Formal group laws and genera
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$\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$ by means of | $\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$ by means of | ||
universal $R$-valued characteristic classes of special type. | universal $R$-valued characteristic classes of special type. | ||
+ | |||
+ | === Construction === | ||
+ | Let $BU=\lim\limits_{n\to\infty}BU(n)$. Then $H^*(BU)$ is | ||
+ | isomorphic to the graded ring of formal power series | ||
+ | $\mathbb Z[[c_1,c_2,\ldots]]$ in universal Chern classes, $\deg c_k=2k$. | ||
+ | The set of Chern characteristic numbers of a manifold $M$ defines | ||
+ | an element in $\Hom(H^*(BU),\mathbb Z)$, which in fact belongs to the | ||
+ | subgroup $H_*(BU)$ in the latter group. We therefore obtain a | ||
+ | group homomorphism | ||
+ | $$ | ||
+ | \varOmega_U\to H_*(BU). | ||
+ | $$ | ||
+ | Since the multiplication in the ring $H_*(BU)$ is obtained from | ||
+ | the maps $BU_k\times BU_l\to BU_{k+l}$ corresponding to the | ||
+ | Whitney sum of vector bundles, and the Chern classes have the | ||
+ | appropriate multiplicative property, $\varOmega_U\to H_*(BU)$ is a | ||
+ | ring homomorphism. | ||
+ | |||
+ | Part 2 of the [Complex bordism#Structure results|structure theorem for complex bordism] says that | ||
+ | $\varOmega_U\to H_*(BU)$ is a monomorphism, and Part 1 of the same theorem says that the corresponding $\mathbb Q$-map | ||
+ | $\varOmega_U\otimes\mathbb Q\to H_*(BU;\mathbb Q)$ is an isomorphism. It follows | ||
+ | that every homomorphism $\varphi\colon\varOmega_U\otimes\mathbb Q\to | ||
+ | R\otimes\mathbb Q$ can be interpreted as an element of | ||
+ | $$ | ||
+ | \Hom_{\mathbb Q}(H_*(BU;\mathbb Q),R\otimes\mathbb Q)=H^*(BU;\mathbb Q)\otimes R, | ||
+ | $$ | ||
+ | or as a sequence of homogeneous polynomials | ||
+ | $\{K_i(c_1,\ldots,c_i),\;i\ge0\}$, $\deg K_i=2i$. This sequence of | ||
+ | polynomials cannot be chosen arbitrarily; the fact that $\varphi$ | ||
+ | is a ring homomorphism imposes certain conditions. These | ||
+ | conditions may be described as follows: an identity | ||
+ | $$ | ||
+ | 1+c_1+c_2+\ldots=(1+c'_1+c'_2+\ldots)\cdot(1+c''_1+c''_2+\ldots) | ||
+ | $$ | ||
+ | implies the identity | ||
+ | $$ | ||
+ | \sum_{n\ge0}K_n(c_1,\ldots,c_n)= | ||
+ | \sum_{i\ge0}K_i(c'_1,\ldots,c'_i)\cdot | ||
+ | \sum_{j\ge0}K_j(c''_1,\ldots,c''_j). | ||
+ | $$ | ||
+ | A sequence of homogeneous polynomials | ||
+ | $K=\{K_i(c_1,\ldots,c_i),i\ge0\}$ with $K_0=1$ satisfying these | ||
+ | identities is called a <i>multiplicative | ||
+ | Hirzebruch sequence</i>. | ||
+ | |||
+ | A multiplicative sequence $K$ is completely determined by the | ||
+ | series | ||
+ | $ | ||
+ | Q(x)=1+q_1x+q_2x^2+\ldots\in R\otimes\Q[[x]], | ||
+ | $ | ||
+ | where $x=c_1$, and $q_i=K_i(1,0,\ldots,0)$; moreover, every series | ||
+ | $Q(x)$ as above determines a multiplicative sequence. Indeed, by | ||
+ | considering the identity | ||
+ | $$ | ||
+ | 1+c_1+\ldots+c_n=(1+x_1)\cdots(1+x_n) | ||
+ | $$ | ||
+ | we obtain from that | ||
+ | $$ | ||
+ | Q(x_1)\cdots Q(x_n)=1+K_1(c_1)+K_2(c_1,c_2)+\ldots+ | ||
+ | K_n(c_1,\ldots,c_n)+K_{n+1}(c_1,\ldots,c_n,0)+\ldots. | ||
+ | $$ | ||
+ | Along with the series $Q(x)$ it is convenient to consider the | ||
+ | series $f(x)\in R\otimes\mathbb Q[[x]]$ given by the identity | ||
+ | $$ | ||
+ | Q(x)=\frac x{f(x)};\quad f(x)=x+f_1x+f_2x^2+\ldots. | ||
+ | $$ | ||
