Formal group laws and genera
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Contents |
1 Introduction
The theory of formal group laws, originally appeared in algebraic geometry, has been brought into the bordism theory in the pioneering work [Novikov1967]. The applications of formal group laws in algebraic topology are closely connected with the Hirzebruch genera [Hirzebruch1966], one of the most important class of invariants of bordism classes of manifolds.
2 Elements of the theory of formal group laws
Let be a commutative ring with unit.
A formal power series is called a (commutative one-dimensional) formal group law over if it satisfies the following equations:
- , ;
- ;
- .
The original example of a formal group law over a field is provided by the expansion near the unit of the multiplication map in a one-dimensional algebraic group over . This also explains the terminology.
A formal group law over is called linearisable if there exists a coordinate change such that
Note that every formal group law over determines a formal group law over .
Theorem 2.1. Every formal group law is linearisable over .
Proof. Consider the series . Then
We therefore have . Set
then . This implies that . Since and , we get . Thus, .
A series satisfying the equation is called the logarithm of the formal group law ; the above Theorem shows that a formal group law over always has a logarithm. Its functional inverse series is called the exponential of the formal group law, so that we have over . If does not have torsion (i.e. is monic), the latter formula shows that a formal group law (as a series with coefficients in ) is fully determined by its logarithm (which is a series with coefficients in ).
Let be a formal group law over a ring and a ring homomorphism. Denote by the formal series ; then is a formal group law over .
A formal group law over a ring is universal if for any formal group law over any ring there exists a unique homomorphism such that .
Proposition 2.2. Assume that the universal formal group law over exists. Then
- The ring is multiplicatively generated by the coefficients of the series ;
- The universal formal group law is unique: if is another universal formal group law over , then there is an isomorphism such that .
Proof. To prove the first statement, denote by the subring in generated by the coefficients of . Then there is a monomorphism satisfying . On the other hand, by universality there exists a homomorphism satisfying . It follows that . This implies that by the uniqueness requirement in the definition of . Thus . The second statement is proved similarly.
Theorem 2.3 ([Lazard1955]). The universal formal group law exists, and its coefficient ring is isomorphic to the polynomial ring on an infinite number of generators.
3 Formal group law of geometric cobordisms
The applications of the formal group laws in the cobordism theory build upon the following basic example.
Let be a cell complex and two geometric cobordisms corresponding to elements respectively. Denote by the geometric cobordism corresponding to the cohomology class .
Proposition 3.1. The following relation holds in :
where the coefficients do not depend on . The series is a formal group law over the ring .
Proof. We first do calculations with the universal example . Then
where are canonical geometric cobordisms given by the projections of onto its factors. We therefore have the following relation in :
where .
Now let the geometric cobordisms be given by maps respectively. Then , and , where . Applying the -module map to the universal formula for above gives the required expression. The fact that is a formal group law follows directly from the properties of the group multiplication .
The series is called the formal group law of geometric cobordisms.
The geometric cobordism is the first Chern-Conner-Floyd class of the complex line bundle over obtained by pulling back the canonical bundle along the map . It follows that the formal group law of geometric cobordisms gives an expression of the first class of the tensor product of two complex line bundles over in terms of the classes and of the factors:
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
Tex syntax error
where () are [[Complex bordism#Milnor hypersurfaces|Milnor hypersurfaces]] and .
Proof. Set in Proposition~\ref{u2x}. Consider the \emph{Poincar\'e-Atiyah duality} map and the map induced by the projection . Then the composition \[
\varepsilon D\colon U^2(\C P^i\times\C P^j)\to\varOmega_{2(i+j)-2}^U
\] takes geometric cobordisms to the bordism classes of the corresponding submanifolds. In particular, , . Applying to~\eqref{fglgc} we obtain \[
[H_{ij}]=\sum_{k,\,l}\alpha_{kl}[\C P^{i-k}][\C P^{j-l}].
\] Therefore, \[
\sum_{i,j}[H_{ij}]u^iv^j=\Bigl(\sum_{k,\,l}\alpha_{kl}u^kv^l\Bigr) \Bigl(\sum_{i\ge k}[\C P^{i-k}]u^{i-k}\Bigr) \Bigl(\sum_{j\ge l}[\C P^{j-l}]v^{j-l}\Bigr),
\] which implies the required formula. \end{proof}
Theorem 3.3 {Mishchenko~[novi67, Appendix~1].} The logarithm of the formal group law of geometric cobordisms is given by \[
g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\Qu.
\]
\begin{proof} By~\eqref{log}, \[
dg_{\mathcal F}(u)=\frac{du}{\frac{\partial\mathcal F(u,v)}{\partial v}\Bigl|_{v=0}}.
