Bordism

Revision as of 14:07, 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 [[Wikipedia: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 bordism == <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 $\overline{M}$ denotes the manifold with the opposite structure. The universal homotopical framework for geometric bordisms with additional structure is provided by the theory of [[B-Bordism|B-bordism]]s. The simplest additional structure is an orientation. The ''oriented bordism'' relation arises accordingly. The ''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 ''stably complex'' (also known as ''weakly almost complex'', ''stably almost complex'' or ''quasicomplex'') structures. Let ${\mathcal T}\!M$ denote the tangent bundle of $M$, and $\underline{\mathbb R}^k$ the product vector bundle $M\times\mathbb R^k$ over $M$. A ''tangential stably complex structure'' on $M$ is determined by a choice of an isomorphism $$ c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi $$ 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 Chapters II and VII of \cite{Stong1968}). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A ''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 $\mathcal T\!M\oplus\nu(M)\cong\underline{\mathbb R}^N$. We therefore may restrict our attention to tangential structures only. A ''stably complex manifold'' is 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 ''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 $c_{\mathcal T}$ with $k=0$). {{beginthm|Example}} Let $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism $$ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta} $$ where $\eta$ is the Hopf line bundle. On the other hand, the isomorphism $$ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\mathbb C}^2 $$ determines a trivial stably complex structure on $\mathbb C P^1$. {{endthm}} 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 ''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 $M\times\mathbb C^k$). The sphere $S^n$ provides an example of such a 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 $$ {\mathcal T}\!M\oplus\underline{\mathbb R}^k\oplus\underline{\mathbb R}^2\stackrel{c_{\mathcal T}\oplus e}{\relbar\joinrel\hspace{-1pt}\relbar\joinrel\hspace{-1pt}\longrightarrow}\xi\oplus\underline{\mathbb C} $$ where $e\colon\mathbb R^2\to\mathbb C$ is given by $e(x,y)=x-iy$. An abbreviated notation $[M]$ for the complex bordism class will be used whenever the stably complex structure $c_{\mathcal T}$ is clear from the context. The ''groups of complex bordisms'' and ''cobordisms'' of a space $X$ are defined similarly to the [[#Unoriented bordism|unoriented]] case: $$ \begin{aligned} U_n(X)&=\lim_{k\to\infty}\pi_{2k+n}((X_+)\wedge MU(k)),\ U^n(X)&=\lim_{k\to\infty}[\Sigma^{2k-n}(X_+),MU(k)] \end{aligned} $$ where $MU(k)$ is the Thom space of the universal complex $k$-plane bundle $EU(k)\to BU(k)$. These groups are $\varOmega_*^U$-modules and give rise to a multiplicative [[Wikipedia:Homology_theory|(co)homology theory]]. In particular, $U^*(X)=\oplus_n U^n(X)$ is a graded ring. The graded ring $\varOmega^*_U$ with $\varOmega^{n}_U=\varOmega_{-n}^U$ is called the ''complex cobordism ring''; it has nontrivial elements only