisms is equivalent to the calculation of the homotopy groups of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703040.png" />, and who could determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703042.png" /> in this way (see [[#References|[2]]] for a detailed account of his studies, and
+
More complex variants comprise the bordism of smooth closed manifolds with a supplementary structure. For instance, two oriented manifolds  $  M _ {0} $
 +
and  $  M $
 +
are orientedly bordant if they are bordant in the sense explained above, if the "membrane"  $  W $
 +
is oriented and if, using the above notation, the orientation induced by the orientation of  $  W $
 +
on  $  N _ {0} $
 +
and  $  N $ (which are parts of the boundary) goes over under the diffeomorphisms  $  g _ {0} $
 +
and  $  g $,
 +
respectively, to the original orientation on  $  M $
 +
and the opposite orientation on  $  M _ {0} $.  
 +
One then speaks of oriented bordism; if it is desired to stress the difference between such bordism and bordism in the sense given above, then one speaks of the latter as non-oriented bordism. In analogy with  $  \mathfrak N _ {n} $
 +
and  $  \mathfrak N _ {*} $
 +
one introduces the groups of oriented bordism  $  \Omega _ {n} $
 +
and the ring  $  \Omega _ {*} = \sum \Omega _ {n} $.
  
for an introductory text). Non-oriented and oriented bordisms were introduced in 1951–1953 by V.A. Rokhlin [[#References|[3]]], who calculated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703045.png" />. It had been previously shown by Pontryagin [[#References|[1]]] that if two manifolds are bordant, their characteristic numbers are identical (Stiefel–Whitney numbers for non-oriented manifolds, Stiefel–Whitney and Pontryagin numbers for oriented manifolds). It was subsequently found that the converse proposition is also true.
+
Historically, the first example was that of the bordism of nested manifolds, introduced in 1938 by L.S. Pontryagin, who showed that the classification of such bordisms is equivalent to the calculation of the homotopy groups of the spheres  $  \pi _ {i} (S  ^ {n} ) $,
 +
and who could determine  $  \pi _ {n+1} (S  ^ {n} ) $
 +
and  $  \pi _ {n+2} (S  ^ {n} ) $
 +
in this way (see [[#References|[2]]] for a detailed account of his studies, and for an introductory text). Non-oriented and oriented bordisms were introduced in 1951–1953 by V.A. Rokhlin [[#References|[3]]], who calculated $  \mathfrak N _ {n} $
 +
and $  \Omega _ {n} $
 +
for $  n \leq  4 $.  
 +
It had been previously shown by Pontryagin [[#References|[1]]] that if two manifolds are bordant, their characteristic numbers are identical (Stiefel–Whitney numbers for non-oriented manifolds, Stiefel–Whitney and Pontryagin numbers for oriented manifolds). It was subsequently found that the converse proposition is also true.
  
Modern methods of algebraic topology were first applied in the theory of bordism in 1954 by R. Thom [[#References|[5]]], [[#References|[6]]], who rediscovered (for the case of both oriented and non-oriented bordisms) the connection between bordisms and certain homotopy problems. Thus, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703046.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703047.png" /> for a sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703048.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703049.png" /> is the [[Thom space|Thom space]] of the universal vector bundle with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703050.png" />. Owing to this connection, Thom was able to make a complete computation of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703051.png" /> and to contribute substantially to the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703052.png" />, which was subsequently continued by other workers. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703053.png" /> proved to be the ring of polynomials over the field of residues modulo 2 in the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703054.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703056.png" /> runs through all positive numbers not equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703058.png" />; a geometric realization of these generators is known (i.e. concrete manifolds whose bordism classes are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703059.png" /> have been given [[#References|[7]]]).
+
Modern methods of algebraic topology were first applied in the theory of bordism in 1954 by R. Thom [[#References|[5]]], [[#References|[6]]], who rediscovered (for the case of both oriented and non-oriented bordisms) the connection between bordisms and certain homotopy problems. Thus, the group $  \mathfrak N _ {n} $
 +
is isomorphic to the group $  \pi _ {n+r} (  \mathop{\rm TBO} (r)) $
 +
for a sufficiently large $  r $;  
 +
here, $  \mathop{\rm TBO} (r) $
 +
is the [[Thom space|Thom space]] of the universal vector bundle with structure group $  O(r) $.  
 +
Owing to this connection, Thom was able to make a complete computation of the ring $  \mathfrak N _ {*} $
 +
and to contribute substantially to the study of $  \Omega _ {*} $,  
 +
which was subsequently continued by other workers. Thus, $  \mathfrak N _ {*} $
 +
proved to be the ring of polynomials over the field of residues modulo 2 in the generators $  x _ {i} $
 +
of dimension $  i $,  
 +
where $  i $
 +
runs through all positive numbers not equal to $  {2  ^ {s} } - 1 $,  
 +
$  s \geq  1 $;  
 +
a geometric realization of these generators is known (i.e. concrete manifolds whose bordism classes are $  x _ {i} $
 +
have been given [[#References|[7]]]).
  
