Bordism

Revision as of 12:17, 10 March 2010

1 Introduction

The theory of bordism is is one of the most deep and influential parts of the algebraic topology, which experienced a spectacular development in the 1960s. The main introductory reference is the monograph [Stong1968].

Basic geometric constructions of bordisms and cobordisms, as well as homotopical definitions are summarised here. For the more specific information, see B-Bordism and pages on specific bordism theories, such as unoriented, oriented and complex.

2 The bordism relation

All manifolds here are assumed to be smooth, compact and closed (without boundary), unless otherwise specified. Given two ${{Authors|Taras Panov}} == Introduction == The theory of bordism is is one of the most deep and influential parts of the algebraic topology, which experienced a spectacular development in the 1960s. The main introductory reference is the monograph \cite{Stong1968}. Basic geometric constructions of bordisms and cobordisms, as well as homotopical definitions are summarised here. For the more specific information, see [[B-Bordism]] and pages on specific bordism theories, such as [[unoriented bordism|unoriented]], [[oriented bordism|oriented]] and [[complex bordism|complex]]. == The bordism relation == <wikitex>; All manifolds here are assumed to be smooth, compact and closed (without boundary), unless otherwise specified. Given two $n$-dimensional manifolds $M_1$ and $M_2$, a ''bordism'' between them is an $(n+1)$-dimensional manifold $W$ with boundary, whose boundary is the disjoint union of $M_1$ and $M_2$, that is, $\partial W=M_1\sqcup M_2$. If such $W$ exists, $M_1$ and $M_2$ are called ''bordant''. The bordism relation splits manifolds into equivalence classes (see Figure), which are called ''bordism classes''. [[Image:trcob.jpg|thumb|300px|Transitivity of the bordism relation]] </wikitex> == Unoriented bordism == <wikitex>; We denote the bordism class of $M$ by $[M]$, and denote by $\varOmega_n^O$ the set of bordism classes of $n$-dimensional manifolds. Then $\varOmega_n^O$ is an abelian group with respect to the disjoint union operation: $[M_1]+[M_2]=[M_1\sqcup M_2]$. Zero is represented by the bordism class of an empty set (which is counted as a manifold in any dimension), or by the bordism class of any manifold which bounds. We also have $-[M]=[M]$, so that $\varOmega_n^O$ is a 2-torsion group. Set $\varOmega _*^O:=\bigoplus _{n \ge 0}\varOmega _n^O$. The product of bordism classes, namely $[M_1]\times [M_2]=[M_1 \times M_2]$, makes $\varOmega_*^O$ a graded commutative ring known as the ''unoriented bordism ring''. For any (good) space $X$ the bordism relation can be extended to maps of $n$-dimensional manifolds to $X$: two maps $M_1\to X$ and $M_2\to X$ are ''bordant'' if there is a bordism $W$ between $M_1$ and $M_2$ and the map $M_1\sqcup M_2\to X$ extends to a map $W\to X$. The set of bordism classes of maps $M\to X$ forms an abelian group called the ''group of $n$-dimensional unoriented bordisms of $X$'' and denoted $O_n(X)$ (other notations: $N_n(X)$, $MO_n(X)$). The assignment $X\mapsto O_*(X)$ defines a [[generalized homology theory|generalised homology theory]], that is, satisfies the homotopy invariance, has the excision property and exact sequences of pairs. For this theory we have $O_*(pt)=\varOmega_*^O$, and $O_*(X)$ is an $\varOmega_*^O$-module. The ''Pontrjagin--Thom construction'' reduces the calculation of the bordism groups to a homotopical problem: $$ O_n(X)=\lim_{k\to\infty}\pi_{k+n}\bigl((X_+)\wedge MO(k)\bigr) $$ where $X_+=X\sqcup pt$, and $MO(k)$ is the ''Thom space'' of the universal vector $k$-plane bundle $EO(k)\to BO(k)$. The ''cobordism groups'' are defined dually: $$ O^n(X)=\lim_{k\to\infty}[\Sigma^{k-n}(X_+),MO(k)] $$ where $[X,Y]$ denotes the set of homotopy classes of maps from $X$ to $Y$. The resulting generalised cohomology theory is multiplicative, which implies that $O^*(X)=\oplus_n O^n(X)$ is a graded commutative ring. It follows from the definitions that $O^n(pt)=O_{-n}(pt)$. The graded ring $\varOmega^*_O$ with $\varOmega^{-n}_O:=O^{-n}(pt)=\varOmega_n^O$ is called the ''unoriented cobordism ring''. It has nonzero elements only in nonpositively graded components. The bordism ring $\varOmega^O_*$ and the cobordism ring $\varOmega_O^*$ differ only by their gradings, so the notions of the "bordism class" and "cobordism class" of a manifold $M$ are interchangeable. The difference between bordism and cobordism appears only for nontrivial spaces $X$. </wikitex> == Oriented and complex == <wikitex>; The most important examples of bordism theories arise from