Cobordism

cobordism theory

A generalized cohomology theory determined by spectra of Thom spaces and related to various structures in the stable tangent or normal bundle to a manifold. Cobordism theory is dual (in the sense of $ S $- duality) to the theory of bordism.

The simplest example of cobordism is orthogonal or non-oriented cobordism. Let $ O _ {r} $ by the group of orthogonal transformations of the Euclidean space $ \mathbf R ^ {n} $, and $ {BO } _ {r} $ its classifying space. The standard imbedding $ O _ {r} \rightarrow O _ {r+} 1 $ defines a mapping $ j _ {r} : BO _ {r} \rightarrow {BO } _ {r+} 1 $ taking the universal fibre bundle $ \gamma _ {r+} 1 $ over $ {BO } _ {r+} 1 $ into the bundle $ \gamma _ {r} \oplus \theta $, where $ \theta $ is the one-dimensional trivial bundle over $ {BO } _ {r} $. If $ {TBO } _ {r} $ is the Thom space of $ \gamma _ {r} $, then one obtains a mapping $ s _ {r} : STBO _ {r} \rightarrow {TBO } _ {r+} 1 $ induced by $ j _ {r} $, where $ S $ is suspension. The sequence $ \{ TBO _ {r} , s _ {r} \} $ forms a spectrum of spaces and therefore defines a cohomology theory, called the theory of orthogonal cobordism or non-oriented cobordism or $ O $- cobordism; it is denoted by $ O ^ {*} $. The group $ O ^ {n} ( X , A ) $ of $ n $- dimensional $ O $- cobordism of the pair $ ( X , A ) $ is defined as

$$ \lim\limits _ {i \rightarrow \infty } \ [ S ^ {i} ( X / A ) ,\ T {BO } _ {i+ n} ] , $$

where $ [ P , Q ] $ is the set of homotopy classes of mappings from $ P $ into $ Q $. Here $ O ^ {n} ( x) = O ^ {n} ( X , \emptyset ) $, $ \emptyset $ is the empty set, and by $ X / \emptyset = X ^ {+} $ one means the disjoint union of $ X $ and a point. The group $ O ^ {n} ( X , x _ {0} ) $, where $ x _ {0} \in X $, is called the reduced group of $ n $- dimensional $ O $- cobordism $ \widetilde{O} {} ^ {n} ( X) $ of $ X $. The generalized homology theory dual to the $ O $- cobordism theory is called $ O $- bordism theory. The groups $ O _ {n} ( X , A ) $ of $ n $- dimensional bordism of the pair $ ( X , A ) $ are defined as

$$ \lim\limits _ {i \rightarrow \infty } \ \pi _ {i+ n} ( ( X / A ) \wedge T {BO } _ {i} ) . $$

The groups of $ n $- dimensional $ O $- bordism of a point are denoted by $ \Omega _ {O} ^ {n} $ and the $ n $- dimensional $ O $- bordism of a point by $ \Omega _ {n} ^ {O} $; the latter can be described purely geometrically. Furthermore, $ \Omega _ {O} ^ {-} n \approx \Omega _ {n} ^ {O} \approx \pi _ {n+ N} ( T {BO } _ {N} ) $, $ N \gg n $, so that it can be interpreted both as a cobordism group and a bordism group (see bordism, where it is denoted by $ \mathfrak N _ {n} $). The total coefficient group of $ O $- cobordism theory, the graded group $ \Omega _ {O} = \oplus _ {- \infty } ^ {+ \infty } \Omega _ {O} ^ {n} $, is a ring: multiplication is induced by the Cartesian product of manifolds. Furthermore, for any finite $ \mathop{\rm CW} $- complex $ X $ the group $ O ( X) = \oplus _ {n= - \infty } ^ {+ \infty } O ^ {n} ( X) $ is a natural ring with respect to $ X $ since the mapping $ {BO } _ {m} \times {BO } _ {n} \rightarrow {BO } _ {m+ n} $ induced by the imbedding $ O _ {m} \times O _ {n} \rightarrow O _ {m+ n} $ defines a mapping $ T {BO } _ {m} \wedge T {BO } _ {n} \rightarrow T {BO } _ {m+ n} $, so that $ \{ T {BO } _ {r} \} $ is a multiplicative spectrum of spaces.

The general situation is described as follows. By a structural series $ ( B , \phi ) $ one means a sequence of bundles $ \phi _ {r} : B _ {r} \rightarrow {BO } _ {r} $ and mappings $ i _ {r} : B _ {r} \rightarrow B _ {r+} 1 $ such that $ \phi _ {r+} 1 \circ i _ {r} = j _ {r} \circ \phi _ {r} $. The mapping $ \phi _ {r} $ defines a vector bundle $ \xi _ {r} = \phi ^ {*} \gamma _ {r} $ over $ B _ {r} $, where $ i _ {r} ^ {*} \xi _ {r+1 }= \xi _ {r} + \phi _ {r} ^ {*} \theta $. Let $ TB _ {r} $ be the Thom space of the bundle $ \xi _ {r} $; the above equality defines a mapping $ s _ {r} : STB _ {r} \rightarrow TB _ {r+ 1} $ such that the sequence $ T ( B , \phi ) = \{ TB _ {r} , s _ {r} \} $ is a spectrum of spaces, hence defines a cohomology theory. It is called $ ( B , \phi ) $- cobordism theory and is denoted by $ ( B , \phi ) ^ {*} $. Thus,

$$ ( B \phi ) ^ {i} ( X , A ) = \ \lim\limits _ {N \rightarrow \infty } \ [ S ^ {N} ( X / A ) ,\ T B _ {i+ N} ] . $$

The coefficient group of the $ ( B , \phi ) $- cobordism theory is denoted by $ \Omega _ {( B , \phi ) } $. Here, $ \Omega _ {i} ^ {( B , \phi ) } = \Omega _ {( B , \phi ) } ^ {-} i = \pi _ {i+} N ( T B _ {N} ) $, $ N \gg i $, where $ \Omega _ {i} ^ {( B , \phi ) } $ is the coefficient group of the dual $ ( B , \phi ) $- bordism theory, which admits of a geometric definition using the concept of a so-called $ ( B , \phi ) $- structure: $ ( B , \phi ) $- bordancy is defined and the elements of $ \Omega ^ {( B , \phi ) } $ are interpreted as classes of $ ( B , \phi ) $- bordant manifolds.