\] Using the formula of Theorem~\ref{buchth} and the identity , we calculate \[
dg_{\mathcal F}(u)=\frac{1+\sum_{k>0}[\C P^k]u^k} {1+\sum_{i>0}([H_{i1}]-[\C P^1][\C P^{i-1}])u^i}.
\] Now (see Exercise~\ref{h1i}; we have already seen that in the Remark preceding Theorem~\ref{hijgen}). Therefore, , which implies the required formula. \end{proof}
Theorem 3.4 {Quillen~[quil69, Th.~2].} The formal group law of geometric cobordisms is universal.
\begin{proof} Let be the universal formal group law over a ring~. Then there is a homomorphism which takes to . The series , viewed as a formal group law over the ring , has the universality properties for all formal group laws over -algebras. By theorem~\ref{logth}, such a formal group law is determined by its logarithm, which can be any series starting from~. It follows that if we write the logarithm of as then the ring is the polynomial ring . By Theorem~3.3, . Since , this implies that is an isomorphism.
By Theorem~\ref{lazardth} the ring does not have torsion, so is a monomorphism. On the other hand, Theorem~\ref{buchth} implies that the image contains the bordism classes , . Since these classes generate the whole (Theorem~\ref{hijgen}), is onto and thus an isomorphism. \end{proof}
4 Hirzebruch genera
5 References
- [Buchstaber1970] V. M. Buhštaber, The Chern-Dold character in cobordisms. I, Mat. Sb. (N.S.) 83 (125) (1970), 575–595. MR0273630 (42 #8507) Zbl 0219.57027
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Lazard1955] M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251–274. MR0073925 (17,508e) Zbl 0068.25703
- [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
- [novi67] Template:Novi67
- [quil69] Template:Quil69
This page has not been refereed. The information given here might be incomplete or provisional. |
A formal power series is called a (commutative one-dimensional) formal group law over if it satisfies the following equations:
- , ;
- ;
- .
The original example of a formal group law over a field is provided by the expansion near the unit of the multiplication map in a one-dimensional algebraic group over . This also explains the terminology.
A formal group law over is called linearisable if there exists a coordinate change such that
Note that every formal group law over determines a formal group law over .
Theorem 2.1. Every formal group law is linearisable over .
Proof. Consider the series . Then
We therefore have . Set
then . This implies that . Since and , we get . Thus, .
A series satisfying the equation is called the logarithm of the formal group law ; the above Theorem shows that a formal group law over always has a logarithm. Its functional inverse series is called the exponential of the formal group law, so that we have over . If does not have torsion (i.e. is monic), the latter formula shows that a formal group law (as a series with coefficients in ) is fully determined by its logarithm (which is a series with coefficients in ).
Let be a formal group law over a ring and a ring homomorphism. Denote by the formal series ; then is a formal group law over .
A formal group law over a ring is universal if for any formal group law over any ring there exists a unique homomorphism such that .
Proposition 2.2. Assume that the universal formal group law over exists. Then
- The ring is multiplicatively generated by the coefficients of the series ;
- The universal formal group law is unique: if is another universal formal group law over , then there is an isomorphism such that .
Proof. To prove the first statement, denote by the subring in generated by the coefficients of . Then there is a monomorphism satisfying . On the other hand, by universality there exists a homomorphism satisfying . It follows that . This implies that by the uniqueness requirement in the definition of . Thus . The second statement is proved similarly.
Theorem 2.3 ([Lazard1955]). The universal formal group law exists, and its coefficient ring is isomorphic to the polynomial ring on an infinite number of generators.
3 Formal group law of geometric cobordisms
The applications of the formal group laws in the cobordism theory build upon the following basic example.
Let be a cell complex and two geometric cobordisms corresponding to elements respectively. Denote by the geometric cobordism corresponding to the cohomology class .
Proposition 3.1. The following relation holds in :
where the coefficients do not depend on . The series is a formal group law over the ring .
Proof. We first do calculations with the universal example . Then
where are canonical geometric cobordisms given by the projections of onto its factors. We therefore have the following relation in :
where .
Now let the geometric cobordisms be given by maps respectively. Then , and , where . Applying the -module map to the universal formula for above gives the required expression. The fact that is a formal group law follows directly from the properties of the group multiplication .
The series is called the formal group law of geometric cobordisms.
The geometric cobordism is the first Chern-Conner-Floyd class of the complex line bundle over obtained by pulling back the canonical bundle along the map . It follows that the formal group law of geometric cobordisms gives an expression of the first class of the tensor product of two complex line bundles over in terms of the classes and of the factors:
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
Tex syntax error
where () are [[Complex bordism#Milnor hypersurfaces|Milnor hypersurfaces]] and .
