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 ${{Authors|Taras Panov}} == Introduction == The theory of formal group laws, 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 Hirzebruch genera \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) formal group law 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 linearisable 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, $g(F(u,v))=g(u)+g(v)$. $\square$ 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 $R\otimes\mathbb Q$. 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 universal if for any formal group law $F$ over any ring $R$ there exists a unique homomorphism $r\colon A\to R$ such that $F=r(F_U)$. {{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}} 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. $\square$ {{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 == <wikitex>; The applications of the formal group laws in the cobordism theory build upon the following basic example. Let $X$ be a cell complex and $u,v\in U^2(X)$ two [[Complex bordism#Geometric cobordisms|geometric cobordisms]] corresponding to elements $x,y\in H^2(X)$ respectively. Denote by $u+_{\!{}_H}\!v$ the geometric cobordism corresponding to the cohomology class $x+y$. {{beginthm|Proposition}} The following relation holds in $U^2(X)$: $$ u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l, $$ where the coefficients $\alpha_{kl}\in\varOmega_U^{-2(k+l-1)}$ do not depend on $X$. The series $\mathcal F(u,v)$ is a formal group law over the ring $\varOmega_U=\varOmega^*_U$. {{endthm}} Proof. We first do calculations with the universal example $X=\mathbb C P^\infty\times\mathbb C P^\infty$. Then $$ U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]], $$ where $\underline u,\underline v$ are canonical geometric cobordisms given by the projections of $\mathbb C P^\infty\times\mathbb C P^\infty$ onto its factors. We therefore have the following relation in $U^2(\mathbb C P^\infty\times\mathbb C P^\infty)$: $$ \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l, $$ where $\alpha_{kl}\in\varOmega_U^{-2(k+l-1)}$. Now let the geometric cobordisms $u,v\in U^2(X)$ be given by maps $f_u,f_v\colon X\to\mathbb C P^\infty$ respectively. Then $u=(f_u\times f_v)^*(\underline u)$, $v=(f_u\times f_v)^*(\underline v)$ and $u+_{\!{}_H}\!v=(f_u\times f_v)^*(\underline u+_{\!{}_H}\!\underline v)$, where $f_u\times f_v\colon X\to\mathbb C P^\infty\times\mathbb C P^\infty$. Applying the $\varOmega^*_U$-module map $(f_u\times f_v)^*$ to the universal formula for $\underline u+_{\!{}_H}\!\underline v$ above gives the required expression. The fact that $\mathcal F(u,v)$ is a formal group law follows directly from the properties of the group multiplication $\mathbb C P^\infty\times\mathbb C P^\infty\to \mathbb C P^\infty$. $\square$ The series $\mathcal F(u,v)$ is called the formal group law of geometric cobordisms. The geometric cobordism $u\in U^2(X)$ is the first Chern-Conner-Floyd class of the complex line bundle $\xi$ over $X$ obtained by pulling back the canonical bundle along the map $f_u\colon X\to\mathbb C P^\infty$. It follows that the formal group law of geometric cobordisms gives an expression of the first class $c_1^U(\xi\otimes\eta)\in U^2(X)$ of the tensor product of two complex line bundles over $X$ in terms of the classes $u=c_1^U(\xi)$ and $v=c_1^U(\eta)$ of the factors: $$ c_1^U(\xi\otimes\eta)=\mathcal F(u,v). $$ The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms. {{beginthm|Theorem|\cite[Th. 4.8]{Buchstaber1970}}} $$ \mathcal F(u,v)=\frac{\sum_{i,j\ge0}[H_{ij}]u^iv^j} {\bigl(\sum_{r\ge0}[\mathbb C P^r]u^r\bigr)\bigl(\sum_{s\ge0}[\mathbb C P^s]v^s\bigr)}, $$ where $H_{ij}$ (<script type="math/tex">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:

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

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 linearisable if there exists a coordinate change $u\mapsto g_F(u)=u+\sum_{i>0}g_iu^i\in R[[u]]$ such that

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

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

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

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

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

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

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

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, $g(F(u,v))=g(u)+g(v)$ . $\square$

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 $R\otimes\mathbb Q$ . 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 $F_U$ $F_U$ over a ring

Tex syntax error

$A$ 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 $F_U$ $F_U$ over

Tex syntax error

$A$ exists.

Then

The ring
```
Tex syntax error
```
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 $F'_U=r(F_U)$ .

Proof. To prove the first statement, denote by $A'$ $A'$ the subring in

Tex syntax error

$A$

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. $\square$

The universal formal group law $F_U$ exists, and its coefficient

ring

Tex syntax error

$A$ 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.