The first examples of cobordism theory arose from series of linear groups. For example, the series of orthogonal groups $ \{ O _ {r} \} $ defines the structural series $ \{ B _ {r} , \phi _ {r} \} $, where $ B _ {r} = {BO } _ {r} $, $ \phi _ {r} = \mathop{\rm id} $. The series $ \{ SO _ {r} \} $ defines the structural series $ \{ B _ {r} , \phi _ {r} \} $, where $ B _ {r} = B SO _ {r} $ and $ \phi _ {r} : B SO _ {r} \rightarrow {BO } _ {r} $ is the universal two-sheeted covering corresponding to the inclusion $ SO _ {r} \subset O _ {r} $. The corresponding cobordism theory is called the theory of oriented cobordism; it is denoted by $ SO ^ {*} $. The series of unitary groups $ \{ U _ {r} \} $ defines a theory of unitary or complex cobordism, quasi-complex cobordism, almost-complex cobordism; it is denoted by $ U ^ {*} $. Here the series $ \{ B , \phi \} $ is constructed in the following way: $ B _ {2r} = B _ {2r+ 1} = {BU } _ {r} $ is classifying space of $ U _ {r} $ and the $ \phi _ {r} $, $ \phi _ {2r+ 1} $ are the mappings of the classifying spaces $ BU _ {r} \rightarrow {BO } _ {2r} $ and $ {BU } _ {r} \rightarrow {BO } _ {2r} \rightarrow BO _ {2r+ 1} $, respectively, induced by the natural imbeddings $ U _ {r} \subset O _ {2r} \subset O _ {2r+ 1 }$. The series of symplectic groups $ \{ \mathop{\rm Sp} _ {r} \} $ defines a theory of symplectic cobordism, $ \mathop{\rm Sp} ^ {*} $, where $ B _ {4r} = B _ {4r+ 1} = B _ {4r+ 2} = B _ {4r+ 3} = B \mathop{\rm Sp} _ {r} $, and the $ \phi _ {r} $ are constructed in the same way as for the unitary case. There are also cobordism theories corresponding to the series of groups $ \{ \mathop{\rm Spin} _ {r} \} $, $ \{ SU _ {r} \} $, etc. Finally, the series of identity groups $ \{ E _ {r} \} $, where $ \phi _ {r} : B _ {r} \rightarrow {BO } _ {r} $ is a fibre bundle with contractible $ B _ {r} $, defines a cobordism theory which is the same as the theory of stable cohomotopy groups, and therefore the dual bordism theory is isomorphic to the theory of stable homotopy groups, $ E _ {i} ( X) \approx \pi _ {i+ N} ( S ^ {N} X ) $, $ N \gg i $. An $ E $- manifold is said to be framed (trivialized) since the $ E $- structure is precisely a frame (trivialization) of the stable normal bundle. $ E $- cobordism theory is called trivialized or framed cobordism theory, its $ i $- dimensional coefficient group being denoted by $ \Omega _ { \mathop{\rm fr} } ^ {i} $, so that $ \Omega _ { \mathop{\rm fr} } ^ {-} i = \Omega _ {i} ^ { \mathop{\rm fr} } = \pi _ {i+} N ( S ^ {N} ) $. This is the first example of a cobordism; it was due to L.S. Pontryagin, who interpreted the stable homotopy groups of the sphere as (geometrically defined) groups of framed cobordism of a point of $ \Omega _ { \mathop{\rm fr} } ^ {i} $, with the aim of computing the group $ \pi _ {i+ N }( S ^ {N} ) $.

All these cobordism theories arising from series of linear groups are multiplicative, and therefore for any finite \mathop{\rm CW}$- complex $X$, the total (graded) cobordism group is a ring. For example, for the series of groups $ \{ U _ {r} \} $ there is an imbedding $ U _ {m} \times U _ {n} \rightarrow U _ {m+n} $ inducing a mapping '"`UNIQ-MathJax5-QINU`"' and therefore a mapping $ T {BU } _ {m} \wedge T {BU } _ {n} \rightarrow T {BU } _ {m+ n} $. The spectrum $ \{ M _ {r} \} $ representing the theory $ U ^ {*} $ has the form $ M _ {2r} = T {BU } _ {r} $, $ M _ {2r+ 1} = S T BU _ {r} $, hence there exist mappings $ M _ {r} \wedge M _ {s} \rightarrow M _ {r+ s} $ so that the spectrum of spaces $ \{ M _ {r} \} $ is multiplicative. The development of cobordism theory started from the geometric definition and calculation of the groups $ \Omega _ {E} $, $ \Omega _ {O } $, $ \Omega _ {SO } $. An important role was played by Pontryagin's theorem stating that $ O $- bordant manifolds have the same Stiefel number. The study of cobordism theory was advanced by R. Thom. He introduced the spaces $ T {BO } _ {N} $, $ TB SO _ {N} $ and proved the isomorphism $ \pi _ {i+ N} ( T {BO } _ {N} ) \approx \Omega _ {SO} ^ {- i} $, enabling one to bring into the calculation of the cobordism rings some of the methods of homotopic topology. Thom's constructions stimulated the introduction of $ T {BU } _ {n} $, $ T B \mathop{\rm Sp} _ {n} $, etc., and the corresponding cobordisms. The fundamental problem of the first stage of the development of cobordism theory was the calculation of the cobordism rings of a point. In the study of the cobordism of a point a big part is played by the characteristic classes: Chern classes for $ \Omega _ {U} $, Stiefel classes for $ \Omega _ {O} $, Pontryagin and Stiefel classes for $ \Omega _ {SO} $( cf. [[Characteristic class|Characteristic class]]; [[Chern class|Chern class]]; [[Pontryagin class|Pontryagin class]]). In general, given any structural series $ ( B , \phi ) $ and any multiplicative cohomology theory $ h ^ {*} $ in which all bundles $ \xi _ {r} $ over $ B _ {r} $ are orientable, one can define the characteristic classes as elements of the group $ h ^ {*} ( B) $, where $ B = \lim\limits ( B _ {r} , j _ {r} ) $. Furthermore, the corresponding characteristic numbers, which are elements of the rings $ h ^ {*} ( \mathop{\rm pt} ) $ are invariant with respect to $ ( B , \phi ) $- bordancy. Let $ \omega = ( i _ {1} \dots i _ {k} ) $ be a partition of $ n $ and let $ S _ \omega $ be the symmetric function of $ n $ variables corresponding to $ \omega $. The characteristic class $ S _ \omega ( c _ {1} \dots c _ {n} ) $( see [[Chern class|Chern class]]) is denoted by $ S _ \omega ^ {c} $. The analogous constructions for the Pontryagin and Stiefel classes are denoted by $ S _ \omega ^ {p} $ and $ S _ \omega ^ {s} $, respectively. 