Complex bordism

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1 Introduction

Complex bordism (also known as unitary bordism) is the bordism theory of stably complex manifolds. It is one of the most important theories of bordism with additional structure, or B-bordism.

The theory of complex bordism is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordism or other bordism theories with additional structure (B-bordism). Thanks to this, the complex cobordism theory has found the most striking and important applications in algebraic topology and beyond. Many of these applications, including the formal group techniques and Adams-Novikov spectral sequence were outlined in the pioneering work [Novikov1967].

2 Stably complex structures

A direct attempt to define the bordism relation on complex manifolds fails because the manifold $== Introduction == <wikitex>;  ''Complex bordism'' (also known as ''unitary bordism'') is the [[Bordism|bordism theory]] of [[#Stably complex structures|stably complex manifolds]]. It is one of the most important theories of bordism with additional structure, or [[B-Bordism|B-bordism]]. The theory of complex bordism is much richer than its [[Bordism#Unoriented bordism|unoriented]] analogue, and at the same time is not as complicated as [[Oriented bordism|oriented bordism]] or other bordism theories with additional structure ([[B-Bordism|B-bordism]]). Thanks to this, the complex cobordism theory has found the most striking and important applications in algebraic topology and beyond. Many of these applications, including the [[Formal group laws and genera|formal group techniques]] and [[#Adams-Novikov spectral sequence|Adams-Novikov spectral sequence]] were outlined in the pioneering work \cite{Novikov1967}. </wikitex> == Stably complex structures == <wikitex>; A direct attempt to define the [[Bordism#The bordism relation|bordism relation]] on complex manifolds fails because the manifold $\,W$ is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of a complex structure. This leads directly to considering ''stably complex'' (also known as ''weakly almost complex'', ''stably almost complex'' or ''quasicomplex'') manifolds. 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 $M\hookrightarrow\mathbb R^N$. 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 of 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 [[Wikipedia:Hopf_bundle|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}} </wikitex> == Definition of bordism and cobordism == <wikitex>; The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordism, 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 ''$n$-dimensional complex bordism group'' 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 ''complex bordism group'' $U_n(X)$ and ''cobordism group'' $U^n(X)$ of a space $X$ may also be defined geometrically, at least for the case when $X$ is a manifold. This can be done along the lines suggested by \cite{Quillen1971a} and \cite{Dold1978} by considering special "stably complex" maps of manifolds $M$ to $X$. However, nowadays the homotopical approach to bordism takes over, and the (co)bordism groups are usually defined using the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom construction]] similarly to the [[Bordism#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> == Geometric cobordisms == <wikitex>; There is one important case when certain cobordism classes can be represented very explicitly by maps of manifolds. For any cell complex $X$ the cohomology group $H^2(X)$ can be identified with the set $[X,\mathbb C P^\infty]$ of homotopy classes of maps into $\mathbb C P^\infty$. Since $\mathbb C P^\infty=MU(1)$, every element $x\in H^2(X)$ also determines a cobordism class $u_x\in U^2(X)$. The elements of $U^2(X)$ obtained in this way are called ''geometric cobordisms'' of $X$. We therefore may view $H^2(X)$ as a subset in $U^2(X)$, however the group operation in $H^2(X)$ is not obtained by restricting the group operation in $U^2(X)$ (see [[#Formal group laws and genera|Formal group laws and genera]] for the relationship between the two operations). When $X$ is a manifold, geometric cobordisms may be described by submanifolds $M\subset X$ of codimension 2 with a fixed complex structure in the normal bundle. Indeed, every $x\in H^2(X)$ corresponds to a homotopy class of maps $f_x\colon X\to\mathbb C P^\infty$. The image $f_x(X)$ is contained in some $\mathbb C P^N\subset\mathbb C P^\infty$, and we may assume that $f_x(X)$ is transversal to a certain hyperplane $H\subset\mathbb C P^N$. Then $M_x:=f_x^{-1}(H)$ is a codimension 2 submanifold in $X$ whose normal bundle acquires a complex structure by restriction of the complex structure in the normal bundle of $H\subset\mathbb C P^N$. Changing the map $f_x$ within its homotopy class does not affect the bordism class of embedding $M_x\to X$. Conversely, assume given a submanifold $M\subset X$ of codimension 2 whose normal bundle is endowed with a complex structure. Then the composition $$ X\to M(\nu)\to MU(1)=\mathbb C P^\infty $$ of the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom collapse map]] $X\to M(\nu)$ and the map of Thom spaces corresponding the the classifying map $M\to BU(1)$ of $\nu$ defines and element $x_M\in H^2(X)$, and therefore a geometric cobordism. If $X$ is an oriented manifold, then a choice of complex structure in the normal bundle of a codimension 2 embedding $M\subset X$ is equivalent to orienting $M$. The image of the fundamental class of $M$ in the homology of $X$ is Poincare dual to $x_M\in H^2(X)$. </wikitex> == Structure results == <wikitex>; Complex bordism ring $\varOmega_*^U$ is described as follows. {{beginthm|Theorem}} #$\varOmega_*^U\otimes\mathbb Q$ is a polynomial ring over $\mathbb Q$ generated by the bordism classes of complex projective spaces $\mathbb C P^i$, $i\ge1$. #Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers. #$\varOmega_*^U$ is a polynomial ring over $\mathbb Z$ with one generator $a_i$ in every even dimension i$, where $i\ge1$. {{endthm}} Part 1 can be proved by the methods of \cite{Thom1954}. Part 2 follows from the results of \cite{Milnor1960} and \cite{Novikov1960}. Part 3 is the most difficult one; it was done by \cite{Novikov1960} using Adams spectral sequence and structure theory of Hopf algebras (see also \cite{Novikov1962} for a more detailed account) and Milnor (unpublished, but see \cite{Thom1995}) in 1960. Another more geometric proof was given by \cite{Stong1965}, see also \cite{Stong1968}. </wikitex> == Multiplicative generators == === Preliminaries: characteristic number <i>s<sub>n</sub></i> === <wikitex>; To describe a set of multiplicative generators for the ring $\varOmega_*^U$ we shall need a special characteristic class of complex vector bundles. Let $\xi$ be a complex $k$-plane bundle over a manifold~$M$. Write formally its total Chern class as follows: $$ c(\xi )=1+c_1(\xi )+\ldots +c_k(\xi )=(1+x_1)\dots(1+x_k), $$ so that $c_i(\xi )=\sigma_i(x_1,\ldots,x_k)$ is the $i$th elementary symmetric function in formal indeterminates. These indeterminates acquire a geometric meaning if $\xi$ is a sum $\xi_1\oplus\ldots\oplus\xi_k$ of line bundles; then $x_j=c_1(\xi_j)$, \,W$ is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of a complex structure. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.