Other variants of bordism of manifolds with a supplementary structure comprise the very important bordisms of quasi-complex manifolds (also called unitary bordism or complex cobordism [[#References|[8]]], [[#References|[9]]]), and bordism of manifolds acted upon by a group of transformations [[#References|[10]]]. There are also variations of another kind (for piecewise-linear or topological manifolds, for Poincaré complexes, etc. [[#References|[11]]]). Special kinds of bordisms include foliated bordism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703061.png" />-bordism (previously referred to as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703063.png" />-equivalences); these are used for relating differential and homotopy topological properties [[#References|[12]]].
+
Other variants of bordism of manifolds with a supplementary structure comprise the very important bordisms of quasi-complex manifolds (also called unitary bordism or complex cobordism [[#References|[8]]], [[#References|[9]]]), and bordism of manifolds acted upon by a group of transformations [[#References|[10]]]. There are also variations of another kind (for piecewise-linear or topological manifolds, for Poincaré complexes, etc. [[#References|[11]]]). Special kinds of bordisms include foliated bordism and $  h $-bordism (previously referred to as $  J $-equivalences); these are used for relating differential and homotopy topological properties [[#References|[12]]].
  
Further development of bordism theory is related to the bordism groups of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703064.png" /> (bordism of spaces for short, cf. [[#References|[13]]]). They are defined for different variants of bordisms (the simplest example is given below). A singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703066.png" />-dimensional (sub) manifold of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703069.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703071.png" /> is a closed smooth manifold, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703072.png" /> is a continuous mapping. Two such pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703074.png" /> are bordant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703076.png" /> are bordant in the ordinary sense, and (using previous notations) if there exists a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703077.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703079.png" />. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703081.png" /> are identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703083.png" />, one can simply say that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703084.png" /> induces the given mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703086.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703087.png" />.) The bordism classes of singular manifolds in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703088.png" /> form the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703089.png" />-dimensional bordism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703090.png" /> of this space (the group operation is generated by the union of manifolds).
+
Further development of bordism theory is related to the bordism groups of a topological space $  X $ (bordism of spaces for short, cf. [[#References|[13]]]). They are defined for different variants of bordisms (the simplest example is given below). A singular $  n $-dimensional (sub) manifold of a space $  X $
 +
is a pair $  (M  ^ {n} , f) $,  
 +
where $  M  ^ {n} $
 +
is a closed smooth manifold, and $  f: M  ^ {n} \rightarrow X $
 +
is a continuous mapping. Two such pairs $  (M _ {0} , f _ {0} ) $,
 +
$  (M  ^ {n} , f) $
 +
are bordant if $  M _ {0} $
 +
and $  M $
 +
are bordant in the ordinary sense, and (using previous notations) if there exists a continuous mapping $  h: W \rightarrow X $