1) Unitary cobordism. The ring $ \Omega _ {U} $ is the free graded polynomial algebra in a countable number of homogeneous generators '"`UNIQ-MathJax6-QINU`"' The set $ \{ x _ {n} \} $, $ \mathop{\rm deg} x _ {n} = - 2 n $, is a system of polynomial generators if and only if '"`UNIQ-MathJax7-QINU`"' where $ ( n) $ is the partition of $ n $ consisting of a single term. One of the systems of polynomial generators of $ \Omega _ {U} $ can be described as follows. Let $ \mathbf C P ^ {n} $ be $ n $- dimensional complex projective space. The complex algebraic hypersurface of bidegree $ ( 1 , 1 ) $ in $ \mathbf C P ^ {i} \times \mathbf C P ^ {j} $ is a complex manifold. Its unitary cobordism class is denoted by $ H _ {i,j} $, $ \mathop{\rm dim} _ {\mathbf R} H _ {i,j} = 2 ( i + j - 1 ) $. It turns out that '"`UNIQ-MathJax8-QINU`"' so that an appropriate linear integer combination of elements of $ H _ {i,j} $ defines a generator of $ \Omega _ {U} $ of degree $ 2 ( 1 - j - i ) $. Since $ \Omega _ {U} $ is torsion-free and $ H ^ {*} ( {BU } ; \mathbf Z ) = \mathbf Z ( c _ {1} \dots c _ {n} ,\dots ) $, where the $ c _ {n} $ are the Chern classes, $ \mathop{\rm deg} c _ {n} = 2 n $( cf. [[Chern class]]), it follows that the Chern numbers (cf. [[Chern number]]) completely determine the unitary cobordism class of an almost-complex manifold. Let $ n $ be a positive integer and let $ ( i _ {1} \dots i _ {k} ) $, $ i _ {s} > 0 $, $ \sum i _ {s} = n $, be a partition of it. There corresponds to each $ 2n $- dimensional (real dimension) almost-complex manifold $ M $ a set $ \{ a _ {i _ {1} \dots i _ {k} } \} = \{ c _ {i _ {1} } \dots c _ {i _ {k} } ( M) \} $ of integers, where the multi-index $ i _ {1} \dots i _ {k} $ runs through all the partitions of $ n $. A set of such integers $ \{ b _ {i _ {1} } \dots b _ {i _ {k} } \} $ is realized as the set of Chern numbers of some almost-complex manifold in the following situation. Let $ S _ \omega ^ {c} ( e) \in H ^ {**} ( {BU } ; \mathbf Q ) $ be the [[Characteristic class|characteristic class]] given by replacing the Wu generators $ x _ {i} $ in the representation of $ S _ \omega ^ {c} $ with the variables $ e ^ {x _ {i} } - 1 $, $ i = 1 \dots | \omega | $, and let $ T \in H ^ {**} ( {BU } ; \mathbf Q ) $ be the characteristic class given by the product of the functions $ x _ {i} / ( e ^ {x _ {i} } - 1 ) $. Let $ x ( M) $ be the value of the characteristic class $ x \in H ^ {n} ( {BU } ; \mathbf Q ) $ on the fundamental class $ [ M] \in H _ {n} ( M , \mathbf Z ) $ of the almost-complex manifold $ M $ with tangent bundle $ T M $. There exists for a homomorphism $ \phi : H ^ {n} ( {BU } ; \mathbf Q ) \rightarrow \mathbf Q $ a closed almost-complex manifold $ M $ such that $ \phi ( x) = x ( M) $ for all $ x \in H ^ {n} ( {BU } ; \mathbf Q ) $ if and only if $ \phi $ takes integer values on all the $ n $- dimensional components of each characteristic class $ S _ \omega ^ {c} ( e) T $( Stong's theorem, see [[#References|[1]]], Chapt. 7). Equivalently, the Hurewicz homomorphism '"`UNIQ-MathJax9-QINU`"' where $ N \gg k $, is a monomorphism onto a direct summand (Hattori's theorem). Here $ \widetilde{K} $ denotes reduced [[K-theory| $ K $- theory]]. 2) Non-oriented, or orthogonal, cobordism. Each element of the ring $ \Omega _ {O} $ has order $ 2 $, and '"`UNIQ-MathJax10-QINU`"' that is, $ \Omega _ {O} $ is a free polynomial $ \mathbf Z _ {2} $- algebra. One can choose as generator $ x _ {i} $ any element $ [ M] $ with $ S _ {(} i) ^ {w} ( M) \neq 0 $, for example, $ x _ {2i} = \mathbf R P ^ {2i} $. In this theory there are analogues of the manifolds $ H _ {i,j} $, obtained by replacing $ \mathbf C P ^ {k} $ by $ \mathbf R P ^ {k} $; a suitable manifold $ H _ {i,j} $ can serve as a generator of degree $ 1 - i - j $. The Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) completely define the non-orientable cobordism class of the manifold. The following theorem gives relations among the Stiefel numbers: Given a homomorphism $ \phi : H ^ {n} ( {BO } , \mathbf Z _ {2} ) \rightarrow \mathbf Z _ {2} $, there exists a closed $ n $- dimensional manifold $ M $ such that $ \phi ( x) = x ( M) $ for all $ x \in H ^ {n} ( {BO } ; \mathbf Z _ {2} ) $ if and only if $ \phi ( S q b + v b ) = 0 $ for all $ b \in H ^ {*} ( {BO } ; \mathbf Z _ {2} ) $, where $ v = S q ^ {-} 1 w $. Here $ S q = S q ^ {1} + S q ^ {2} + \dots $ is the full [[Steenrod operation|Steenrod operation]] and $ w = w _ {1} + w _ {2} + \dots $ is the full Stiefel class. The ring $ ( \Omega _ {O} ) ^ {2} $ is the image of the homomorphism $ \Omega _ {U} \rightarrow \Omega _ {O} $. 