Bordism

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.

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