 +
such that $  h g _ {0}  ^ {-1} = f _ {0} $,  
 +
$  h g  ^ {-1} = f _ {0} $.  
 +
(If $  M _ {0} $
 +
and $  M $
 +
are identified with $  N _ {0} $
 +
and $  N $,  
 +
one can simply say that the mapping $  h $
 +
induces the given mappings of $  M _ {0} $
 +
and $  M $
 +
into $  X $.)  
 +
The bordism classes of singular manifolds in a space $  X $
 +
form the $  n $-dimensional bordism group $  \mathfrak N _ {n} (X) $
 +
of this space (the group operation is generated by the union of manifolds).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703091.png" /> is a manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703092.png" />, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703093.png" /> can be visualized as submanifolds, as can the corresponding "membranes" ; in this respect bordism of spaces resembles the original attempts at introducing homologies. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703094.png" /> is a point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703095.png" /> is reduced to the previous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703096.png" />. To a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703097.png" /> correspond homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703098.png" />, generated by the transition from a singular manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b01703099.png" /> to another one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030100.png" />. The functor by means of which each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030101.png" /> is brought into correspondence with the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030102.png" /> and each mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030103.png" /> with the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030104.png" /> is a generalized homology theory. In the present case it reduces to ordinary homology, viz., for any cellular polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030105.png" />
+
If $  X $
 +
is a manifold of dimension > 2n $,  
 +
the elements of $  \mathfrak N _ {n} (X) $
 +
can be visualized as submanifolds, as can the corresponding "membranes" ; in this respect bordism of spaces resembles the original attempts at introducing homologies. If $  X $
 +
is a point, $  \mathfrak N _ {n} (X) $
 +
is reduced to the previous $  \mathfrak N _ {n} $.  
 +
To a mapping $  \phi : X \rightarrow Y $
 +
correspond homomorphisms $  \phi _ {n} : \mathfrak N _ {n} (X) \rightarrow \mathfrak N _ {n} (Y) $,  
 +
generated by the transition from a singular manifold $  (M, f) $
 +
to another one $  (M, \phi f) $.  
 +
The functor by means of which each space $  X $
 +
is brought into correspondence with the groups $  \mathfrak N _ {n} (X) $
 +
and each mapping $  \phi $
 +
with the mappings $  \phi _ {n} $
 +
is a generalized homology theory. In the present case it reduces to ordinary homology, viz., for any cellular polyhedron $  X $
  
<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/b/b017/b017030/b017030106.png" /></td> </tr></table>
+
$$
 +
\sum \mathfrak N _ {n} (X)  \simeq  H _ {*} (X, \mathbf Z _ {2} )
 +
\otimes \mathfrak N _ {*}  $$
  
(the tensor product of graded modules on the right-hand side of this expression is over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030107.png" />), but this is generally not true for other bordisms (oriented, etc.). Many generalized homology theories may be obtained by way of a so-called bordism with singularities .
+
(the tensor product of graded modules on the right-hand side of this expression is over $  \mathbf Z _ {2} = \mathbf Z /(2) $),  
 +
but this is generally not true for other bordisms (oriented, etc.). Many generalized homology theories may be obtained by way of a so-called bordism with singularities.
  