in nonpositively graded components. </wikitex> == Structure results == <wikitex>; The theory of unoriented (co)bordism was completed by \cite{Thom1954}: the coefficient ring $\varOmega_*^O$ was calculated, and the bordism groups $O_*(X)$ of cell complexes $X$ were reduced to homology groups of $X$ with coefficients in $\varOmega_*^O$. The corresponding results are summarised as follows. {{beginthm|Theorem|(Thom)}} \ \begin{itemize} \item[1.] Two manifolds are unorientedly bordant if and only if they have identical sets of Stiefel--Whitney characteristic classes. \item[2.] $\varOmega_*^O$ is a polynomial ring over $\mathbb Z/2$ with one generator $a_i$ in every positive dimension $i\ne 2^k-1$. \item[3.] For every cell complex $X$ the module $O_*(X)$ is a free graded $\varOmega_*^O$-module isomorphic to $H_*(X;\mathbb Z/2)\otimes_{\mathbb Z/2}\varOmega_*^O$. \end{itemize} {{endthm}} Calculating the complex bordism ring $\varOmega_*^U$ turned out to be a much more difficult problem, which was solved by \cite{Novikov1960} \cite{Novikov1962} and Milnor (unpublished) in 1960. Here is the summary of these results. {{beginthm|Theorem}} \ \begin{itemize} \item[1.] $\varOmega_*^U\otimes\mathbb Q$ is a polynomial ring over $\mathbb Q$ generated by the bordism classes of complex projective spaces $\mathbb C P^i$, \ $i\ge1$. \item[2.] Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic classes. \item[3.] $\varOmega_*^U$ is a polynomial ring over $\mathbb Z$ with one generator $a_i$ in every even dimension i$, where $i\ge1$. \end{itemize} {{endthm}} Note that part 3 of Theorem~\ref{uocob} does not generalise to complex bordisms; $U_*(X)$ is not a free $\varOmega_*^U$-module in general. Therefore, unlike the unoriented bordisms, the calculation of complex bordisms of a space $X$ does not reduce to calculating the coefficient ring $\varOmega^U_*$ and homology groups $H_*(X)$. The theory of complex (co)bordisms is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordisms or other bordisms with additional structure. Thanks to this, the complex cobordism theory found the most stricking and important applications in algebraic topology and beyond. Many of these applications were outlined in the pioneering work of Novikov~\cite{novi67}. The calculation of the oriented bordism ring was also completed by Novikov in~\cite{novi60}, with important contributions made by Rokhlin, Averbuch, Wall and Milnor. Unlike complex bordisms, the ring $\varOmega_*^{SO}$ has additive torsion. We give only partial result here (which does not fully describe the torsion elements). \begin{theorem}\ \begin{itemize} \item[1.] $\varOmega_*^{SO}\otimes\Q$ is a polynomial ring over $\Q$ generated by the bordism classes of complex projective spaces $\C P^{2i}$, \ $i\ge1$. \item[2.] The subring $\mathrm{Tors}\subset\varOmega_*^{SO}$ of torsion elements contains only elements of order~2. The ring $\varOmega_*^{SO}/\mathrm{Tors}$ is a polynomial ring over $\Z$ with one generator $a_i$ in every dimension i$, where $i\ge1$. \item[3.] Two oriented manifolds are bordant if and only if they have identical sets of Pontrjagin and Stiefel--Whitney characteristic classes. \end{itemize} \end{theorem} </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