extending the bordism relation to manifolds endowed with some additional structure. To take account of this structure in the definition of bordism one requires that $\partial W=M_1\sqcup\overline{M}_2$, where the structure on $\partial W$ is induced from that on $W$, and $\overline{M}_2$ denotes the manifold with the opposite structure. The universal homotopical frameworks for geometric bordisms with additional structure is provided by the theory of [[B-bordism]]s. The simplest additional structure is an orientation. The \emph{oriented bordism} relation arises accordingly. The \emph{oriented bordism ring} $\varOmega_*^{SO}$ is defined similarly to $\varOmega_*^O$, with the only difference that $-[M]=[\overline{M}]$. Elements of $\varOmega_*^{SO}$ generally do not have order~2. Complex structure gives another important example of an additional structure on manifolds. However, a direct attempt to define the bordism relation on complex manifolds fails because the manifold $W$ is odd-dimensional and therefore cannot be complex. This can be remedied by considering \emph{stably complex} (also known as \emph{stably almost complex} or \emph{quasicomplex}) structures. Let ${\mathcal T}\!M$ denote the tangent bundle of~$M$, and $\underline{\R}^k$ the product vector bundle $M\times\R^k$ over~$M$. A \emph{tangential stably complex structure} on $M$ is determined by a choice of an isomorphism \begin{equation}\label{scs} c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\R}^k\to \xi \end{equation} between the ``stable'' tangent bundle and a complex vector bundle~$\xi$ over~$M$. Some of the choices of such isomorphisms are deemed to be equivalent, i.e. determining the same stably complex structures (see details in~\cite[Ch.~II,~VII]{ston68}). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A \emph{normal stably complex structure} on $M$ is determined by a choice of a complex bundle structure in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A tangential and normal stably complex structures on $M$ determine each other by means of the canonical isomorphism $\mathcal T\!M\oplus\nu(M)\cong\underline{\R}^N$. We therefore may restrict our attention to tangential structures only. By a \emph{stably complex manifold} we mean a pair $(M,c_{\mathcal T})$ consisting of a manifold $M$ and a stably complex structure $c_{\mathcal T}$ on it. This is a generalisation to a complex and \emph{almost complex} manifold (where the latter means a manifold with a choice of a complex structure on ${\mathcal T}\!M$, i.e. a stably complex structure~\eqref{scs} with $k=0$). %The following %example shows that a manifold may admit many different stably %complex structures. \begin{example}\label{2cp1} Let $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism \[ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta} \] where $\eta$ is the Hopf line bundle. On the other hand, the isomorphism \[ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\C}^2 \] determines a trivial stably complex structure on~$\mathbb C P^1$. \end{example} The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordisms, the set of bordism classes $[M,c_{\mathcal T}]$ of stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the \emph{group of $n$-dimensional complex bordisms} and denoted $\varOmega^U_n$. A zero is represented by the bordism class of any manifold $M$ which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle $M\times\C^k$). The sphere $S^n$ provides an example of such manifold. The opposite element to the bordism class $[M,c_{\mathcal T}]$ in the group $\varOmega^U_n$ may be represented by the same manifold $M$ with the stably complex structure determined by the isomorphism \[ {\mathcal T}\!M\oplus\underline{\R}^k\oplus\underline{\R}^2\stackrel{c_{\mathcal T}\oplus e}{\lllra}\xi\oplus\underline{\C} \] where $c_{\mathcal T}$ is given as in~\eqref{scs}, and $e\colon\R^2\to\C$, $e(x,y)=x-iy$. We shall use the abbreviated notation $[M]$ for the complex bordism class whenever the stably complex structure $c_{\mathcal T}$ is clear from the context. </wikitex> <!-- COMMENT: To achieve a unified layout, along with using the template below, please observe the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Theory]] {{Stub}}n$-dimensional manifolds $M_1$ and $M_2$, a bordism between them is an $(n+1)$-dimensional manifold $W$ with boundary, whose boundary is the disjoint union of $M_1$ and $M_2$, that is, $\partial W=M_1\sqcup M_2$. If such $W$ exists, $M_1$ and $M_2$ are called bordant. The bordism relation splits manifolds into equivalence classes (see Figure), which are called bordism classes.