Proof. Set in Proposition~\ref{u2x}. Consider the \emph{Poincar\'e-Atiyah duality} map and the map induced by the projection . Then the composition \[
\varepsilon D\colon U^2(\C P^i\times\C P^j)\to\varOmega_{2(i+j)-2}^U
\] takes geometric cobordisms to the bordism classes of the corresponding submanifolds. In particular, , . Applying to~\eqref{fglgc} we obtain \[
[H_{ij}]=\sum_{k,\,l}\alpha_{kl}[\C P^{i-k}][\C P^{j-l}].
\] Therefore, \[
\sum_{i,j}[H_{ij}]u^iv^j=\Bigl(\sum_{k,\,l}\alpha_{kl}u^kv^l\Bigr) \Bigl(\sum_{i\ge k}[\C P^{i-k}]u^{i-k}\Bigr) \Bigl(\sum_{j\ge l}[\C P^{j-l}]v^{j-l}\Bigr),
\] which implies the required formula. \end{proof}
Theorem 3.3 {Mishchenko~[novi67, Appendix~1].} The logarithm of the formal group law of geometric cobordisms is given by \[
g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\Qu.
\]
\begin{proof} By~\eqref{log}, \[
dg_{\mathcal F}(u)=\frac{du}{\frac{\partial\mathcal F(u,v)}{\partial v}\Bigl|_{v=0}}.
\] Using the formula of Theorem~\ref{buchth} and the identity , we calculate \[
dg_{\mathcal F}(u)=\frac{1+\sum_{k>0}[\C P^k]u^k} {1+\sum_{i>0}([H_{i1}]-[\C P^1][\C P^{i-1}])u^i}.
\] Now (see Exercise~\ref{h1i}; we have already seen that in the Remark preceding Theorem~\ref{hijgen}). Therefore, , which implies the required formula. \end{proof}
Theorem 3.4 {Quillen~[quil69, Th.~2].} The formal group law of geometric cobordisms is universal.
\begin{proof} Let be the universal formal group law over a ring~. Then there is a homomorphism which takes to . The series , viewed as a formal group law over the ring , has the universality properties for all formal group laws over -algebras. By theorem~\ref{logth}, such a formal group law is determined by its logarithm, which can be any series starting from~. It follows that if we write the logarithm of as then the ring is the polynomial ring . By Theorem~3.3, . Since , this implies that is an isomorphism.
By Theorem~\ref{lazardth} the ring does not have torsion, so is a monomorphism. On the other hand, Theorem~\ref{buchth} implies that the image contains the bordism classes , . Since these classes generate the whole (Theorem~\ref{hijgen}), is onto and thus an isomorphism. \end{proof}
4 Hirzebruch genera
5 References
- [Buchstaber1970] V. M. Buhštaber, The Chern-Dold character in cobordisms. I, Mat. Sb. (N.S.) 83 (125) (1970), 575–595. MR0273630 (42 #8507) Zbl 0219.57027
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Lazard1955] M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251–274. MR0073925 (17,508e) Zbl 0068.25703
- [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
- [novi67] Template:Novi67
- [quil69] Template:Quil69
This page has not been refereed. The information given here might be incomplete or provisional. |
A formal power series is called a (commutative one-dimensional) formal group law over if it satisfies the following equations:
- , ;
- ;
- .
The original example of a formal group law over a field is provided by the expansion near the unit of the multiplication map in a one-dimensional algebraic group over . This also explains the terminology.
A formal group law over is called linearisable if there exists a coordinate change such that
Note that every formal group law over determines a formal group law over .
Theorem 2.1. Every formal group law is linearisable over .
Proof. Consider the series . Then
We therefore have . Set
then . This implies that . Since and , we get . Thus, .
A series satisfying the equation is called the logarithm of the formal group law ; the above Theorem shows that a formal group law over always has a logarithm. Its functional inverse series is called the exponential of the formal group law, so that we have over . If does not have torsion (i.e. is monic), the latter formula shows that a formal group law (as a series with coefficients in ) is fully determined by its logarithm (which is a series with coefficients in ).
Let be a formal group law over a ring and a ring homomorphism. Denote by the formal series ; then is a formal group law over .
A formal group law over a ring is universal if for any formal group law over any ring there exists a unique homomorphism such that .
Proposition 2.2. Assume that the universal formal group law over exists. Then
- The ring is multiplicatively generated by the coefficients of the series ;
- The universal formal group law is unique: if is another universal formal group law over , then there is an isomorphism such that .