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$\varOmega_n^O$ 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

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between bordism and cobordism appears only for nontrivial spaces $X$.

4 Oriented and complex bordism

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 $\overline{M}$ denotes the manifold with the opposite structure. The universal homotopical framework for geometric bordisms with additional structure is provided by the theory of B-bordisms.

The simplest additional structure is an orientation. The oriented bordism relation arises accordingly. The 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 stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) structures.

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determined by a choice of an isomorphism

$$\displaystyle c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi$$

between the "stable" tangent bundle and a complex vector

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are deemed to be equivalent, i.e. determining the same stably complex structures (see details in Chapters II and VII of [Stong1968]). In particular, two stably complex structures are equivalent if they

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structure in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A tangential and normal stably

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canonical isomorphism $\mathcal T\!M\oplus\nu(M)\cong\underline{\mathbb R}^N$. We therefore may restrict our attention to tangential structures only.

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$c_{\mathcal T}$ on it. This is a generalisation to a complex and 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 $c_{\mathcal T}$ with $k=0$).

Example 4.1.

Let $M=\mathbb{C}P^1$. The standard complex structure on

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$M$ is

equivalent to a stably complex structure determined by the isomorphism

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

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

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

determines a trivial stably complex structure on $\mathbb C P^1$.

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 group of $n$-dimensional complex bordisms and denoted $\varOmega^U_n$. A zero is represented by the bordism

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bundle is trivial (and therefore isomorphic to a product complex vector bundle $M\times\mathbb C^k$). The sphere $S^n$ provides an example of such a manifold. The opposite element to the bordism class $[M,c_{\mathcal T}]$ in the group $\varOmega^U_n$ may be

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structure determined by the isomorphism

$$\displaystyle {\mathcal T}\!M\oplus\underline{\mathbb R}^k\oplus\underline{\mathbb R}^2\stackrel{c_{\mathcal T}\oplus e}{\relbar\joinrel\hspace{-1pt}\relbar\joinrel\hspace{-1pt}\longrightarrow}\xi\oplus\underline{\mathbb C}$$

where $e\colon\mathbb R^2\to\mathbb C$ is given by $e(x,y)=x-iy$.

An abbreviated notation $[M]$ for the complex bordism class will be used whenever the stably complex structure $c_{\mathcal T}$ is clear from the context.

The groups of complex bordisms and cobordisms of a space $X$ are defined similarly to the unoriented case:

$$\displaystyle \begin{aligned} U_n(X)&=\lim_{k\to\infty}\pi_{2k+n}((X_+)\wedge MU(k)),\\ U^n(X)&=\lim_{k\to\infty}[\Sigma^{2k-n}(X_+),MU(k)] \end{aligned}$$

where $MU(k)$ is the Thom space of the universal complex $k$-plane bundle $EU(k)\to BU(k)$. These groups are $\varOmega_*^U$-modules and give rise to a multiplicative (co)homology theory. In particular, $U^*(X)=\oplus_n U^n(X)$ is a graded ring. The graded ring $\varOmega^*_U$ with $\varOmega^{n}_U=\varOmega_{-n}^U$ is called the complex cobordism ring; it has nontrivial elements only in nonpositively graded components.

5 Structure results

The theory of unoriented (co)bordism was completed by [Thom1954]: the coefficient ring $\varOmega_*^O$ was calculated, and the bordism groups $O_*(X)$ of cell complexes $X$ were reduced to homology groups of $X$ with coefficients in $\varOmega_*^O$. The corresponding results are summarised as follows.

Calculating the complex bordism ring $\varOmega_*^U$ turned out to be a much more difficult problem, which was solved by [Novikov1960] [Novikov1962] and Milnor (unpublished) in 1960. Here is the summary of these results.

Note that part 3 of Theorem~\ref{uocob} does not generalise to complex bordisms; $U_*(X)$ is not a free $\varOmega_*^U$-module in general. Therefore, unlike the unoriented bordisms, the calculation of complex bordisms of a space $X$ does not reduce to calculating the coefficient ring $\varOmega^U_*$ and homology groups $H_*(X)$. The theory of complex (co)bordisms is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordisms or other bordisms with additional structure. Thanks to this, the complex cobordism theory found the most stricking and important applications in algebraic topology and beyond. Many of these applications were outlined in the pioneering work of Novikov~[novi67].

The calculation of the oriented bordism ring was also completed by Novikov in~[novi60], with important contributions made by Rokhlin, Averbuch, Wall and Milnor. Unlike complex bordisms, the ring $\varOmega_*^{SO}$ has additive torsion. We give only partial result here (which does not fully describe the torsion elements).

Theorem 5.3.\ \begin{itemize}

\item[1.]

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$\varOmega_*^{SO}\otimes\Q$ is a polynomial ring over $\Q$ generated by the bordism classes of complex projective spaces

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$\C P^{2i}$, \ $i\ge1$.

\item[2.] The subring $\mathrm{Tors}\subset\varOmega_*^{SO}$ of torsion elements contains only elements of order~2. The ring $\varOmega_*^{SO}/\mathrm{Tors}$ is a polynomial ring over $\Z$ with one generator $a_i$ in every dimension $4i$, where $i\ge1$.

\item[3.] Two oriented manifolds are bordant if and only if they have identical sets of Pontrjagin and Stiefel--Whitney characteristic classes. \end{itemize}

6 References

• [Novikov1960] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
• [Novikov1962] S. P. Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442. MR0157381 (28 #615) Zbl 0193.51801
• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
• [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
• [novi60] Template:Novi60
• [novi67] Template:Novi67