Complex bordism

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1 Introduction

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

The theory of complex 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 (B-bordisms). Thanks to this, the complex cobordism theory found the most stricking 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 $W$${{Authors|Taras Panov}} == Introduction == <wikitex>; <!--A unitary struture $\bar \nu$ on a manifold $M$ is a choice of weak complex structure on the stable normal bundle of $M$. By the [[B-Bordism#Pontrjagin-Thom isomorphism|Pontrjagin-Thom isomorphism]] the bordism groups of closed unitary manifolds $(M, \bar \nu)$ are isomorphic to the homotopy groups of the Thom spectrum $MU$, $\Omega_*^{U} \cong \pi_n(MU)$.--> ''Complex bordisms'' (also known as ''unitary bordisms'') is the [[Bordism and cobordism|bordism theory]] of [[#Stably complex structures|stably complex manifolds]]. It is one of the most important theory of bordisms with additional structure, or [[B-Bordism|B-bordisms]]. The theory of complex bordisms is much richer than its [[Bordism and cobordism#Unoriented bordism|unoriented]] analogue, and at the same time is not as complicated as [[Oriented bordism|oriented bordisms]] or other bordisms with additional structure ([[B-Bordism|B-bordisms]]). Thanks to this, the complex cobordism theory found the most stricking 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 and cobordism#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 the complex structures. 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 $$ between the "stable" tangent bundle and a complex vector bundle $\xi$ over $M$. Some of the choices of such isomorphisms are deemed to be equivalent, i.e. determining the same stably complex structures (see details in Chapters II and VII of \cite{Stong1968}). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A ''normal stably complex structure'' on $M$ is determined by a choice of a complex bundle structure in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A tangential and normal stably complex structures on $M$ determine each other by means of the canonical isomorphism $\mathcal T\!M\oplus\nu(M)\cong\underline{\mathbb R}^N$. We therefore may restrict our attention to tangential structures only. A ''stably complex manifold'' is a pair $(M,c_{\mathcal T})$ consisting of a manifold $M$ and a stably complex structure $c_{\mathcal T}$ on it. This is a generalisation to a complex and ''almost complex'' manifold (where the latter means a manifold with a choice of a complex structure on ${\mathcal T}\!M$, i.e. a stably complex structure $c_{\mathcal T}$ with $k=0$). {{beginthm|Example}} Let $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism $$ {\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 $$ determines a trivial stably complex structure on $\mathbb C P^1$. {{endthm}} </wikitex> == Definition of bordism and cobordism == <wikitex>; The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordisms, the set of bordism classes $[M,c_{\mathcal T}]$ of stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the ''group of $n$-dimensional complex bordisms'' and denoted $\varOmega^U_n$. A zero is represented by the bordism class of any manifold $M$ which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle $M\times\mathbb C^k$). The sphere $S^n$ provides an example of such a manifold. The opposite element to the bordism class $[M,c_{\mathcal T}]$ in the group $\varOmega^U_n$ may be represented by the same manifold $M$ with the stably complex structure determined by the isomorphism $$ {\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} $$ where $e\colon\mathbb R^2\to\mathbb C$ is given by $e(x,y)=x-iy$. An abbreviated notation $[M]$ for the complex bordism class will be used whenever the stably complex structure $c_{\mathcal T}$ is clear from the context. The ''groups of complex bordisms'' $U_n(X)$ and ''cobordisms'' $U^n(X)$ of a space $X$ may also be defined geometrically, at least for the case when $X$ is a manifold. This can be done along the lines suggested by \cite{Quillen1971} and \cite{Dold1978} by considering special "stably complex" maps of manifolds $M$ to $X$. However, nowadays the homotopical approach to bordisms takes over, and the (co)bordism groups are usually defined using the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom construction]] similarly to the [[Bordism and cobordism#Unoriented bordism|unoriented]] case: $$ \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,\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 $$ 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 Poincar\'e dual to $x_M\in H^2(X)$. </wikitex> == Structure results == == Multiplicative generators == == Formal group laws and genera == == Adams-Novikov spectral sequence == <wikitex>; The main references here are \cite{Novikov1967} and \cite{Ravenel1986} </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Bordism]] {{Stub}}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 the complex structures. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.

