Formal group laws and genera

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== Elements of the theory of formal group laws ==
== Elements of the theory of formal group laws ==
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<wikitex>;
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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
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it satisfies the following equations:
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# $F(u,0)=u$, $F(0,v)=v$;
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# $F(F(u,v),w)=F(u,F(v,w))$;
+
# $F(u,v)=F(v,u)$.
+
+
The original example of a formal group law over a field $\mathbb k$ is
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provided by the expansion near the unit of the multiplication map
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$G\times G\to G$ in a one-dimensional algebraic group over $\mathbb k$.
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This also explains the terminology.
+
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A formal group law $F$ over $R$ is called <i>linearisable</i> if
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there exists a coordinate change $u\mapsto
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g_F(u)=u+\sum_{i>0}g_iu^i\in R[[u]]$ such that
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$$
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g_F(F(u,v))=g_F(u)+g_F(v).
+
$$
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Note that every formal group law over $R$ determines a formal
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group law over $R\otimes\mathbb Q$.
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{{beginthm|Theorem}}
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Every formal group law $F$ is linearisable over $R\otimes\Q$.
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{{endthm}}
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''Proof.'' Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial
+
w}\Bigl|_{w=0}$. Then
+
$$
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\omega(F(u,v))=\frac{\partial
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F(F(u,v),w)}{\partial w}\Bigl|_{w=0}=\frac{\partial
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F(F(u,w),v)}{\partial F(u,w)}\cdot\frac{\partial F(u,w)}{\partial
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w}\Bigl|_{w=0}=\frac{\partial F(u,v)}{\partial u}\omega(u).
+
$$
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We therefore have
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$\frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}$. Set
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$$
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g(u)=\int_0^u\frac{dv}{\omega(v)};
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$$
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then $dg(u)=dg(F(u,v))$. This implies that $g(F(u,v))=g(u)+C$.
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Since $F(0,v)=v$ and $g(0)=0$, we get $C=g(v)$. Thus,
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$g(F(u,v))=g(u)+g(v)$.
+
+
A series $g(u)$ satisfying the equation $g(F(u,v))=g(u)+g(v)$ is called
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the <i>logarithm</i> of the formal group law $F$; the above Theorem
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shows that a formal group law over $R\otimes\mathbb Q$ always has a logarithm. Its functional inverse series
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$f(t)\in R\otimes\mathbb Q[[t]]$ is called the <i>exponential</i> of
+
the formal group law, so that we have $F(u,v)=f(g(u)+g(v))$
+
over $R\otimes\Q$. If $R$ does not have torsion (so that $R\to
+
R\otimes\mathbb Q$ 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
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for any formal group law $F$ over any ring $R$ there exists a
+
unique homomorphism $r\colon A\to R$ such that $F=r(F_U)$.
+
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{{beginthm|Proposition}}
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Assume that the universal formal group law $F_U$ over $A$ exists.
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Then
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# The ring $A$ is multiplicatively generated by the coefficients of the series $F_U$;
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# The universal formal group law is unique: if
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$F'_U$ is another universal formal group law over $A'$, then there
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is an isomorphism $r\colon A\to A'$ such that $F'_U=r(F_U)$.
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{{endthm}}
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<i>Proof</i>
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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 $ir(F_U)=F_U$.
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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
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statement is proved similarly.
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{{beginthm|Theorem|(\cite{Lazard1955})}}
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The universal formal group law $F_U$ exists, and its coefficient
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ring $A$ is isomorphic to the polynomial ring $\mathbb Z[a_1,a_2,\ldots]$
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on an infinite number of generators.
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{{endthm}}
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</wikitex>
== Formal group law of geometric cobordisms ==
== Formal group law of geometric cobordisms ==

Revision as of 19:18, 1 April 2010

An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 09:29, 1 April 2011 and the changes since publication.

The user responsible for this page is Taras Panov. No other user may edit this page at present.

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 R be a commutative ring with unit.

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

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

The original example of a formal group law over a field \mathbb 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 \mathbb k. This also explains the terminology.

A formal group law F over 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]] such that

\displaystyle 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.

Theorem 2.1.

Every formal group law F is linearisable over 
Tex syntax error
.

Proof. Consider the series \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).

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

\displaystyle 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 logarithm 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 exponential of the formal group law, so that we have F(u,v)=f(g(u)+g(v))

over 
Tex syntax error
. If R does not have torsion (so that 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 universal 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).

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

  1. The ring A is multiplicatively generated by the coefficients of the series F_U;
  2. 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).

Proof 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.

Theorem 2.3 ([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.

3 Formal group law of geometric cobordisms

4 Hirzebruch genera

5 References

This page has not been refereed. The information given here might be incomplete or provisional.

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