3) Oriented cobordism with ring $ \Omega _ {SO} $. All the elements of the torsion subgroup $ \mathop{\rm Tors} $ of this ring have order $ 2 $. The ring $ \Omega _ {SO} / \mathop{\rm Tors} $ is the ring of polynomials over $ \mathbf Z $ of classes $ x _ {i} $ of degree $ - 4 i $, the generators being chosen by the condition '"`UNIQ-MathJax11-QINU`"' The $ SO $- cobordism class of a manifold is determined by the Pontryagin and Stiefel numbers (cf. [[Pontryagin number|Pontryagin number]]). The [[Signature|signature]] of the manifold is also an invariant of the cobordism class. The relations among the Stiefel numbers follow from the following fact: The image of the "forgetful" homomorphism $ \Omega _ {SO} \rightarrow \Omega _ {O} $ consists precisely of those cobordism classes for which all numbers containing the class $ w _ {1} $ are zero. For any partition $ \omega = ( i _ {1} \dots i _ {k} ) $, '"`UNIQ-MathJax12-QINU`"' where $ p _ \omega $ is the corresponding Pontryagin number. There do not exist any $ 2 $- prime relations among the Pontryagin numbers. Similarly to the introduction of the classes $ S _ \omega ^ {c} ( e) $ for the unitary cobordism, the classes $ S _ \omega ^ {p} ( e) $ are introduced, which are symmetric functions in $ e ^ {x _ {i} } + e ^ {- x _ {i} } - 2 $. Let $ L $ be the characteristic class defining the Hirzebruch [[L-genus| $ L $- genus]]. All relations among the Pontryagin numbers follow from the fact that the Pontryagin numbers are integers and $ ( S _ \omega ^ {p} ( e) L ) [ M] \in \mathbf Z [ 1/2 ] $. The homomorphism $ \Omega _ {U} \rightarrow \Omega _ {SO} / \mathop{\rm Tors} $ is epimorphic. 4) Special unitary cobordism with ring $ \Omega _ {SU} $. A $ U $- manifold $ M $ has an $ SU $- structure if and only if $ c _ {1} ( M) = 0 $. All the elements of the torsion subgroup $ \mathop{\rm Tors} $ have order 2. The kernel of the homomorphism $ \Omega _ {SU} \rightarrow \Omega _ {U} $ is precisely $ \mathop{\rm Tors} $. The group $ \Omega _ {SU} ^ {n} $ is finitely generated and $ \Omega _ {SU} \otimes \mathbf Q $ is the ring of polynomials over $ \mathbf Q $ of classes $ x _ {i} $ of degree $ - 2 i $, $ i > 1 $. The torsion subgroup $ \mathop{\rm Tors} $ has the form $ \mathop{\rm Tors} ^ {-} n = 0 $ when $ n \neq 8 k + 1 , 8 k + 2 $, while for $ n = 8 k + 1 , 8 k + 2 $, $ \mathop{\rm Tors} ^ {-} n $ is a vector space over $ \mathbf Z _ {2} $ the dimension of which is the number of partitions of $ k $. Two $ SU $- manifolds are bordant if and only if they have the same characteristic number in integer cohomology and in $ KO $- theory. All relations among the Chern numbers for $ n $- dimensional $ SU $- manifolds follows from the following: $ c _ {1} c _ \omega ( M) = 0 $ for all $ \omega $; $ ( S _ \omega ^ {c} ( e) T) [ M] \in \mathbf Z $ for all $ \omega $; if $ n = 4 $ $ \mathop{\rm mod} 8 $, then $ ( S _ \omega ^ {p} ( e) T) [ M] \in 2 \mathbf Z $ for all $ \omega $. The image of the homomorphism $ \Omega _ {SU} \rightarrow \Omega _ {O} $ consists of the classes $ [ M] ^ {2} $, where $ M $ is an oriented manifold all Pontryagin numbers of which containing the class $ p _ {1} $ are even. The rings $ \Omega _ { \mathop{\rm Spin} } $ and $ \Omega _ { \mathop{\rm Spin} \mathbf C } $ have also been completely computed. The rings $ \Omega _ { \mathop{\rm Sp} } $ and $ \Omega _ { \mathop{\rm fr} } $ have to date (1986) not been computed. The ring $ \Omega _ { \mathop{\rm Sp} } \otimes \mathbf Z [ 1 / 2 ] $ is the ring of polynomials on $ ( - 4 i ) $- dimensional generators. All known (1986) elements of $ \mathop{\rm Tors} \Omega _ { \mathop{\rm Sp} } $ have order 2. (However there is an announcement of an element of order 4 in dimensions $ > 100 $.) With regard to $ \Omega _ { \mathop{\rm fr} } $, the main result here is Serre's theorem on the finiteness of these groups. The ring $ \Omega _ {S \mathbf C } $ of self-adjoint cobordism has also been studied, where the objects are almost-complex manifolds with an operator given in the normal bundle which isomorphically maps the complex structure onto the adjoint. The spectrum of $ TBS \mathbf C $ has been constructed; with regard to the groups $ \Omega _ {SO} $ it is known that there is only $ 2 $- prime torsion, but there are elements of order $ 4 ^ {k} $ for any $ k $, namely $ [ \mathbf R P ^ {4k-} 3 ] $. The image $ \mathop{\rm im} ( \Omega _ {S \mathbf C } \rightarrow \Omega _ {O} ) $ has also been calculated using the technique of formal groups (cf. [[Formal group|Formal group]]). A mapping of one cobordism theory into another, for example, $ SU ^ {*} \rightarrow