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

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

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 [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 $M\hookrightarrow\mathbb R^N$ . 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 of 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 $k=0$ ).

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

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

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

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

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

3 Definition of bordism and cobordism

The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordism, the set of bordism classes $[M,c_{\mathcal T}]$ of stably complex manifolds of dimension $n$ is an Abelian group with respect to the disjoint union. This group is called the $n$ -dimensional complex bordism group and denoted $\varOmega^U_n$ . The zero element 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

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

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 complex bordism group $U_n(X)$ and cobordism group $U^n(X)$ of a space $X$ may also be defined geometrically, at least for the case when $X$ is a manifold. This can be done along the lines suggested by [Quillen1971a] and [Dold1978] by considering special "stably complex" maps of manifolds $M$ to $X$ . However, nowadays the homotopical approach to bordism has taken over, and the (co)bordism groups are usually defined using the Pontrjagin-Thom construction similarly to the unoriented case:

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

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.

4 Geometric cobordisms

There is one important case when certain cobordism classes can be represented very explicitly by maps of manifolds.

For any cell complex $X$ the cohomology group $H^2(X)$ can be identified with the set $[X,\mathbb C P^\infty]$ of homotopy classes of maps into $\mathbb C P^\infty$ . Since $\mathbb C P^\infty=MU(1)$ , every element $x\in H^2(X)$ also determines a cobordism class $u_x\in U^2(X)$ . The elements of $U^2(X)$ obtained in this way are called geometric cobordisms of $X$ . We therefore may view $H^2(X)$ as a subset in $U^2(X)$ , however the group operation in $H^2(X)$ is not obtained by restricting the group operation in $U^2(X)$ (see Formal group laws and genera for the relationship between the two operations).

When $X$ is a manifold, geometric cobordisms may be described by submanifolds $M\subset X$ of codimension 2 with a fixed complex structure in the normal bundle.