Transitivity of the bordism relation

3 Unoriented bordism

We denote the bordism class of $M$ by $[M]$, and denote by $\varOmega_n^O$ the set of bordism classes of $n$-dimensional manifolds. Then $\varOmega_n^O$ is an abelian group with respect to the disjoint union operation: $[M_1]+[M_2]=[M_1\sqcup M_2]$. Zero is represented by the bordism class of an empty set (which is counted as a manifold in any dimension), or by the bordism class of any manifold which bounds. We also have $-[M]=[M]$, so that $\varOmega_n^O$ is a 2-torsion group.

Set $\varOmega _*^O:=\bigoplus _{n \ge 0}\varOmega _n^O$. The product of bordism classes, namely $[M_1]\times [M_2]=[M_1 \times M_2]$, makes $\varOmega_*^O$ a graded commutative ring known as the unoriented bordism ring.

For any (good) space $X$ the bordism relation can be extended to maps of $n$-dimensional manifolds to $X$: two maps $M_1\to X$ and $M_2\to X$ are bordant if there is a bordism $W$ between $M_1$ and $M_2$ and the map $M_1\sqcup M_2\to X$ extends to a map $W\to X$. The set of bordism classes of maps $M\to X$ forms an abelian group called the group of $n$-dimensional unoriented bordisms of $X$ and denoted $O_n(X)$ (other notations: $N_n(X)$, $MO_n(X)$).

The assignment $X\mapsto O_*(X)$ defines a generalised homology theory, that is, satisfies the homotopy invariance, has the excision property and exact sequences of pairs. For this theory we have $O_*(pt)=\varOmega_*^O$, and $O_*(X)$ is an $\varOmega_*^O$-module.

The Pontrjagin--Thom construction reduces the calculation of the bordism groups to a homotopical problem:

$$\displaystyle O_n(X)=\lim_{k\to\infty}\pi_{k+n}\bigl((X_+)\wedge MO(k)\bigr)$$

where $X_+=X\sqcup pt$, and $MO(k)$ is the Thom space of the universal vector $k$-plane bundle $EO(k)\to BO(k)$. The cobordism groups are defined dually:

$$\displaystyle O^n(X)=\lim_{k\to\infty}[\Sigma^{k-n}(X_+),MO(k)]$$

where $[X,Y]$ denotes the set of homotopy classes of maps from $X$ to $Y$. The resulting generalised cohomology theory is multiplicative, which implies that $O^*(X)=\oplus_n O^n(X)$ is a graded commutative ring. It follows from the definitions that $O^n(pt)=O_{-n}(pt)$. The graded ring $\varOmega^*_O$ with $\varOmega^{-n}_O:=O^{-n}(pt)=\varOmega_n^O$ is called the unoriented cobordism ring. It has nonzero elements only in nonpositively graded components. The bordism ring $\varOmega^O_*$ and the cobordism ring $\varOmega_O^*$ differ only by their gradings, so the notions of the "bordism class" and "cobordism class" of a manifold $M$ are interchangeable. The difference between bordism and cobordism appears only for nontrivial spaces $X$.