U ^ {*} $, induces a mapping of the spectra $ TB SU \rightarrow T {BU } $. The cone of this mapping in the category of spectra gives a generalized cohomology theory. The ring of the point of the theory so obtained has the following geometric interpretation. Let $ ( U , SU ) $ be a $ U $- manifold on the (possibly empty) boundary of which an $ SU $- structure is fixed. By introducing the appropriate bordism relation for $ ( U , SU ) $- manifolds one obtains the ring $ \Omega _ {U , SU } $. The groups $ \Omega _ {U , \mathop{\rm fr} } $, $ \Omega _ {O , SO } $ etc., are introduced in the same way. So far, smooth manifolds have been considered or, equivalently, linear group representations (the structure series arising from the bundles over $ B SO _ {r} $). It is possible to consider various structures on topological manifolds, that is, to start from a group of homeomorphisms (and even proper homotopy equivalences) of $ \mathbf R ^ {r} $. Here the following examples are known. (Throughout, the letter $ S $ denotes passage to the oriented case.) 5) Piecewise-linear cobordism. The objects are piecewise-linear manifolds. The corresponding bordism relation leads to the groups $ \Omega _ { \mathop{\rm PL} } $, $ \Omega _ {S \mathop{\rm PL} } $. By defining the group $ { \mathop{\rm PL} } _ {n} $( or $ S { \mathop{\rm PL} } _ {n} $) as the group of piecewise-linear homeomorphisms of $ \mathbf R ^ {n} $ onto itself that preserve the origin (or the orientation as well), one can introduce the classifying spaces $ B { \mathop{\rm PL} } _ {n} $( or $ BS { \mathop{\rm PL} } _ {n} $) and the Thom spaces $ TB { \mathop{\rm PL} } _ {n} $( or $ TBS { \mathop{\rm PL} } _ {n} $) and construct a $ { \mathop{\rm PL} } $( or $ S { \mathop{\rm PL} } $) cobordism theory. In this connection, $ \Omega _ { \mathop{\rm PL} } ^ {-} i \approx \pi _ {i} ( TB { \mathop{\rm PL} }) $ and $ \Omega _ {S} { \mathop{\rm PL} } ^ {-} 1 \approx \pi _ {i} ( TBS { \mathop{\rm PL} } ) $. The groups $ \Omega _ { \mathop{\rm PL} } $ have been computed. The cobordism class of a piecewise-linear manifold is completely defined by the characteristic numbers, that is, by the elements of $ H ^ {*} ( B { \mathop{\rm PL} } ; \mathbf Z _ {2} ) $. 6) Topological cobordism. The objects are topological manifolds for which the groups $ \Omega _ { \mathop{\rm Top} } $, $ \Omega _ {S \mathop{\rm Top} } $ are defined. By considering the group $ \mathop{\rm Top} _ {n} $ of homeomorphisms of $ \mathbf R ^ {n} $ onto itself that preserve the origin, one can define the spaces $ B \mathop{\rm Top} $ and $ TP \mathop{\rm Top} _ {n} $. The groups $ \pi _ {i} ( TP \mathop{\rm Top} ) $ and $ H ^ {*} ( B \textrm{ Top } , \mathbf Z _ {2} ) $ have been computed. However, the isomorphism $ \Omega _ { \mathop{\rm Top} } ^ {-} i \approx \pi _ {i} ( TP \mathop{\rm Top} ) $ has been established for all $ i $ except $ i = 4 $. The absence of a proof of this isomorphism is tied up with the fact that the transitivity theorem on which the isomorphism $ \Omega _ {B, \phi } ^ {-} i \approx \pi _ {i} ( T ( B , \phi )) $ is based for topological manifolds, has not been proved in the general case (but it has not been refuted either (1986)). 7) Cobordism of Poincaré complexes $ \Omega _ {G} $, $ \Omega _ {SG} $. The objects are complexes with [[Poincaré duality|Poincaré duality]] and the bordism is the corresponding equivalence relation. Such complexes have a normal spherical bundle induced from the universal bundle over $ B G _ {N} $( or $ B S G _ {N} $). Here $ G _ {N} $( or $ S G _ {N} $) is an [[H-space| $ H $- space]] of homotopy equivalences (of degree 1) of the sphere $ S ^ {N} $ onto itself. The Thom spectra $ T B G $ and $ T B S G $ to which these give rise have finite homotopy groups, whereas the signature defines a non-trivial homomorphism $ \sigma : \Omega _ {SG} ^ {-} 4k \rightarrow \mathbf Z $, so that, a fortiori, the mapping $ \Omega _ {SG} ^ {-} i \rightarrow \pi _ {i+} N ( T B G _ {N} ) $ is not an isomorphism. Yet another series of examples is given by cobordism of manifolds with singularities of a special type. (This is a very good technique for the construction of various cohomology theories with special properties.) One can construct along these lines a cobordism theory that is the same as ordinary singular cohomology theory and one that is the same as connected $ K $- theory. The second stage in the development of cobordism theory is the study of cobordisms as specific generalized cohomology theories. Let $ F $ denote one of the fields $ \mathbf R , \mathbf C $ or the skew-field of quaternions $ \mathbf H $, let $ G F $ be the corresponding series of groups ( $ G \mathbf R _ {n} = O _ {n} $, $ G \mathbf C _ {n} = U _ {n} $, $ G \mathbf H _ {n} = \mathop{\rm Sp} _ {n} $) and let $ G F ^ { * } $ be the corresponding cobordism theory. A multiplicative generalized cohomology theory $ h ^ {*} $ is called $ F $- orientable if any $ F $- vector bundle is $ h ^ {*} $- orientable