Indeed, every $x\in H^2(X)$ corresponds to a homotopy class of maps $f_x\colon X\to\mathbb C P^\infty$ . The image $f_x(X)$ is contained in some $\mathbb C P^N\subset\mathbb C P^\infty$ , and we may assume that $f_x(X)$ is transversal to a certain hyperplane $H\subset\mathbb C P^N$ . Then $M_x:=f_x^{-1}(H)$ is a codimension 2 submanifold in $X$ whose normal bundle acquires a complex structure by restriction of the complex structure in the normal bundle of $H\subset\mathbb C P^N$ . Changing the map $f_x$ within its homotopy class does not affect the bordism class of embedding $M_x\to X$ .

Conversely, assume given a submanifold $M\subset X$ of codimension 2 whose normal bundle is endowed with a complex structure. Then the composition

$\displaystyle X\to M(\nu)\to MU(1)=\mathbb C P^\infty$ $\displaystyle X\to M(\nu)\to MU(1)=\mathbb C P^\infty$

of the Pontrjagin-Thom collapse map $X\to M(\nu)$ and the map of Thom spaces corresponding the the classifying map $M\to BU(1)$ of $\nu$ defines and element $x_M\in H^2(X)$ , and therefore a geometric cobordism.

If $X$ is an oriented manifold, then a choice of complex structure in the normal bundle of a codimension 2 embedding $M\subset X$ is equivalent to orienting $M$ . The image of the fundamental class of $M$ in the homology of $X$ is Poincare dual to $x_M\in H^2(X)$ .

5 Structure results

Complex bordism ring $\varOmega_*^U$ is described as follows.

Theorem 5.1.

$\varOmega_*^U\otimes\mathbb Q$ is a polynomial ring over $\mathbb Q$ generated by the bordism classes of complex projective spaces $\mathbb C P^i$ , $i\ge1$ .
Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
$\varOmega_*^U$ is a polynomial ring over $\mathbb Z$ with one generator $a_i$ in every even dimension $2i$ , where $i\ge1$ .

Part 1 can be proved by the methods of [Thom1954]. Part 2 follows from the results of [Milnor1960] and [Novikov1960]. Part 3 is the most difficult one; it was done by [Novikov1960] using Adams spectral sequence and structure theory of Hopf algebras (see also [Novikov1962] for a more detailed account) and Milnor (unpublished, but see [Thom1995]) in 1960. Another more geometric proof was given by [Stong1965], see also [Stong1968].

6 Multiplicative generators

6.1 Preliminaries: characteristic number s_n

To describe a set of multiplicative generators for the ring $\varOmega_*^U$ we shall need a special characteristic class of complex vector bundles. Let $\xi$ be a complex $k$ -plane bundle over a manifold~ $M$ . Write formally its total Chern class as follows:

$\displaystyle c(\xi )=1+c_1(\xi )+\ldots +c_k(\xi )=(1+x_1)\dots(1+x_k),$ $\displaystyle c(\xi )=1+c_1(\xi )+\ldots +c_k(\xi )=(1+x_1)\dots(1+x_k),$

so that $c_i(\xi )=\sigma_i(x_1,\ldots,x_k)$ is the $i$ th elementary symmetric function in formal indeterminates. These indeterminates acquire a geometric meaning if $\xi$ is a sum $\xi_1\oplus\ldots\oplus\xi_k$ of line bundles; then

$x_j=c_1(\xi_j)$ $x_j=c_1(\xi_j)$ ,

Tex syntax error

$1\le j\le k$ . Consider the polynomial

$\displaystyle P_n(x_1,\ldots x_k)=x_1^n+\ldots +x_k^n$ $\displaystyle P_n(x_1,\ldots x_k)=x_1^n+\ldots +x_k^n$

and express it via the elementary symmetric functions:

$\displaystyle P_n(x_1,\ldots ,x_k)=s_n(\sigma_1,\dots ,\sigma_k).$ $\displaystyle P_n(x_1,\ldots ,x_k)=s_n(\sigma_1,\dots ,\sigma_k).$

Substituting the Chern classes for the elementary symmetric functions we obtain a certain characteristic class of $\xi$ :

$\displaystyle s_n(\xi)=s_n(c_1(\xi),\ldots,c_k(\xi))\in H^{2n}(M).$ $\displaystyle s_n(\xi)=s_n(c_1(\xi),\ldots,c_k(\xi))\in H^{2n}(M).$

This characteristic class plays an important role in detecting the polynomial generators of the complex bordism ring, because of the following properties (which follow immediately from the definition).

Proposition 6.1.

