Formal group laws and genera

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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 $R$${{Authors|Taras Panov}} == Introduction == The theory of <i>formal group laws</i>, originally appeared in algebraic geometry, has been brought into the [[Bordism|bordism theory]] in the pioneering work \cite{Novikov1967}. The applications of formal group laws in algebraic topology are closely connected with the <i>Hirzebruch genera</i> \cite{Hirzebruch1966}, one of the most important class of invariants of bordism classes of manifolds. == Elements of the theory of formal group laws == <wikitex>; Let $R$ be a commutative ring with unit. A formal power series $F(u,v)\in R[[u,v]]$ is called a (commutative one-dimensional) <i>formal group law</i> over $R$ if it satisfies the following equations: # $F(u,0)=u$, $F(0,v)=v$; # $F(F(u,v),w)=F(u,F(v,w))$; # $F(u,v)=F(v,u)$. The original example of a formal group law over a field $\mathbf k$ is provided by the expansion near the unit of the multiplication map $G\times G\to G$ in a one-dimensional algebraic group over $\mathbf k$. This also explains the terminology. A formal group law $F$ over $R$ is called <i>linearisable</i> if there exists a coordinate change $u\mapsto g_F(u)=u+\sum_{i>0}g_iu^i\in R[[u]]$ such that $$ g_F(F(u,v))=g_F(u)+g_F(v). $$ Note that every formal group law over $R$ determines a formal group law over $R\otimes\mathbb Q$. {{beginthm|Theorem}} Every formal group law $F$ is linearisable over $R\otimes\mathbb Q$. {{endthm}} ''Proof.'' Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}$. Then $$ \omega(F(u,v))=\frac{\partial F(F(u,v),w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(F(u,w),v)}{\partial F(u,w)}\cdot\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(u,v)}{\partial u}\omega(u). $$ We therefore have $\frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}$. Set $$ g(u)=\int_0^u\frac{dv}{\omega(v)}; $$ then $dg(u)=dg(F(u,v))$. This implies that $g(F(u,v))=g(u)+C$. Since $F(0,v)=v$ and $g(0)=0$, we get $C=g(v)$. Thus, $g(F(u,v))=g(u)+g(v)$. A series $g(u)$ satisfying the equation $g(F(u,v))=g(u)+g(v)$ is called the <i>logarithm</i> of the formal group law $F$; the above Theorem shows that a formal group law over $R\otimes\mathbb Q$ always has a logarithm. Its functional inverse series $f(t)\in R\otimes\mathbb Q[[t]]$ is called the <i>exponential</i> of the formal group law, so that we have $F(u,v)=f(g(u)+g(v))$ over $R\otimes\mathbb Q$. If $R$ does not have torsion (i.e. $R\to R\otimes\mathbb Q$ is monic), the latter formula shows that a formal group law (as a series with coefficients in $R$) is fully determined by its logarithm (which is a series with coefficients in $R\otimes\mathbb Q$). Let $F=\sum_{k,l}a_{kl}u^kv^l$ be a formal group law over a ring $R$ and $r\colon R\to R'$ a ring homomorphism. Denote by $r(F)$ the formal series $\sum_{k,l}r(a_{kl})u^kv^l\in R'[[u,v]]$; then $r(F)$ is a formal group law over $R'$. A formal group law $F_U$ over a ring $A$ is <i>universal</i> if for any formal group law $F$ over any ring $R$ there exists a unique homomorphism $r\colon A\to R$ such that $F=r(F_U)$. {{beginthm|Proposition}} Assume that the universal formal group law $F_U$ over $A$ exists. Then # The ring $A$ is multiplicatively generated by the coefficients of the series $F_U$; # The universal formal group law is unique: if $F'_U$ is another universal formal group law over $A'$, then there is an isomorphism $r\colon A\to A'$ such that $F'_U=r(F_U)$. {{endthm}} <i>Proof.</i> To prove the first statement, denote by $A'$ the subring in $A$ generated by the coefficients of $F_U$. Then there is a monomorphism $i\colon A'\to A$ satisfying $i(F_U)=F_U$. On the other hand, by universality there exists a homomorphism $r\colon A\to A'$ satisfying $r(F_U)=F_U$. It follows that $ir(F_U)=F_U$. This implies that $ir=\mathrm{id}\colon A\to A$ by the uniqueness requirement in the definition of~$F_U$. Thus $A'=A$. The second statement is proved similarly. {{beginthm|Theorem|(\cite{Lazard1955})}} The universal formal group law $F_U$ exists, and its coefficient ring $A$ is isomorphic to the polynomial ring $\mathbb Z[a_1,a_2,\ldots]$ on an infinite number of generators. {{endthm}} </wikitex> == Formal group law of geometric cobordisms == == Hirzebruch genera == == References == {{#RefList:}} [[Category:Theory]] [[Category:Bordism]] [[Category:Manifolds]] {{Stub}}R$ be a commutative ring with unit.

A formal power series $F(u,v)\in R[[u,v]]$$F(u,v)\in R[[u,v]]$ is called a (commutative one-dimensional) formal group law over $R$$R$ if it satisfies the following equations:

1. $F(u,0)=u$$F(u,0)=u$, $F(0,v)=v$$F(0,v)=v$;
2. $F(F(u,v),w)=F(u,F(v,w))$$F(F(u,v),w)=F(u,F(v,w))$;
3. $F(u,v)=F(v,u)$$F(u,v)=F(v,u)$.