In addition to the bordism of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030108.png" /> there exist generalized cohomology theories which are dual to it. The introduction of these generalized homology and cohomology functors has made it desirable to introduce the changed terminology discussed at the beginning of the present article. Thus, the term "cobordism" is restricted to generalized cohomology theories dual to bordism.
+
In addition to the bordism of a space $  X $
 +
there exist generalized cohomology theories which are dual to it. The introduction of these generalized homology and cohomology functors has made it desirable to introduce the changed terminology discussed at the beginning of the present article. Thus, the term "cobordism" is restricted to generalized cohomology theories dual to bordism.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin,   "Characteristic cycles of differentiable manifolds" ''Mat. Sb.'' , '''21 (63)''' : 2 (1947) pp. 233–284 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin,   "Smooth manifolds and their applications in homology theory" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.A. Rokhlin,   "Theory of inner homology" ''Uspekhi Mat. Nauk'' , '''14''' : 4 (1959) pp. 3–20 (In Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> A.H. Wallace,   "Differentiable topology. First steps" , Benjamin (1968)</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> J.W. Milnor,   "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Thom,   "Quelques propriétés globales des variétés différentiables" ''Comm. Math. Helvetia'' , '''28''' (1954) pp. 17–86</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.W. Milnor,   J.D. Stasheff,   "Characteristic classes" , Princeton Univ. Press &amp; Univ. Tokyo Press (1974)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Dold,   "Erzeugende der Thomschen Algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030109.png" />" ''Math. Z.'' , '''65''' (1956) pp. 25–35</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J. Milnor,   "On the cobordism ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030110.png" /> and a complex analogue I" ''Amer. J. Math.'' , '''82''' (1960) pp. 505–521</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.P. Novikov,   "Homotopic properties of Thom-complexes" ''Mat. Sb.'' , '''57 (99)''' (1962) pp. 406–442 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer (1964)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press (1968)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Milnor,   "Lectures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017030/b017030111.png" />-cobordism theorem" , Princeton Univ. Press (1965)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> M.F. Atiyah,   "Bordism and cobordism" ''Proc. Cambridge Philos. Soc. (2)'' , '''57''' (1961) pp. 200–208</TD></TR><TR><TD valign="top">[14a]</TD> <TD valign="top"> N.A. Baas,   "On bordism theory of manifolds with singularities" ''Math. Scand.'' , '''33''' (1973) pp. 279–302</TD></TR><TR><TD valign="top">[14b]</TD> <TD valign="top"> N.A. Baas,   "On formal groups and singularities in complex bordism theory" ''Math. Scand.'' , '''33''' (1973) pp. 302–313</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Characteristic cycles of differentiable manifolds" ''Mat. Sb.'' , '''21 (63)''' : 2 (1947) pp. 233–284 (In Russian) {{MR|22667}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin, "Smooth manifolds and their applications in homology theory" , Moscow (1976) (In Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> V.A. Rokhlin, "Theory of inner homology" ''Uspekhi Mat. Nauk'' , '''14''' : 4 (1959) pp. 3–20 (In Russian)</TD></TR>
 +
<TR><TD valign="top">[4a]</TD> <TD valign="top"> A.H. Wallace, "Differentiable topology. First steps" , Benjamin (1968) {{ZBL|0164.23805}}</TD></TR>
 +
<TR><TD valign="top">[4b]</TD> <TD valign="top"> J.W. Milnor, "Topology from the differentiable viewpoint" , Univ. Virginia Press (1965)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top"> R. Thom, "Quelques propriétés globales des variétés différentiables" ''Comm. Math. Helvetia'' , '''28''' (1954) pp. 17–86 {{MR|0061823}} {{ZBL|0057.15502}} </TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top"> J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press &amp; Univ. Tokyo Press (1974) {{MR|0440554}} {{ZBL|0298.57008}} </TD></TR>
 +
<TR><TD valign="top">[7]</TD> <TD valign="top"> A. Dold, "Erzeugende der Thomschen Algebra $\mathfrak N$" ''Math. Z.'' , '''65''' (1956) pp. 25–35 {{MR|0079269}} {{ZBL|0071.17601}} </TD></TR>
 +
<TR><TD valign="top">[8]</TD> <TD valign="top"> J. Milnor, "On the cobordism ring $\Omega^\star$ and a complex analogue I" ''Amer. J. Math.'' , '''82''' (1960) pp. 505–521 {{MR|0119209}} {{ZBL|0095.16702}} </TD></TR>
 +
<TR><TD valign="top">[9]</TD> <TD valign="top"> S.P. Novikov, "Homotopic properties of Thom-complexes" ''Mat. Sb.'' , '''57 (99)''' (1962) pp. 406–442 (In Russian)</TD></TR>
 +
<TR><TD valign="top">[10]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR>
 +
<TR><TD valign="top">[11]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR>
 +
<TR><TD valign="top">[12]</TD> <TD valign="top"> J. Milnor, "Lectures on the $h$-cobordism theorem" , Princeton Univ. Press (1965) {{MR|190942}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> M.F. Atiyah, "Bordism and cobordism" ''Proc. Cambridge Philos. Soc. (2)'' , '''57''' (1961) pp. 200–208 {{MR|0126856}} {{ZBL|0104.17405}} </TD></TR>
 +
<TR><TD valign="top">[14a]</TD> <TD valign="top"> N.A. Baas, "On bordism theory of manifolds with singularities" ''Math. Scand.'' , '''33''' (1973) pp. 279–302 {{MR|0346823}} {{MR|0346824}} {{MR|0346825}} {{ZBL|0281.57027}} </TD></TR>
 +
<TR><TD valign="top">[14b]</TD> <TD valign="top"> N.A. Baas, "On formal groups and singularities in complex bordism theory" ''Math. Scand.'' , '''33''' (1973) pp. 302–313 {{ZBL|0281.57028}}</TD></TR>
 +
</table>