$s_n(\xi)=0$ for $2n>\dim M$ .
$s_n(\xi\oplus\eta)=s_n(\xi)+s_n(\eta)$ .

Given a stably complex manifold $(M,c_{\mathcal T})$ of dimension $2n$ , define its characteristic number

$\displaystyle s_n[M]=s_n(\xi)\langle M\rangle\in\mathbb Z$ $\displaystyle s_n[M]=s_n(\xi)\langle M\rangle\in\mathbb Z$

where $\xi$ is the complex bundle from the definition of stably complex structure, and $\langle M\rangle\in H_{2n}(M)$ the fundamental homology class.

If a bordism class $[M]\in\varOmega_{2n}^U$ decomposes as $[M_1]\times[M_2]$ where $\dim M_1>0$ and $\dim M_2>0$ , then $s_n[M]=0$ .

It follows that the characteristic number $s_n$ vanishes on decomposable elements of $\varOmega^U_{2n}$ . It also detects indecomposables that may be chosen as polynomial generators. In fact, the following result is a byproduct of the calculation of $\varOmega^U_*$ :

A bordism class $[M]\in\varOmega_{2n}^U$ may be chosen as a polynomial generator $a_n$ of the ring $\varOmega_*^U$ if and only if

$\displaystyle s_n[M]=\begin{cases} \pm1, &\text{if $n\ne p^k-1$ for any prime $p$;}\\ \pm p, &\text{if $n=p^k-1$ for some prime $p$.} \end{cases}$ $\displaystyle s_n[M]=\begin{cases} \pm1, &\text{if $n\ne p^k-1$ for any prime $p$;}\\ \pm p, &\text{if $n=p^k-1$ for some prime $p$.} \end{cases}$

6.2 Milnor hypersurfaces H_ij

A universal description of connected manifolds representing the polynomial generators $a_n\in\varOmega_*^U$ is unknown. Still, there is a particularly nice family of manifolds whose bordism classes generate the whole ring $\varOmega_*^U$ . This family is superfluous though, so there are algebraic relations between their bordism classes.

Fix a pair of integers $j\ge i\ge0$ and consider the product $\mathbb C P^i\times\mathbb C P^j$ . Its algebraic subvariety

$\displaystyle H_{ij}=\{ (z_0:\ldots :z_i)\times (w_0:\ldots :w_j)\in \mathbb{C}P^i\times \mathbb{C}P^j\colon z_0w_0+\ldots +z_iw_i=0\}$ $\displaystyle H_{ij}=\{ (z_0:\ldots :z_i)\times (w_0:\ldots :w_j)\in \mathbb{C}P^i\times \mathbb{C}P^j\colon z_0w_0+\ldots +z_iw_i=0\}$

is called the Milnor hypersurface. Note that $H_{0j}\cong\mathbb C P^{j-1}$ .

Denote by $p_1$ and $p_2$ the projections $\mathbb C P^i\times\mathbb C P^j$ onto the first and second factors respectively, and by $\eta$ the Hopf line bundle over a complex projective space; then $\bar\eta$ is the hyperplane section bundle. We have

$\displaystyle H^*(\mathbb C P^i\times\mathbb C P^j)=\mathbb Z[x,y]/(x^{i+1}=0,\;y^{j+1}=0)$ $\displaystyle H^*(\mathbb C P^i\times\mathbb C P^j)=\mathbb Z[x,y]/(x^{i+1}=0,\;y^{j+1}=0)$

where $x=p_1^*c_1(\bar\eta)$ , $y=p_2^*c_1(\bar\eta)$ .

The geometric cobordism in $\mathbb C P^i\times\mathbb C P^j$ corresponding to the element $x+y\in H^2(\mathbb C P^i\times\mathbb C P^j)$ is represented by the submanifold $H_{ij}$ . In particular, the image of the fundamental class $\langle H_{ij}\rangle$ in $H_{2(i+j-1)}(\mathbb C P^i\times\mathbb C P^j)$ is Poincare dual to $x+y$ .

See the proof.

Lemma 6.5. We have

$\displaystyle s_{i+j-1}[H_{ij}]=\begin{cases} j,&\text{if \ $i=0$, i.e. $H_{ij}=\mathbb C P^{j-1}$};\\ 2,&\text{if \ $i=j=1$};\\ 0,&\text{if \ $i=1$, $j>1$};\\ -\binom{i+j}i,&\text{if \ $i>1$}. \end{cases}$ $\displaystyle s_{i+j-1}[H_{ij}]=\begin{cases} j,&\text{if \ $i=0$, i.e. $H_{ij}=\mathbb C P^{j-1}$};\\ 2,&\text{if \ $i=j=1$};\\ 0,&\text{if \ $i=1$, $j>1$};\\ -\binom{i+j}i,&\text{if \ $i>1$}. \end{cases}$

See the proof.