or, equivalently, if the canonical one-dimensional $ F $- vector bundle $ \xi \rightarrow F P ^ \infty $, where $ F P ^ \infty $ is a projective space, is $ h ^ {*} $- orientable. By an $ F $- orientation of the theory $ h ^ {*} $ one means an $ h ^ {*} $- orientation $ U _ {h} ( \xi ) \in h ^ {*} ( F P ^ \infty ) $ of the bundle $ \xi $, and a theory with a chosen orientation is called oriented. The $ G F $- cobordism theories have a canonical orientation because of the identification $ F P ^ \infty = T B G F _ {1} $. The theory $ G F ^ {*} $ is universal in the class of $ F $- oriented theories, that is, for any $ F $- oriented theory $ h ^ {*} $ with $ F $- orientation $ U _ {h} ( \xi ) $ there exists a unique multiplicative homomorphism of theories $ \phi ^ {h} : G F ^ { * } \rightarrow h ^ {*} $ under which the canonical orientation of the theory $ G F ^ { * } $ is taken to $ U _ {h} $. Moreover, when $ F _ {O} $ is one of the fields $ \mathbf R , \mathbf C $, there exist for any $ F _ {O} $- oriented theory $ h ^ {*} $ and any finite $ \mathop{\rm CW} $- complex $ X $ spectral sequences $ E _ {p,q} ^ {r} ( X) $ and $ E _ {r} ^ {p,q} ( X) $ with '"`UNIQ-MathJax13-QINU`"' '"`UNIQ-MathJax14-QINU`"' converging to $ h ^ {*} ( X) $ and natural in $ X $ and $ h ^ {*} $, where $ h ^ {*} ( \mathop{\rm pt} ) $ is made into an $ \Omega _ {G F _ {O} } $- module by means of the homomorphism $ \phi ^ {h} ( \mathop{\rm pt} ) $. If $ h _ {*} $ is the homology theory dual to the $ F $- oriented cohomology theory $ h ^ {*} $, then there is a homomorphism $ \phi _ {h} : G F _ {*} \rightarrow h _ {*} $. In the case when $ h _ {*} $ is the ordinary homology theory, it coincides with Steenrod–Thom realization of cycles (see [[Steenrod problem|Steenrod problem]]). The powerful methods of cobordism theory are connected with formal groups (cf. [[Formal group|Formal group]], [[#References|[5]]]). The most important and successful applications of cobordism theory are: the proof of the Atiyah–Singer index theorem for an elliptic operator and the general Riemann–Roch theorem; the study of fixed points of group actions; the classification of smooth (or piecewise-smooth) manifolds of given homotopy type; the proof of the theorem on the topological invariance of rational Pontryagin classes, and the solution of the problem of triangulability of topological manifolds. See also the references in [[Bordism|Bordism]]. ===='"`UNIQ--h-0--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) [https://mathscinet.ams.org/mathscinet/article?mr=0248858 MR0248858] [https://zbmath.org/?q=an%3A0181.26604 Zbl 0181.26604] </td></tr><tr><td valign="top">[2]</td> <td valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) [https://mathscinet.ams.org/mathscinet/article?mr=0176478 MR0176478] [https://zbmath.org/?q=an%3A0125.40103 Zbl 0125.40103] </td></tr><tr><td valign="top">[3]</td> <td valign="top"> S.P. Novikov, "Methods of algebraic topology from the point of view of cobordism theory" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' : 4 (1967) pp. 855–951 (In Russian) [https://mathscinet.ams.org/mathscinet/article?mr=0221509 MR0221509] </td></tr><tr><td valign="top">[4]</td> <td valign="top"> T. Bröcker, T. Tom dieck, "Kobordismentheorie" , Springer (1970) [https://mathscinet.ams.org/mathscinet/article?mr=0275446 MR0275446] [https://zbmath.org/?q=an%3A0211.55501 Zbl 0211.55501] </td></tr><tr><td valign="top">[5]</td> <td valign="top"> V.M. [V.M. Bukhshtaber] Buchstaber, "Cobordisms in problems of algebraic topology" ''J. Soviet Math.'' , '''7''' : 4 (1975) pp. 629–653 ''Itogi Nauk. i Tekh. Algebra. Geom. Topol.'' (1975) pp. 231–272</td></tr></table> ===='"`UNIQ--h-1--QINU`"'Comments==== The letter $ M $ is often used to denote Thom spaces and Thom spectra. Thus, e.g., $ M U ( n) $ is used to denote the Thom space $ T {BU } _ {n} $ and $ M U $ stands for the spectrum of all the $ M U ( n) $; similarly one uses $ M S O ( n) $ for $ TBSO _ {n} $, etc. The corresponding generalized cohomology theories are then indicated by the same symbols as is customary for generalized cohomology theories defined by a spectrum; thus, $ M U ^ {n} ( X) $ is the $ n $- th complex cobordism group of $ X $ and $ M U ^ {*} ( X) $ is its complex cobordism ring. A structural series $ ( B , \phi ) $ as defined above is often called a $ ( B , f ) $- structure (cf. [[B-Phi-structure| $ ( B , \phi ) $- structure]], [[#References|[1]]]). The general theorem that the (co)bordism group of $ n $- dimensional $ ( B , f ) $- manifolds $ \Omega _ {n} ( B , f ) $ is isomorphic to $ \lim\limits _ {r \rightarrow \infty } \pi _ {n+} r ( T B _ {r} , \infty ) $ is known as the Pontryagin–Thom theorem. A complex structure on a real vector bundle $ E $ over a manifold $ M $ is a vector bundle morphism $ J : E \rightarrow E $ such that $ J ^ {2} = - 1 $. If $ M $ is a complex imbedded manifold without boundary $ M \subset \mathbf C ^ {N} $, then multiplication with $ i $ on its normal bundle