Latest revision as of 08:36, 26 March 2023


bordantism

A term used by itself or as a part of standard expressions in a number of similar meanings. The older term, cobordism, is still employed.

The simplest variant is the following. Two smooth closed $ n $-dimensional manifolds $ M _ {0} $ and $ M $ are bordant (cobounding or internally homologous) if there exists a smooth compact $ (n + 1) $-dimensional manifold $ W $ (a "membrane" ) the boundary of which consists of two manifolds $ N _ {0} $ and $ N $ which are diffeomorphic, respectively, to $ M _ {0} $ and $ M $ under certain diffeomorphisms (cf. Diffeomorphism)

$$ g _ {0} : N _ {0} \rightarrow \ M _ {0} \ \textrm{ and } \ \ g: N \rightarrow M. $$

A set of mutually bordant manifolds is called a bordism class, while the triplet $ (W, M _ {0} , M) $ is sometimes called a bordism (it would be more accurate to take $ (W, M _ {0} , M, g _ {0} , g) $). The set of bordism classes of $ n $-dimensional manifolds forms an Abelian group $ \mathfrak N _ {n} $ with respect to unconnected union. The zero element of the group is the bordism class consisting of the manifolds $ M $ which constitute the boundary of a certain manifold $ W $ (one formally speaks of the triplet $ (W, M _ {0} , M) $ with empty $ M _ {0} $; other appellations are: $ M $ is a "bounding manifold" , or $ M $ is "internally homologous to zero" or is "bordantly zero" ). The element of $ \mathfrak N _ {n} $ inverse to a given bordism class is this class itself (since $ M \cup M $ is diffeomorphic to the boundary of the direct product $ M \times [0, 1] $). The direct sum $ \mathfrak N _ {*} $ of the groups $ \mathfrak N _ {n} $ is a commutative graded ring in which the multiplication is induced by the direct product of manifolds with unit element given by the bordism class of a point.

More complex variants comprise the bordism of smooth closed manifolds with a supplementary structure. For instance, two oriented manifolds $ M _ {0} $ and $ M $ are orientedly bordant if they are bordant in the sense explained above, if the "membrane" $ W $ is oriented and if, using the above notation, the orientation induced by the orientation of $ W $ on $ N _ {0} $ and $ N $ (which are parts of the boundary) goes over under the diffeomorphisms $ g _ {0} $ and $ g $, respectively, to the original orientation on $ M $ and the opposite orientation on $ M _ {0} $. One then speaks of oriented bordism; if it is desired to stress the difference between such bordism and bordism in the sense given above, then one speaks of the latter as non-oriented bordism. In analogy with $ \mathfrak N _ {n} $ and $ \mathfrak N _ {*} $ one introduces the groups of oriented bordism $ \Omega _ {n} $ and the ring $ \Omega _ {*} = \sum \Omega _ {n} $.