The bordism classes $\{[H_{ij}],0\le i\le j\}$ multiplicatively generate the complex bordism ring $\varOmega_*^U$ .

Proof. This follows from the fact that

$\displaystyle \mathop{\text{g.c.d.}}\Bigl({\textstyle\binom{n+1}i},\;1\le i\le n\Bigr)= \begin{cases} p, & \text{if \ $n=p^k-1$,}\\ 1, & \text{else,} \end{cases}$ $\displaystyle \mathop{\text{g.c.d.}}\Bigl({\textstyle\binom{n+1}i},\;1\le i\le n\Bigr)= \begin{cases} p, & \text{if \ $n=p^k-1$,}\\ 1, & \text{else,} \end{cases}$

and the previous Lemma.

Example 6.7.

$\varOmega_{2i+1}^U=0$ ;

$\varOmega_0^U=\mathbb Z$ , generated by a point;

$\varOmega_2^U=\mathbb Z$ , generated by $[\mathbb CP^1]$ , as $1=2^1-1$ and $s_1[\mathbb C P^1]=2$ ;

$\varOmega_4^U=\mathbb Z\oplus\mathbb Z$ , generated by $[\mathbb CP^1\times\mathbb CP^1]$ and $[\mathbb CP^2]$ , as $2=3^1-1$ and $s_2[\mathbb C P^2]=3$ ;

$[\mathbb C P^3]$ cannot be taken as the polynomial generator $a_3\in\varOmega_6^U$ , since $s_3[\mathbb C P^3]=4$ , while $s_3(a_3)=\pm2$ . One may take as $a_3$ the bordism class $[H_{22}]+[\mathbb C P^3]$ .

The previous theorem about the multiplicative generators for $\varOmega_*^U$ has the following important specification.

Every bordism class $x\in\varOmega_n^U$ with $n>0$ contains a nonsingular algebraic variety (not necessarily connected).

(The Milnor hypersufaces are algebraic, but one also needs to represent $-[H_{ij}]$ by algebraic varieties!) For the proof see Chapter 7 of [Stong1968].

The following question is still open, even in complex dimension 2.

Describe the set of bordism classes in $\varOmega_*^U$ containing connected nonsingular algebraic varieties.

Every class $k[\mathbb C P^1]\in\varOmega^U_2$ contains a nonsingular algebraic variety, namely, a disjoint union of $k$ copies of $\mathbb C P^1$ for $k>0$ and a Riemannian surface of genus $(1-k)$ for $k\le0$ . Connected algebraic varieties are only contained in the bordism classes $k[\mathbb C P^1]$ with $k\le1$ .

6.3 Toric generators B_ij and quasitoric representatives in cobordism classes

7 Adams-Novikov spectral sequence

The main references here are [Novikov1967] and [Ravenel1986]