defines a complex structure on that bundle (viewed as a real bundle). A weakly-complex manifold (also called a stably (almost) complex manifold) is a real manifold with a complex structure on its stable [[Normal bundle|normal bundle]]; i.e. if $ F $ denotes the normal bundle of $ M $, then there is a complex structure defined on some $ F \oplus \theta ^ {r} $ where $ \theta ^ {r} $ stands for the trivial $ r $- dimensional bundle over $ M $, $ \theta ^ {r} = M \times \mathbf R ^ {r} $. The complex bordism groups of a space $ S $, often denoted by $ \underline{M U } {} _ {*} ( S) $, can now also be interpreted as cobordism classes of mappings $ f : M \rightarrow S $ where $ M $ is a weakly-complex manifold without boundary. There is a similar interpretation of the complex cobordism groups $ \underline{M U } {} ^ {q} ( S) $, cf. [[#References|[a3]]], and for other bordism and cobordism group. The relation between cobordism theory (and other (generalized) cohomology theories) and [[Formal group|formal group]] theory comes about as follows. A generalized cohomology theory $ h $ is complex oriented if it has first Chern classes (in a suitable sense; cf. above and [[#References|[a1]]], p. 121; [[#References|[a5]]], Part II, (2.1)). Let $ \xi $ be the class of the canonical line bundle $ H $ over $ \mathbf C P ^ \infty $, the space of lines in $ \mathbf C ^ \infty $( the fibre of $ H $ at $ x \in \mathbf C P ^ \infty $ is the line $ x $). Then $ h ^ {*} ( \mathbf C P ^ \infty ) = h ^ {*} ( \mathop{\rm pt} ) [ [ \xi ] ] $ and $ h ^ {*} ( \mathbf C P ^ \infty \times \mathbf C P ^ \infty ) = h ^ {*} ( \mathop{\rm pt} ) [ [ \xi \otimes 1 , 1 \otimes \xi ] ] $. Because $ \mathbf C P ^ \infty = \underline{BU } ( 1) $ is classifying for complex line bundles, there is an $ m : \mathbf C P ^ \infty \times \mathbf C P ^ \infty \rightarrow \mathbf C P ^ \infty $ such that $ m ^ {!} ( H) = H \otimes H $ and this induces a ring homomorphism $ h ^ {*} ( \mathbf C P ^ \infty ) \rightarrow h ^ {*} ( \mathbf C P ^ \infty \times \mathbf C P ^ \infty ) $. The image of $ \xi $ is a power series in two variables, here denoted by $ F _ {h} ( X , Y ) = \sum _ {i,j} a _ {ij} X ^ {i} Y ^ {j} $, $ a _ {ij} \in h ^ {*} ( \mathop{\rm pt} ) $, where $ X $ stands for $ \xi \otimes 1 $ and $ Y $ for $ 1 \otimes \xi $. Equivalently, $ F _ {h} $ is the power series with coefficients in $ h ( \mathop{\rm pt} ) $ such that $ c _ {1} ( H \otimes H ) = F _ {h} ( c _ {1} ^ {h} ( H \otimes 1 ) , c _ {1} ^ {h} ( 1 \otimes H ) ) $. The power series $ F _ {h} $ defines a [[Formal group|formal group]] law. Conversely, the question arises whether every (one-dimensional commutative) formal group arises as the formal group of a generalized cohomology theory. Here the study of (co)bordism of manifolds with special singularities is important. It turns out (D. Quillen [[#References|[a4]]], cf. also [[#References|[a12]]]) that for $ h = \underline{M U } $ this formal group law $ F _ {h} $ is a universal formal group law. This universality property shows up in topological terms in the form of the theorem that if $ h ^ {*} $ is any complex oriented generalized cohomology theory, then there is a unique transformation of cohomology theories $ \phi : \underline{M U } {} ^ {*} \rightarrow h ^ {*} $( linear, degree preserving and multiplicative) such that $ \phi _ {*} F _ {\underline{M U } } ( X , Y ) = F _ {h} ( X , Y ) $, where $ \phi _ {*} $ means: apply $ \phi ( \mathop{\rm pt} ) $ to the coefficients of $ F _ {\underline{M U } } ( X , Y ) $. The logarithm of the formal group law $ F _ {\underline{M U } } $ can be calculated (A.S. Mishchenko, cf. [[#References|[a12]]]; cf. [[Formal group|Formal group]] for the notion of logarithm of a formal group law). It is equal to '"`UNIQ-MathJax15-QINU`"' '"`UNIQ-MathJax16-QINU`"' where $ [ \mathbf C P ^ {n} ] \in \underline{M U } {} ^ {*} ( \mathop{\rm pt} ) $ denotes the complex cobordism class of the complex projective space of (complex) dimension $ n $. On the other hand it is possible to write down explicit formulas for the logarithm of a universal formal group law over $ \mathbf Z [ u _ {2} , u _ {3} ,\dots ] $, cf. [[#References|[a2]]], Chapt. 1. There result explicit formulas for the polynomial generators of $ \underline{M U } {} ^ {*} ( \mathop{\rm pt} ) = \mathbf Z [ u _ {1} , u _ {2} ,\dots ] $ in terms of the $ [ \mathbf C P ^ {n} ] $. These formulas take a particularly useful form for the "p-typical" version $ \underline{B P } {} ^ {*} $ of the cohomology theory $ \underline{M U } {} ^ {*} $. The generalized cohomology theory $ \underline{B P } {} ^ {*} $, Brown–Peterson cohomology for a prime number $ p $, cf. [[#References|[a6]]], is defined by a spectrum $ \underline{B P } $ and is such that $ \underline{B P } ( \mathop{\rm pt} ) = \mathbf Z _ {(} p) [ v _ {1} , v _ {2} ,\dots ] $ with $ v _ {i} $ of degree $ - 2 ( p ^ {i} - 1 ) $. It is such that $ \underline{M U } {} ^ {*} ( X) \oplus \mathbf Z _ {(} p) $ is a direct sum of (dimension shifted) copies of $ \underline{B P } {} ^ {*} ( X) $ for each space $ X $, functorially in $ X $. Here $ \mathbf Z _ {(} p) $ stands for the ring of integers localized at $ p $, i.e. $ \mathbf Z _ {(} p) = \{ {a/b \in Q } : {( p , b ) = 1 } \} $. The theory $ \underline{B P } {} ^ {*} $ can also be defined as the image of an idempotent cohomology operator $ \underline{M U } {} ^ {*} \oplus \mathbf Z _ {(} p) \rightarrow \underline{M U } {} ^ {*} \oplus \mathbf Z _ {p} $( cf., e.g., [[#References|[a1]]], Chapt. 