Historically, the first example was that of the bordism of nested manifolds, introduced in 1938 by L.S. Pontryagin, who showed that the classification of such bordisms is equivalent to the calculation of the homotopy groups of the spheres $ \pi _ {i} (S ^ {n} ) $, and who could determine $ \pi _ {n+1} (S ^ {n} ) $ and $ \pi _ {n+2} (S ^ {n} ) $ in this way (see [2] for a detailed account of his studies, and for an introductory text). Non-oriented and oriented bordisms were introduced in 1951–1953 by V.A. Rokhlin [3], who calculated $ \mathfrak N _ {n} $ and $ \Omega _ {n} $ for $ n \leq 4 $. It had been previously shown by Pontryagin [1] that if two manifolds are bordant, their characteristic numbers are identical (Stiefel–Whitney numbers for non-oriented manifolds, Stiefel–Whitney and Pontryagin numbers for oriented manifolds). It was subsequently found that the converse proposition is also true.

Modern methods of algebraic topology were first applied in the theory of bordism in 1954 by R. Thom [5], [6], who rediscovered (for the case of both oriented and non-oriented bordisms) the connection between bordisms and certain homotopy problems. Thus, the group $ \mathfrak N _ {n} $ is isomorphic to the group $ \pi _ {n+r} ( \mathop{\rm TBO} (r)) $ for a sufficiently large $ r $; here, $ \mathop{\rm TBO} (r) $ is the Thom space of the universal vector bundle with structure group $ O(r) $. Owing to this connection, Thom was able to make a complete computation of the ring $ \mathfrak N _ {*} $ and to contribute substantially to the study of $ \Omega _ {*} $, which was subsequently continued by other workers. Thus, $ \mathfrak N _ {*} $ proved to be the ring of polynomials over the field of residues modulo 2 in the generators $ x _ {i} $ of dimension $ i $, where $ i $ runs through all positive numbers not equal to $ {2 ^ {s} } - 1 $, $ s \geq 1 $; a geometric realization of these generators is known (i.e. concrete manifolds whose bordism classes are $ x _ {i} $ have been given [7]).

Other variants of bordism of manifolds with a supplementary structure comprise the very important bordisms of quasi-complex manifolds (also called unitary bordism or complex cobordism [8], [9]), and bordism of manifolds acted upon by a group of transformations [10]. There are also variations of another kind (for piecewise-linear or topological manifolds, for Poincaré complexes, etc. [11]). Special kinds of bordisms include foliated bordism and $ h $-bordism (previously referred to as $ J $-equivalences); these are used for relating differential and homotopy topological properties [12].

Further development of bordism theory is related to the bordism groups of a topological space $ X $ (bordism of spaces for short, cf. [13]). They are defined for different variants of bordisms (the simplest example is given below). A singular $ n $-dimensional (sub) manifold of a space $ X $ is a pair $ (M ^ {n} , f) $, where $ M ^ {n} $ is a closed smooth manifold, and $ f: M ^ {n} \rightarrow X $ is a continuous mapping. Two such pairs $ (M _ {0} , f _ {0} ) $, $ (M ^ {n} , f) $ are bordant if $ M _ {0} $ and $ M $ are bordant in the ordinary sense, and (using previous notations) if there exists a continuous mapping $ h: W \rightarrow X $ such that $ h g _ {0} ^ {-1} = f _ {0} $, $ h g ^ {-1} = f _ {0} $. (If $ M _ {0} $ and $ M $ are identified with $ N _ {0} $ and $ N $, one can simply say that the mapping $ h $ induces the given mappings of $ M _ {0} $ and $ M $ into $ X $.) The bordism classes of singular manifolds in a space $ X $ form the $ n $-dimensional bordism group $ \mathfrak N _ {n} (X) $ of this space (the group operation is generated by the union of manifolds).