The original example of a formal group law over a field $\mathbf k$$\mathbf k$ is provided by the expansion near the unit of the multiplication map $G\times G\to G$$G\times G\to G$ in a one-dimensional algebraic group over $\mathbf k$$\mathbf k$. This also explains the terminology.

A formal group law $F$$F$ over $R$$R$ is called linearisable if there exists a coordinate change $u\mapsto g_F(u)=u+\sum_{i>0}g_iu^i\in R[[u]]$$u\mapsto g_F(u)=u+\sum_{i>0}g_iu^i\in R[[u]]$ such that

$\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).$ $$\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).$$

Note that every formal group law over $R$$R$ determines a formal group law over $R\otimes\mathbb Q$$R\otimes\mathbb Q$.

Theorem 2.1. Every formal group law $F$$F$ is linearisable over $R\otimes\mathbb Q$$R\otimes\mathbb Q$.

Proof. Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}$$\omega(u)=\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}$. Then

$\displaystyle \omega(F(u,v))=\frac{\partial F(F(u,v),w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(F(u,w),v)}{\partial F(u,w)}\cdot\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(u,v)}{\partial u}\omega(u).$ $$\displaystyle \omega(F(u,v))=\frac{\partial F(F(u,v),w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(F(u,w),v)}{\partial F(u,w)}\cdot\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(u,v)}{\partial u}\omega(u).$$

We therefore have $\frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}$$\frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}$. Set

$\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};$ $$\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};$$

then $dg(u)=dg(F(u,v))$$dg(u)=dg(F(u,v))$. This implies that $g(F(u,v))=g(u)+C$$g(F(u,v))=g(u)+C$. Since $F(0,v)=v$$F(0,v)=v$ and $g(0)=0$$g(0)=0$, we get $C=g(v)$$C=g(v)$. Thus, $g(F(u,v))=g(u)+g(v)$$g(F(u,v))=g(u)+g(v)$.

A series $g(u)$$g(u)$ satisfying the equation $g(F(u,v))=g(u)+g(v)$$g(F(u,v))=g(u)+g(v)$ is called the logarithm of the formal group law $F$$F$; the above Theorem shows that a formal group law over $R\otimes\mathbb Q$$R\otimes\mathbb Q$ always has a logarithm. Its functional inverse series $f(t)\in R\otimes\mathbb Q[[t]]$$f(t)\in R\otimes\mathbb Q[[t]]$ is called the exponential of the formal group law, so that we have $F(u,v)=f(g(u)+g(v))$$F(u,v)=f(g(u)+g(v))$ over $R\otimes\mathbb Q$$R\otimes\mathbb Q$. If $R$$R$ does not have torsion (i.e. $R\to R\otimes\mathbb Q$$R\to R\otimes\mathbb Q$ is monic), the latter formula shows that a formal group law (as a series with coefficients in $R$$R$) is fully determined by its logarithm (which is a series with coefficients in $R\otimes\mathbb Q$$R\otimes\mathbb Q$).

Let $F=\sum_{k,l}a_{kl}u^kv^l$$F=\sum_{k,l}a_{kl}u^kv^l$ be a formal group law over a ring $R$$R$ and $r\colon R\to R'$$r\colon R\to R'$ a ring homomorphism. Denote by $r(F)$$r(F)$ the formal series $\sum_{k,l}r(a_{kl})u^kv^l\in R'[[u,v]]$$\sum_{k,l}r(a_{kl})u^kv^l\in R'[[u,v]]$; then $r(F)$$r(F)$ is a formal group law over $R'$$R'$.

A formal group law $F_U$$F_U$ over a ring $A$$A$ is universal if for any formal group law $F$$F$ over any ring $R$$R$ there exists a unique homomorphism $r\colon A\to R$$r\colon A\to R$ such that $F=r(F_U)$$F=r(F_U)$.

Proposition 2.2. Assume that the universal formal group law $F_U$$F_U$ over $A$$A$ exists. Then

1. The ring $A$$A$ is multiplicatively generated by the coefficients of the series $F_U$$F_U$;
2. The universal formal group law is unique: if

$F'_U$$F'_U$ is another universal formal group law over $A'$$A'$, then there is an isomorphism $r\colon A\to A'$$r\colon A\to A'$ such that $F'_U=r(F_U)$$F'_U=r(F_U)$.

Proof. To prove the first statement, denote by $A'$$A'$ the subring in $A$$A$ generated by the coefficients of $F_U$$F_U$. Then there is a monomorphism $i\colon A'\to A$$i\colon A'\to A$ satisfying $i(F_U)=F_U$$i(F_U)=F_U$. On the other hand, by universality there exists a homomorphism $r\colon A\to A'$$r\colon A\to A'$ satisfying $r(F_U)=F_U$$r(F_U)=F_U$. It follows that $ir(F_U)=F_U$$ir(F_U)=F_U$. This implies that $ir=\mathrm{id}\colon A\to A$$ir=\mathrm{id}\colon A\to A$ by the uniqueness requirement in the definition of~$F_U$$F_U$. Thus $A'=A$$A'=A$. The second statement is proved similarly.

Theorem 2.3 ([Lazard1955]). The universal formal group law $F_U$$F_U$ exists, and its coefficient ring $A$$A$ is isomorphic to the polynomial ring $\mathbb Z[a_1,a_2,\ldots]$$\mathbb Z[a_1,a_2,\ldots]$ on an infinite number of generators.

5 References

• [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

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