Proof. To prove the first statement, denote by the subring in generated by the coefficients of . Then there is a monomorphism satisfying . On the other hand, by universality there exists a homomorphism satisfying . It follows that . This implies that by the uniqueness requirement in the definition of . Thus . The second statement is proved similarly.
Theorem 2.3 ([Lazard1955]). The universal formal group law exists, and its coefficient ring is isomorphic to the polynomial ring on an infinite number of generators.
3 Formal group law of geometric cobordisms
The applications of the formal group laws in the cobordism theory build upon the following basic example.
Let be a cell complex and two geometric cobordisms corresponding to elements respectively. Denote by the geometric cobordism corresponding to the cohomology class .
Proposition 3.1. The following relation holds in :
where the coefficients do not depend on . The series is a formal group law over the ring .
Proof. We first do calculations with the universal example . Then
where are canonical geometric cobordisms given by the projections of onto its factors. We therefore have the following relation in :
where .
Now let the geometric cobordisms be given by maps respectively. Then , and , where . Applying the -module map to the universal formula for above gives the required expression. The fact that is a formal group law follows directly from the properties of the group multiplication .
The series is called the formal group law of geometric cobordisms.
The geometric cobordism is the first Chern-Conner-Floyd class of the complex line bundle over obtained by pulling back the canonical bundle along the map . It follows that the formal group law of geometric cobordisms gives an expression of the first class of the tensor product of two complex line bundles over in terms of the classes and of the factors:
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
Tex syntax error
where () are [[Complex bordism#Milnor hypersurfaces|Milnor hypersurfaces]] and .
Proof. Set in Proposition~\ref{u2x}. Consider the \emph{Poincar\'e-Atiyah duality} map and the map induced by the projection . Then the composition \[
\varepsilon D\colon U^2(\C P^i\times\C P^j)\to\varOmega_{2(i+j)-2}^U
\] takes geometric cobordisms to the bordism classes of the corresponding submanifolds. In particular, , . Applying to~\eqref{fglgc} we obtain \[
[H_{ij}]=\sum_{k,\,l}\alpha_{kl}[\C P^{i-k}][\C P^{j-l}].
\] Therefore, \[
\sum_{i,j}[H_{ij}]u^iv^j=\Bigl(\sum_{k,\,l}\alpha_{kl}u^kv^l\Bigr) \Bigl(\sum_{i\ge k}[\C P^{i-k}]u^{i-k}\Bigr) \Bigl(\sum_{j\ge l}[\C P^{j-l}]v^{j-l}\Bigr),
\] which implies the required formula. \end{proof}
Theorem 3.3 {Mishchenko~[novi67, Appendix~1].} The logarithm of the formal group law of geometric cobordisms is given by \[
g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\Qu.
\]
\begin{proof} By~\eqref{log}, \[
dg_{\mathcal F}(u)=\frac{du}{\frac{\partial\mathcal F(u,v)}{\partial v}\Bigl|_{v=0}}.
\] Using the formula of Theorem~\ref{buchth} and the identity , we calculate \[
dg_{\mathcal F}(u)=\frac{1+\sum_{k>0}[\C P^k]u^k} {1+\sum_{i>0}([H_{i1}]-[\C P^1][\C P^{i-1}])u^i}.
\] Now (see Exercise~\ref{h1i}; we have already seen that in the Remark preceding Theorem~\ref{hijgen}). Therefore, , which implies the required formula. \end{proof}
Theorem 3.4 {Quillen~[quil69, Th.~2].} The formal group law of geometric cobordisms is universal.
\begin{proof} Let be the universal formal group law over a ring~. Then there is a homomorphism which takes to . The series , viewed as a formal group law over the ring , has the universality properties for all formal group laws over -algebras. By theorem~\ref{logth}, such a formal group law is determined by its logarithm, which can be any series starting from~. It follows that if we write the logarithm of as then the ring is the polynomial ring . By Theorem~3.3, . Since , this implies that is an isomorphism.
By Theorem~\ref{lazardth} the ring does not have torsion, so is a monomorphism. On the other hand, Theorem~\ref{buchth} implies that the image contains the bordism classes , . Since these classes generate the whole (Theorem~\ref{hijgen}), is onto and thus an isomorphism. \end{proof}
4 Hirzebruch genera
5 References
- [Buchstaber1970] V. M. Buhštaber, The Chern-Dold character in cobordisms. I, Mat. Sb. (N.S.) 83 (125) (1970), 575–595. MR0273630 (42 #8507) Zbl 0219.57027
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Lazard1955] M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251–274. MR0073925 (17,508e) Zbl 0068.25703
- [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
- [novi67] Template:Novi67
- [quil69] Template:Quil69
This page has not been refereed. The information given here might be incomplete or provisional. |