+ | It follows that the ring homomorphisms | ||
+ | $\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$ are in | ||
+ | one-to-one correspondence with the series $f(x)\in R\otimes\mathbb Q[[x]]$. | ||
+ | By this correspondence, the value of $\varphi$ on an | ||
+ | $2n$-dimensional bordism class $[M]\in\varOmega_U$ is given | ||
+ | by | ||
+ | $$ | ||
+ | \varphi[M]=\Bigl(\prod^n_{i=1}\frac{x_i}{f(x_i)}, | ||
+ | \langle M\rangle\Bigr) | ||
+ | $$ | ||
+ | where one needs to plug in the Chern classes $c_1,\ldots,c_n$ for the elementary symmetric functions | ||
+ | in $x_1,\ldots,x_n$ and then calculate the value of the resulting characteristic class on the fundamental class $\langle M\rangle$. | ||
+ | |||
+ | The homomorphism $\varphi\colon\varOmega_U\to R\otimes\mathbb Q$ given by | ||
+ | the formula above is called the <i>Hirzebruch genus</i> | ||
+ | associated to the series $f(x)=x+f_1x+f_2x^2+\ldots\in | ||
+ | R\otimes\mathbb Q[[x]]$. Thus, there is a one-two-one correspondence | ||
+ | between series $f(x)\in R\otimes\mathbb Q[[x]]$ starting from $x$ and | ||
+ | genera $\varphi\colon\varOmega_U\to R\otimes\Q$. | ||
+ | |||
+ | We shall also denote the | ||
+ | characteristic class $\prod^n_{i=1}\frac{x_i}{f(x_i)}$ of a | ||
+ | complex vector bundle $\xi$ by $\varphi(\xi)$; so that | ||
+ | $\varphi[M]=\varphi({\mathcal T}\!M)\langle M\rangle$. | ||
</wikitex> | </wikitex> | ||
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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 bordism 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 complex bordism
ring
.
See the proof.
The series is called the formal group law of
geometric cobordisms; nowadays it is also usually referred to as "complex cobordism formal group law".
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 coefficients of the formal group law of geometric cobordisms and its logarithm may be described geometrically by the following results.
See the proof.
Theorem 3.3 (Mishchenko, see [Novikov1967]). The logarithm of the formal group law of geometric cobordisms is given by the series
![\displaystyle g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\mathbb C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\mathbb Q[[u]].](/images/math/d/1/d/d1db6dc3034e8a8e12ce289fee9dd6ee.png)
See the proof.
Using these calculations the following most important property of the formal group law can be easily established:
Theorem 3.4 ([Quillen1969]).
The formal group law of geometric cobordisms is universal.
See the proof.
The earliest applications of the formal group laws in cobordism concerned finite group actions on manifolds, or "differentiable periodic maps", see [Novikov1967], [Buchstaber&Novikov1971], [Buchstaber&Mishchenko&Novikov1971]. For instance, a theorem of [Novikov1967] describes the complex cobordism ring of the classifying space of the group as
![\displaystyle U^*(B\mathbb Z/p)\cong\varOmega_U[[u]]/[u]_p,](/images/math/b/d/7/bd7da689ff14861c5d795b6b619a1156.png)
where denotes the ring of power series in one generator
of degree 2 with coefficients in
, and
denotes the
th power in the formal group law of geometric cobordisms. This result extended and unified many earlier calculations of bordisms with
-actions from [Conner&Floyd1964].
The universality of the formal group law of geometric cobordism has important consequences for the stable homotopy theory: it implies that complex bordism is the universal complex oriented cohomology theory.