4 Oriented and complex

The most important examples of bordism theories arise from extending the bordism relation to manifolds endowed with some additional structure. To take account of this structure in the definition of bordism one requires that $\partial W=M_1\sqcup\overline{M}_2$, where the structure on $\partial W$ is induced from that on $W$, and $\overline{M}_2$ denotes the manifold with the opposite structure. The universal homotopical frameworks for geometric bordisms with additional structure is provided by the theory of B-bordisms.

The simplest additional structure is an orientation. The \emph{oriented bordism} relation arises accordingly. The \emph{oriented bordism ring} $\varOmega_*^{SO}$ is defined similarly to $\varOmega_*^O$, with the only difference that $-[M]=[\overline{M}]$. Elements of $\varOmega_*^{SO}$ generally do not have order~2.

Complex structure gives another important example of an additional structure on manifolds. However, a direct attempt to define the bordism relation on complex manifolds fails because the manifold $W$ is odd-dimensional and therefore cannot be complex. This can be remedied by considering \emph{stably complex} (also known as \emph{stably almost complex} or \emph{quasicomplex}) structures.

Let ${\mathcal T}\!M$ denote the tangent bundle of~$M$, and

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over~$M$. A \emph{tangential stably complex structure} on $M$ is determined by a choice of an isomorphism

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between the ``stable tangent bundle and a complex vector bundle~$\xi$ over~$M$. Some of the choices of such isomorphisms are deemed to be equivalent, i.e. determining the same stably complex structures (see details in~[ston68, Ch.~II,~VII]). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A \emph{normal stably complex structure} on $M$ is determined by a choice of a complex bundle structure in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A tangential and normal stably complex structures on $M$ determine each other by means of the

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our attention to tangential structures only.

By a \emph{stably complex manifold} we mean a pair $(M,c_{\mathcal T})$ consisting of a manifold $M$ and a stably complex structure $c_{\mathcal T}$ on it. This is a generalisation to a complex and \emph{almost complex} manifold (where the latter means a manifold with a choice of a complex structure on ${\mathcal T}\!M$, i.e. a stably complex structure~\eqref{scs} with $k=0$). %The following %example shows that a manifold may admit many different stably %complex structures.

\begin{example} Let $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism \[

 {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\R}^2\stackrel{\cong}{\longrightarrow}
 \overline{\eta}\oplus \overline{\eta}

\] where $\eta$ is the Hopf line bundle. On the other hand, the isomorphism \[

 {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\R}^2\stackrel{\cong}{\longrightarrow}
 \eta\oplus \overline{\eta}\cong \underline{\C}^2

\] determines a trivial stably complex structure on~$\mathbb C P^1$. \end{example}

The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordisms, the set of bordism classes $[M,c_{\mathcal T}]$ of stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the \emph{group of $n$-dimensional complex bordisms} and denoted $\varOmega^U_n$. A zero is represented by the bordism class of any manifold $M$ which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex

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of such manifold. The opposite element to the bordism class $[M,c_{\mathcal T}]$ in the group $\varOmega^U_n$ may be represented by the same manifold $M$ with the stably complex structure determined by the isomorphism \[

 {\mathcal T}\!M\oplus\underline{\R}^k\oplus\underline{\R}^2\stackrel{c_{\mathcal T}\oplus
 e}{\lllra}\xi\oplus\underline{\C}

\] where $c_{\mathcal T}$ is given as in~\eqref{scs}, and

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We shall use the abbreviated notation $[M]$ for the complex bordism class whenever the stably complex structure $c_{\mathcal T}$ is clear from the context.

5 References

• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
• [ston68] Template:Ston68