If $ X $ is a manifold of dimension $ > 2n $, the elements of $ \mathfrak N _ {n} (X) $ can be visualized as submanifolds, as can the corresponding "membranes" ; in this respect bordism of spaces resembles the original attempts at introducing homologies. If $ X $ is a point, $ \mathfrak N _ {n} (X) $ is reduced to the previous $ \mathfrak N _ {n} $. To a mapping $ \phi : X \rightarrow Y $ correspond homomorphisms $ \phi _ {n} : \mathfrak N _ {n} (X) \rightarrow \mathfrak N _ {n} (Y) $, generated by the transition from a singular manifold $ (M, f) $ to another one $ (M, \phi f) $. The functor by means of which each space $ X $ is brought into correspondence with the groups $ \mathfrak N _ {n} (X) $ and each mapping $ \phi $ with the mappings $ \phi _ {n} $ is a generalized homology theory. In the present case it reduces to ordinary homology, viz., for any cellular polyhedron $ X $

$$ \sum \mathfrak N _ {n} (X) \simeq H _ {*} (X, \mathbf Z _ {2} ) \otimes \mathfrak N _ {*} $$

(the tensor product of graded modules on the right-hand side of this expression is over $ \mathbf Z _ {2} = \mathbf Z /(2) $), but this is generally not true for other bordisms (oriented, etc.). Many generalized homology theories may be obtained by way of a so-called bordism with singularities.

In addition to the bordism of a space $ X $ there exist generalized cohomology theories which are dual to it. The introduction of these generalized homology and cohomology functors has made it desirable to introduce the changed terminology discussed at the beginning of the present article. Thus, the term "cobordism" is restricted to generalized cohomology theories dual to bordism.

References

[1] L.S. Pontryagin, "Characteristic cycles of differentiable manifolds" Mat. Sb. , 21 (63) : 2 (1947) pp. 233–284 (In Russian) MR22667
[2] L.S. Pontryagin, "Smooth manifolds and their applications in homology theory" , Moscow (1976) (In Russian)
[3] V.A. Rokhlin, "Theory of inner homology" Uspekhi Mat. Nauk , 14 : 4 (1959) pp. 3–20 (In Russian)
[4a] A.H. Wallace, "Differentiable topology. First steps" , Benjamin (1968) Zbl 0164.23805
[4b] J.W. Milnor, "Topology from the differentiable viewpoint" , Univ. Virginia Press (1965)
[5] R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helvetia , 28 (1954) pp. 17–86 MR0061823 Zbl 0057.15502
[6] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press & Univ. Tokyo Press (1974) MR0440554 Zbl 0298.57008
[7] A. Dold, "Erzeugende der Thomschen Algebra $\mathfrak N$" Math. Z. , 65 (1956) pp. 25–35 MR0079269 Zbl 0071.17601
[8] J. Milnor, "On the cobordism ring $\Omega^\star$ and a complex analogue I" Amer. J. Math. , 82 (1960) pp. 505–521 MR0119209 Zbl 0095.16702
[9] S.P. Novikov, "Homotopic properties of Thom-complexes" Mat. Sb. , 57 (99) (1962) pp. 406–442 (In Russian)
[10] P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[11] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[12] J. Milnor, "Lectures on the $h$-cobordism theorem" , Princeton Univ. Press (1965) MR190942
[13] M.F. Atiyah, "Bordism and cobordism" Proc. Cambridge Philos. Soc. (2) , 57 (1961) pp. 200–208 MR0126856 Zbl 0104.17405
[14a] N.A. Baas, "On bordism theory of manifolds with singularities" Math. Scand. , 33 (1973) pp. 279–302 MR0346823 MR0346824 MR0346825 Zbl 0281.57027
[14b] N.A. Baas, "On formal groups and singularities in complex bordism theory" Math. Scand. , 33 (1973) pp. 302–313 Zbl 0281.57028
How to Cite This Entry:
Bordism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bordism&oldid=18713
This article was adapted from an original article by D.V. AnosovM.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article