8 References

[Dold1978] A. Dold, Geometric cobordism and the fixed point transfer, in Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), Lecture Notes in Math. 673, Springer, Berlin, (1978), 32–87. MR517084 (80g:57052) Zbl 0386.57005
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\le j\le k$. Consider the polynomial $$ P_n(x_1,\ldots x_k)=x_1^n+\ldots +x_k^n $$ and express it via the elementary symmetric functions: $$ P_n(x_1,\ldots ,x_k)=s_n(\sigma_1,\dots ,\sigma_k). $$ Substituting the Chern classes for the elementary symmetric functions we obtain a certain characteristic class of $\xi$: $$ s_n(\xi)=s_n(c_1(\xi),\ldots,c_k(\xi))\in H^{2n}(M). $$ This characteristic class plays an important role in detecting the polynomial generators of the complex bordism ring, because of the following properties (which follow immediately from the definition). {{beginthm|Proposition}} # $s_n(\xi)=0$ for n>\dim M$. # $s_n(\xi\oplus\eta)=s_n(\xi)+s_n(\eta)$. {{endthm}} Given a stably complex manifold $(M,c_{\mathcal T})$ of dimension n$, define its characteristic number $$ s_n[M]=s_n(\xi)\langle M\rangle\in\mathbb Z $$ where $\xi$ is the complex bundle from the definition of [[#Stably complex structures|stably complex structure]], and $\langle M\rangle\in H_{2n}(M)$ the fundamental homology class. {{beginthm|Corollary}} If a bordism class $[M]\in\varOmega_{2n}^U$ decomposes as $[M_1]\times[M_2]$ where $\dim M_1>0$ and $\dim M_2>0$, then $s_n[M]=0$. {{endthm}} It follows that the characteristic number $s_n$ vanishes on decomposable elements of $\varOmega^U_{2n}$. It also detects indecomposables that may be chosen as polynomial generators. In fact, the following result is a byproduct of the calculation of $\varOmega^U_*$: {{beginthm|Theorem}} A bordism class $[M]\in\varOmega_{2n}^U$ may be chosen as a polynomial generator $a_n$ of the ring $\varOmega_*^U$ if and only if $$ s_n[M]=\begin{cases} \pm1, &\text{if $n\ne p^k-1$ for any prime $p$;}\ \pm p, &\text{if $n=p^k-1$ for some prime $p$.} \end{cases} $$ {{endthm}} === Milnor hypersurfaces H_ij === ; A universal description of connected manifolds representing the polynomial generators $a_n\in\varOmega_*^U$ is unknown. Still, there is a particularly nice family of manifolds whose bordism classes generate the whole ring $\varOmega_*^U$. This family is superfluous though, so there are algebraic relations between their bordism classes. Fix a pair of integers $j\ge i\ge0$ and consider the product $\mathbb C P^i\times\mathbb C P^j$. Its algebraic subvariety $$ H_{ij}=\{ (z_0:\ldots :z_i)\times (w_0:\ldots :w_j)\in \mathbb{C}P^i\times \mathbb{C}P^j\colon z_0w_0+\ldots +z_iw_i=0\} $$ is called the ''Milnor hypersurface''. Note that $H_{0j}\cong\mathbb C P^{j-1}$. Denote by $p_1$ and $p_2$ the projections $\mathbb C P^i\times\mathbb C P^j$ onto the first and second factors respectively, and by $\eta$ the Hopf line bundle over a complex projective space; then $\bar\eta$ is the hyperplane section bundle. We have $$ H^*(\mathbb C P^i\times\mathbb C P^j)=\mathbb Z[x,y]/(x^{i+1}=0,\;y^{j+1}=0) $$ where $x=p_1^*c_1(\bar\eta)$, $y=p_2^*c_1(\bar\eta)$. {{beginthm|Proposition}} The [[#Geometric cobordisms|geometric cobordism]] in $\mathbb C P^i\times\mathbb C P^j$ corresponding to the element $x+y\in H^2(\mathbb C P^i\times\mathbb C P^j)$ is represented by the submanifold $H_{ij}$. In particular, the image of the fundamental class $\langle H_{ij}\rangle$ in $H_{2(i+j-1)}(\mathbb C P^i\times\mathbb C P^j)$ is Poincare dual to $x+y$. {{endthm}} See the [[Media:complex_cobordims_proof64.pdf|proof]]. {{beginthm|Lemma}} We have $$ s_{i+j-1}[H_{ij}]=\begin{cases} j,&\text{if \ $i=0$, i.e. $H_{ij}=\mathbb C P^{j-1}$};\ 2,&\text{if \ $i=j=1$};\ 0,&\text{if \ $i=1$, $j>1$};\ -\binom{i+j}i,&\text{if \ $i>1$}. \end{cases} $$ {{endthm}} See the [[Media:complex_cobordims_proof65.pdf|proof]]. {{beginthm|Theorem}} The bordism classes $\{[H_{ij}],0\le i\le j\}$ multiplicatively generate the complex bordism ring $\varOmega_*^U$. {{endthm}} ''Proof.'' This follows from the fact that $$ \mathop{\text{g.c.d.}}\Bigl({\textstyle\binom{n+1}i},\;1\le i\le n\Bigr)= \begin{cases} p, & \text{if \ $n=p^k-1$,}\ 1, & \text{else,} \end{cases} $$ and the previous Lemma. {{beginthm|Example}} * $\varOmega_{2i+1}^U=0$; * $\varOmega_0^U=\mathbb Z$, generated by a point; * $\varOmega_2^U=\mathbb Z$, generated by $[\mathbb CP^1]$, as \,W is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of a complex structure. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.