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
![\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).](/images/math/7/9/c/79c476308fa074091cdd70a3bbc6a0dc.png)
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
![\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).](/images/math/7/5/0/7501bdbad3dcddbac402d33108d8b625.png)
We therefore have
. Set
![\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};](/images/math/b/b/d/bbd2161c870c820ada46478e53ec003c.png)
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 :
![\displaystyle u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,](/images/math/7/f/7/7f7618c94644c88b8c4466c7acf22751.png)
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
![\displaystyle U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]],](/images/math/d/2/1/d213e18ac11c23f4fc53e3d524395efd.png)
where are canonical geometric
cobordisms given by the projections of
onto its factors. We therefore have the following
relation in
:
![\displaystyle \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l,](/images/math/9/9/9/9996c4d56d4112b9877eb5c0a1323569.png)
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:
![\displaystyle c_1^U(\xi\otimes\eta)=\mathcal F(u,v).](/images/math/5/f/1/5f1a96ca83cce28b92deb47497bf2dcb.png)
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
![\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)},](/images/math/0/4/7/047277326106a8d596df93ae07b21b2e.png)
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
![\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).](/images/math/7/9/c/79c476308fa074091cdd70a3bbc6a0dc.png)
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
![\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).](/images/math/7/5/0/7501bdbad3dcddbac402d33108d8b625.png)
We therefore have
. Set
![\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};](/images/math/b/b/d/bbd2161c870c820ada46478e53ec003c.png)
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 :
![\displaystyle u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,](/images/math/7/f/7/7f7618c94644c88b8c4466c7acf22751.png)
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
![\displaystyle U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]],](/images/math/d/2/1/d213e18ac11c23f4fc53e3d524395efd.png)
where are canonical geometric
cobordisms given by the projections of
onto its factors. We therefore have the following
relation in
:
![\displaystyle \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l,](/images/math/9/9/9/9996c4d56d4112b9877eb5c0a1323569.png)
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:
![\displaystyle c_1^U(\xi\otimes\eta)=\mathcal F(u,v).](/images/math/5/f/1/5f1a96ca83cce28b92deb47497bf2dcb.png)
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
![\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)},](/images/math/0/4/7/047277326106a8d596df93ae07b21b2e.png)
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
![\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).](/images/math/7/9/c/79c476308fa074091cdd70a3bbc6a0dc.png)
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
![\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).](/images/math/7/5/0/7501bdbad3dcddbac402d33108d8b625.png)
We therefore have
. Set
![\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};](/images/math/b/b/d/bbd2161c870c820ada46478e53ec003c.png)
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 :
![\displaystyle u+_{\!{}_H}\!v=\mathcal F(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,](/images/math/7/f/7/7f7618c94644c88b8c4466c7acf22751.png)
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
![\displaystyle U^*(\mathbb C P^\infty\times\mathbb C P^\infty)=\varOmega^*_U[[\underline u,\underline v]],](/images/math/d/2/1/d213e18ac11c23f4fc53e3d524395efd.png)
where are canonical geometric
cobordisms given by the projections of
onto its factors. We therefore have the following
relation in
:
![\displaystyle \underline u+_{\!{}_H}\!\underline v= \sum_{k,l\ge0} \alpha_{kl}\,\underline u^k\underline v^l,](/images/math/9/9/9/9996c4d56d4112b9877eb5c0a1323569.png)
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:
![\displaystyle c_1^U(\xi\otimes\eta)=\mathcal F(u,v).](/images/math/5/f/1/5f1a96ca83cce28b92deb47497bf2dcb.png)
The next statement describes manifolds representing the coefficients of the formal group law of geometric cobordisms.
Theorem 3.2 [Buchstaber1970, Th. 4.8].
![\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)},](/images/math/0/4/7/047277326106a8d596df93ae07b21b2e.png)
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
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