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 $X$ be a cell complex and $u,v\in U^2(X)$ two geometric cobordisms corresponding to elements $x,y\in H^2(X)$ respectively. Denote by $u+_{\!{}_H}\!v$ the geometric cobordism corresponding to the cohomology class $x+y$ .

The following relation holds in $U^2(X)$ :

$\displaystyle u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,$ $\displaystyle u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,$

where the coefficients $\alpha_{kl}\in\varOmega_U^{-2(k+l-1)}$ do not depend on $X$ . The series $\mathcal F(u,v)$ is a formal group law over the ring $\varOmega_U=\varOmega^*_U$ .

Proof. We first do calculations with the universal example $X=\mathbb C P^\infty\times\mathbb C P^\infty$ . Then

$\displaystyle U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]],$ $\displaystyle U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]],$

where $\underline u,\underline v$ are canonical geometric cobordisms given by the projections of $\mathbb C P^\infty\times\mathbb C P^\infty$ onto its factors. We therefore have the following relation in $U^2(\mathbb C P^\infty\times\mathbb C P^\infty)$ :

$\displaystyle \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l,$ $\displaystyle \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l,$

where $\alpha_{kl}\in\varOmega_U^{-2(k+l-1)}$ .

Now let the geometric cobordisms $u,v\in U^2(X)$ be given by maps $f_u,f_v\colon X\to\mathbb C P^\infty$ respectively. Then $u=(f_u\times f_v)^*(\underline u)$ , $v=(f_u\times f_v)^*(\underline v)$ and $u+_{\!{}_H}\!v=(f_u\times f_v)^*(\underline u+_{\!{}_H}\!\underline v)$ , where $f_u\times f_v\colon X\to\mathbb C P^\infty\times\mathbb C P^\infty$ . Applying the $\varOmega^*_U$ -module map $(f_u\times f_v)^*$ to the universal formula for $\underline u+_{\!{}_H}\!\underline v$ above gives the required expression. The fact that $\mathcal F(u,v)$ is a formal group law follows directly from the properties of the group multiplication $\mathbb C P^\infty\times\mathbb C P^\infty\to \mathbb C P^\infty$ . $\square$

The series $\mathcal F(u,v)$ is called the formal group law of geometric cobordisms.

The geometric cobordism $u\in U^2(X)$ is the first Chern-Conner-Floyd class of the complex line bundle $\xi$ over $X$ obtained by pulling back the canonical bundle along the map $f_u\colon X\to\mathbb C P^\infty$ . It follows that the formal group law of geometric cobordisms gives an expression of the first class $c_1^U(\xi\otimes\eta)\in U^2(X)$ of the tensor product of two complex line bundles over $X$ in terms of the classes $u=c_1^U(\xi)$ and $v=c_1^U(\eta)$ of the factors:

$\displaystyle c_1^U(\xi\otimes\eta)=\mathcal F(u,v).$ $\displaystyle c_1^U(\xi\otimes\eta)=\mathcal F(u,v).$

The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.

Theorem 3.2 [Buchstaber1970].

$\displaystyle \mathcal F(u,v)=\frac{\sum_{i,j\ge0}[H_{ij}]u^iv^j} {\bigl(\sum_{r\ge0}[\mathbb C P^r]u^r\bigr)\bigl(\sum_{s\ge0}[\mathbb C P^s]v^s\bigr)},$ $\displaystyle \mathcal F(u,v)=\frac{\sum_{i,j\ge0}[H_{ij}]u^iv^j} {\bigl(\sum_{r\ge0}[\mathbb C P^r]u^r\bigr)\bigl(\sum_{s\ge0}[\mathbb C P^s]v^s\bigr)},$

where $H_{ij}$ ( $0\le i\le j$ ) are Milnor hypersurfaces and $H_{ji}=H_{ij}$ .

Proof. Set $X=\mathbb C P^i\times\mathbb C P^j$ in Proposition~\ref{u2x}. Consider the \emph{Poincar\'e-Atiyah duality} map $D\colon U^2(\C P^i\times\C P^j)\to U_{2(i+j)-2}(\C P^i\times\C P^j)$ and the map $\varepsilon\colon U_*(\C P^i\times\C P^j)\to U_*(pt)=\varOmega_*^U$ induced by the projection $\C P^i\times\C P^j\to pt$ . 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, $\varepsilon D(u+_{\!{}_H}\!v)=[H_{ij}]$ , $\varepsilon D(u^kv^l)=[\C P^{i-k}][\C P^{j-l}]$ . Applying $\varepsilon D$ 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 $H_{i0}=\C P^{i-1}$ , 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 $[H_{i1}]=[\C P^1][\C P^{i-1}]$ (see Exercise~\ref{h1i}; we have already seen that $H_{11}=\C P^1$ in the Remark preceding Theorem~\ref{hijgen}). Therefore, $dg_{\mathcal F}(u)=1+\sum_{k>0}[\C P^k]u^k$ , which implies the required formula. \end{proof}