4). This operation corresponds to $ p $- typification in formal group theory. The Hazewinkel generators ([[#References|[a1]]], pp. 137, 369-370) $ v _ {1} , v _ {2} ,\dots $ of $ \underline{B P } ^ {*} ( \mathop{\rm pt} ) $ are defined recursively by '"`UNIQ-MathJax17-QINU`"' '"`UNIQ-MathJax18-QINU`"' They arise from the explicit construction of a $ p $- typical universal formal group [[#References|[a8]]]. A different set of generators $ \overline{v}\; _ {1} \dots \overline{v}\; _ {i} \equiv v _ {i} $ $ \mathop{\rm mod} p $ has been given by S. Araki [[#References|[a7]]], the Araki generators. In a certain precise sense, $ \underline{B P } $- theory is $ \underline{M U } $- theory for one prime at the time, and currently a great deal of complex cobordism theory is written in terms of $ \underline{B P } $ rather than $ \underline{M U } $ itself. Combined with the theory of cohomology operations, formal group theory (the rings of operations $ \underline{M U } {} ^ {*} ( \underline{M U } ) $ and $ \underline{B P } {} ^ {*} ( \underline{B P } ) $ of $ \underline{M U } $ and $ \underline{B P } $ also have interpretations in terms of formal groups, cf. [[#References|[a1]]], [[#References|[a9]]], [[#References|[a10]]]), and spectral sequences, notably the Adams–Novikov spectral sequence and the chromatic spectral sequence (cf. [[#References|[a1]]], [[#References|[a11]]]), complex cobordism and Brown–Peterson cohomology have become a strong calculation tool in algebraic topology, e.g. for the stable homotopy groups of the spheres. ===='"`UNIQ--h-2--QINU`"'References==== <table><tr><td valign="top">[a1]</td> <td valign="top"> D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986) [https://mathscinet.ams.org/mathscinet/article?mr=0860042 MR0860042] [https://zbmath.org/?q=an%3A0608.55001 Zbl 0608.55001] </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) [https://mathscinet.ams.org/mathscinet/article?mr=0506881 MR0506881] [https://mathscinet.ams.org/mathscinet/article?mr=0463184 MR0463184] [https://zbmath.org/?q=an%3A0454.14020 Zbl 0454.14020] </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> D. Quillen, "Elementary proofs of some results of cobordism theory using Steenrod operations" ''Adv. Math.'' , '''7''' (1971) pp. 29–56 [https://mathscinet.ams.org/mathscinet/article?mr=0290382 MR0290382] [https://zbmath.org/?q=an%3A0214.50502 Zbl 0214.50502] </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> D. Quillen, "On the formal group laws of unoriented and complex cobordism theory" ''Bull. Amer. Math. Soc'' , '''75''' (1969) pp. 1293–1298 [https://mathscinet.ams.org/mathscinet/article?mr=0253350 MR0253350] [https://zbmath.org/?q=an%3A0199.26705 Zbl 0199.26705] </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12 [https://mathscinet.ams.org/mathscinet/article?mr=0402720 MR0402720] [https://zbmath.org/?q=an%3A0309.55016 Zbl 0309.55016] </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> E.H. Brown, F.P. Peterson, "A spectrum whose $\ZZ_p$-cohomology is the algebra of reduced $p$-th powers" ''Topology'' , '''5''' (1966) pp. 149–154</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> S. Araki, "Typical formal groups in complex cobordism and K-theory" , Kinokuniya (1973) [https://mathscinet.ams.org/mathscinet/article?mr=0375354 MR0375354] </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> M. Hazewinkel, "Constructing formal groups III: applications to complex cobordism and Brown–Peterson cohomology" ''J. Pure Appl. Algebra'' , '''10''' (1977) pp. 1–18 [https://mathscinet.ams.org/mathscinet/article?mr=0463182 MR0463182] [https://mathscinet.ams.org/mathscinet/article?mr=0463183 MR0463183] [https://mathscinet.ams.org/mathscinet/article?mr=0463184 MR0463184] [https://zbmath.org/?q=an%3A0363.14013 Zbl 0363.14013] </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> P.S. Landweber, "$BP^\ast(BP)$ and typical formal groups" Osaka J. Math. , 12 (1975) pp. 357–363 MR377945 Zbl 0311.55003 [a10] P.S. Landweber, "Invariant regular ideals in Brown–Peterson cohomology" Duke Math. J. , 42 (1975) pp. 499–505[a11] H.R. Miller, D.C. Ravenel, W.S. Wilson, "Periodic phenomena of the Adams–Novikov spectral sequence" Ann. of Math. , 106 (1977) pp. 469–516 MR458423 [a12] V.M. Bukhshtaber, A.S. Mishchenko, S.P. Novikov, "Formal groups and their role in the apparatus of algebraic topology" Russ. Math. Surveys , 26 (1971) pp. 63–90 Uspekhi Mat. Nauk , 26 (1971) pp. 131–154 Zbl 0224.57006