Let ${\mathcal T}\!M$${\mathcal T}\!M$ denote the tangent bundle of $M$$M$, and $\underline{\mathbb R}^k$$\underline{\mathbb R}^k$ the product vector bundle $M\times\mathbb R^k$$M\times\mathbb R^k$ over $M$$M$. A tangential stably complex structure on $M$$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$$\xi$ over $M$$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$$M$ is determined by a choice of a complex bundle structure in the normal bundle $\nu(M)$$\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$$M\hookrightarrow\mathbb R^N$. A tangential and normal stably complex structures on $M$$M$ determine each other by means of the canonical isomorphism $\mathcal T\!M\oplus\nu(M)\cong\underline{\mathbb R}^N$$\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})$$(M,c_{\mathcal T})$ consisting of a manifold $M$$M$ and a stably complex structure $c_{\mathcal T}$$c_{\mathcal T}$ on it. This is a generalisation to a complex and almost complex manifold (where the latter means a manifold with a choice of a complex structure on ${\mathcal T}\!M$${\mathcal T}\!M$, i.e. a stably complex structure $c_{\mathcal T}$$c_{\mathcal T}$ with $k=0$$k=0$).

Example 2.1. Let $M=\mathbb{C}P^1$$M=\mathbb{C}P^1$. The standard complex structure on $M$$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$$\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$$\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 bordisms, the set of bordism classes $[M,c_{\mathcal T}]$$[M,c_{\mathcal T}]$ of stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the group of $n$$n$-dimensional complex bordisms and denoted $\varOmega^U_n$$\varOmega^U_n$. A zero is represented by the bordism class of any manifold $M$$M$ which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle $M\times\mathbb C^k$$M\times\mathbb C^k$). The sphere $S^n$$S^n$ provides an example of such a manifold. The opposite element to the bordism class $[M,c_{\mathcal T}]$$[M,c_{\mathcal T}]$ in the group $\varOmega^U_n$$\varOmega^U_n$ may be represented by the same manifold $M$$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$$e\colon\mathbb R^2\to\mathbb C$ is given by $e(x,y)=x-iy$$e(x,y)=x-iy$.

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

The groups of complex bordisms $U_n(X)$$U_n(X)$ and cobordisms $U^n(X)$$U^n(X)$ of a space $X$$X$ may also be defined geometrically, at least for the case when $X$$X$ is a manifold. This can be done along the lines suggested by [Quillen1971] and [Dold1978] by considering special "stably complex" maps of manifolds $M$$M$ to $X$$X$. However, nowadays the homotopical approach to bordisms takes 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)$$MU(k)$ is the Thom space of the universal complex $k$$k$-plane bundle $EU(k)\to BU(k)$$EU(k)\to BU(k)$. These groups are $\varOmega_*^U$$\varOmega_*^U$-modules and give rise to a multiplicative (co)homology theory. In particular, $U^*(X)=\oplus_n U^n(X)$$U^*(X)=\oplus_n U^n(X)$ is a graded ring.

The graded ring $\varOmega^*_U$$\varOmega^*_U$ with $\varOmega^{n}_U=\varOmega_{-n}^U$$\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$$X$ the cohomology group $H^2(X)$$H^2(X)$ can be identified with the set

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$[X,\C P^\infty]$ of homotopy classes of

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

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

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

Conversely, assume given a submanifold $M\subset X$$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)$$X\to M(\nu)$ and the map of Thom spaces corresponding the the classifying map $M\to BU(1)$$M\to BU(1)$ of $\nu$$\nu$ defines and element $x_M\in H^2(X)$$x_M\in H^2(X)$, and therefore a geometric cobordism.

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

8 Adams-Novikov spectral sequence

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

9 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
• [Novikov1967] S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Math. USSR, Izv. 1, (1967) 827–913. MR0221509 (36 #4561) Zbl 0176.52401
• [Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
• [Ravenel1986] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press Inc., Orlando, FL, 1986. MR860042 (87j:55003) Zbl 1073.55001
• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010

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