} The formal group law $\mathcal F$ of geometric cobordisms $\mathcal F$ is universal.

\begin{proof}

Let $F_U$ $F_U$ be the universal formal group law over a ring~

Tex syntax error

$A$ . Then

there is a homomorphism $r\colon A\to\varOmega_U$ which takes $F_U$ to $\mathcal F$ . The series $F_U$ , viewed as a formal group law over the ring $A\otimes\Q$ , has the universality properties for all formal group laws over $\Q$ -algebras. By theorem~\ref{logth}, such a formal group law is determined by its logarithm, which can be any series starting from~ $u$ . It follows that if we write the logarithm of $F_U$ as $\sum b_k\frac{u^{k+1}}{k+1}$ then the ring $A\otimes\Q$ is the polynomial ring $\Q[b_1,b_2,\ldots]$ . By Theorem~3.3, $r(b_k)=[\C P^k]\in\varOmega_U$ . Since $\varOmega_U\otimes\Q\cong\Q[[\C P^1],[\C P^2],\ldots]$ , this implies that $r\otimes\Q$ is an isomorphism.

By Theorem~\ref{lazardth} the ring

Tex syntax error

$A$ does not have torsion, so

Tex syntax error

$r$ is a monomorphism. On the other hand, Theorem~\ref{buchth}

implies that the image $r(A)$ contains the bordism classes $[H_{ij}]\in\varOmega_U$ , $0\le i\le j$ . Since these classes

generate the whole $\varOmega_U$ $\varOmega_U$ (Theorem~\ref{hijgen}),

Tex syntax error

$r$ 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.

\le i\le j$) are [[Complex bordism#Milnor hypersurfaces|Milnor hypersurfaces]] and $H_{ji}=H_{ij}$. {{endthm}} Proof. Set $X=\mathbb C P^i\times\mathbb C P^j$ in Proposition~\ref{u2x}. Consider the \emph{Poincar\'e-Atiyah duality} map $D\colon U^2(\C P^i\times\C P^j)\to U_{2(i+j)-2}(\C P^i\times\C P^j)$ and the map $\varepsilon\colon U_*(\C P^i\times\C P^j)\to U_*(pt)=\varOmega_*^U$ induced by the projection $\C P^i\times\C P^j\to pt$. 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, $\varepsilon D(u+_{\!{}_H}\!v)=[H_{ij}]$, $\varepsilon D(u^kv^l)=[\C P^{i-k}][\C P^{j-l}]$. Applying $\varepsilon D$ 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} \begin{theorem}[{Mishchenko~\cite[Appendix~1]{novi67}}]\label{mishth} The logarithm of the formal group law of geometric cobordisms is given by \[ g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\Q[[u]]. \] \end{theorem} \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 $H_{i0}=\C P^{i-1}$, 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 $[H_{i1}]=[\C P^1][\C P^{i-1}]$ (see Exercise~\ref{h1i}; we have already seen that $H_{11}=\C P^1$ in the Remark preceding Theorem~\ref{hijgen}). Therefore, $dg_{\mathcal F}(u)=1+\sum_{k>0}[\C P^k]u^k$, which implies the required formula. \end{proof} \begin{theorem}[{Quillen~\cite[Th.~2]{quil69}}] The formal group law $\mathcal F$ of geometric cobordisms $\mathcal F$ is universal. \end{theorem} \begin{proof} Let $F_U$ be the universal formal group law over a ring~$A$. Then there is a homomorphism $r\colon A\to\varOmega_U$ which takes $F_U$ to $\mathcal F$. The series $F_U$, viewed as a formal group law over the ring $A\otimes\Q$, has the universality properties for all formal group laws over $\Q$-algebras. By theorem~\ref{logth}, such a formal group law is determined by its logarithm, which can be any series starting from~$u$. It follows that if we write the logarithm of $F_U$ as $\sum b_k\frac{u^{k+1}}{k+1}$ then the ring $A\otimes\Q$ is the polynomial ring $\Q[b_1,b_2,\ldots]$. By Theorem~\ref{mishth}, $r(b_k)=[\C P^k]\in\varOmega_U$. Since $\varOmega_U\otimes\Q\cong\Q[[\C P^1],[\C P^2],\ldots]$, this implies that $r\otimes\Q$ is an isomorphism. By Theorem~\ref{lazardth} the ring $A$ does not have torsion, so $r$ is a monomorphism. On the other hand, Theorem~\ref{buchth} implies that the image $r(A)$ contains the bordism classes $[H_{ij}]\in\varOmega_U$, $R$ 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:

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