Bordism

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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 $n$${{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)] $$ where $[X,Y]$ denotes the set of homotopy classes of maps from $X$ to $Y$. The resulting generalised cohomology theory is multiplicative, which implies that $O^*(X)=\oplus_n O^n(X)$ is a graded commutative ring. It follows from the definitions that $O^n(pt)=O_{-n}(pt)$. The graded ring $\varOmega^*_O$ with $\varOmega^{-n}_O:=O^{-n}(pt)=\varOmega_n^O$ is called the ''unoriented cobordism ring''. It has nonzero elements only in nonpositively graded components. The bordism ring $\varOmega^O_*$ and the cobordism ring $\varOmega_O^*$ differ only by their gradings, so the notions of the "bordism class" and "cobordism class" of a manifold $M$ are interchangeable. The difference between bordism and cobordism appears only for nontrivial spaces $X$. </wikitex> <!-- COMMENT: To achieve a unified layout, along with using the template below, please observe the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Theory]] {{Stub}} [[File:[[File:Example.jpg]]]]n$-dimensional manifolds $M_1$$M_1$ and $M_2$$M_2$, a bordism between them is an $(n+1)$$(n+1)$-dimensional manifold $W$$W$ with boundary, whose boundary is the disjoint union of $M_1$$M_1$ and $M_2$$M_2$, that is, $\partial W=M_1\sqcup M_2$$\partial W=M_1\sqcup M_2$. If such $W$$W$ exists, $M_1$$M_1$ and $M_2$$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$$M$ by $[M]$$[M]$, and denote by $\varOmega_n^O$$\varOmega_n^O$ the set of bordism classes of $n$$n$-dimensional manifolds. Then $\varOmega_n^O$$\varOmega_n^O$ is an abelian group with respect to the disjoint union operation: $[M_1]+[M_2]=[M_1\sqcup M_2]$$[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]$$-[M]=[M]$, so that $\varOmega_n^O$$\varOmega_n^O$ is a 2-torsion group.

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

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

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

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

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

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

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

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

4 References

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

This page has not been refereed. The information given here might be incomplete or provisional.

[[File:File:Example.jpg]]