4 Hirzebruch genera
Every homomorphism from the
complex cobordism ring to a commutative ring
with unit can be
regarded as a multiplicative characteristic of manifolds which is
invariant on cobordism classes. Such a homomorphism is called
a (complex)
-genus.
Assume that the ring does not have additive torsion. Then
every
-genus
is fully determined by the corresponding
homomorphism
, which we shall
also denote by
. The following famous construction of [Hirzebruch1966] allows us to describe homomorphisms
by means of
universal
-valued characteristic classes of special type.
Construction
Let . Then
is
isomorphic to the graded ring of formal power series
in universal Chern classes,
.
The set of Chern characteristic numbers of a manifold
defines
an element in
, which in fact belongs to the
subgroup
in the latter group. We therefore obtain a
group homomorphism
![\displaystyle \varOmega_U\to H_*(BU).](/images/math/f/6/2/f624910c5bb97513b82195bff5d1eb54.png)
Since the multiplication in the ring is obtained from
the maps
corresponding to the
Whitney sum of vector bundles, and the Chern classes have the
appropriate multiplicative property,
is a
ring homomorphism.
Part 2 of the [Complex bordism#Structure results|structure theorem for complex bordism] says that
is a monomorphism, and Part 1 of the same theorem says that the corresponding
-map
is an isomorphism. It follows
that every homomorphism
can be interpreted as an element of
![\displaystyle \Hom_{\mathbb Q}(H_*(BU;\mathbb Q),R\otimes\mathbb Q)=H^*(BU;\mathbb Q)\otimes R,](/images/math/e/e/c/eecca17722f6fdc856a3cde67992b5ba.png)
or as a sequence of homogeneous polynomials
,
. This sequence of
polynomials cannot be chosen arbitrarily; the fact that
is a ring homomorphism imposes certain conditions. These
conditions may be described as follows: an identity
![\displaystyle 1+c_1+c_2+\ldots=(1+c'_1+c'_2+\ldots)\cdot(1+c''_1+c''_2+\ldots)](/images/math/4/4/f/44f74eb767fedd1c683390090a2cb4de.png)
implies the identity
![\displaystyle \sum_{n\ge0}K_n(c_1,\ldots,c_n)= \sum_{i\ge0}K_i(c'_1,\ldots,c'_i)\cdot \sum_{j\ge0}K_j(c''_1,\ldots,c''_j).](/images/math/c/d/d/cdde65a37ce330d554c546403e8c5dd9.png)
A sequence of homogeneous polynomials
with
satisfying these
identities is called a multiplicative
Hirzebruch sequence.
A multiplicative sequence is completely determined by the
series
Tex syntax error
where , and
; moreover, every series
as above determines a multiplicative sequence. Indeed, by
considering the identity
![\displaystyle 1+c_1+\ldots+c_n=(1+x_1)\cdots(1+x_n)](/images/math/5/a/1/5a17609b12e31b22d8746e5bfd4ec596.png)
we obtain from that
![\displaystyle Q(x_1)\cdots Q(x_n)=1+K_1(c_1)+K_2(c_1,c_2)+\ldots+ K_n(c_1,\ldots,c_n)+K_{n+1}(c_1,\ldots,c_n,0)+\ldots.](/images/math/2/6/d/26db6dd1b255d5976cdefc99bc0b3b98.png)
Along with the series it is convenient to consider the
series
given by the identity
![\displaystyle Q(x)=\frac x{f(x)};\quad f(x)=x+f_1x+f_2x^2+\ldots.](/images/math/7/6/7/767432fe0fa9b0a0bae1f384dd09c645.png)
It follows that the ring homomorphisms
are in
one-to-one correspondence with the series
.
By this correspondence, the value of
on an
-dimensional bordism class
is given
by
![\displaystyle \varphi[M]=\Bigl(\prod^n_{i=1}\frac{x_i}{f(x_i)}, \langle M\rangle\Bigr)](/images/math/2/3/4/234d2f4052f422a1b5dd5ed93933832a.png)
where one needs to plug in the Chern classes for the elementary symmetric functions
in
and then calculate the value of the resulting characteristic class on the fundamental class
.
The homomorphism given by
the formula above is called the Hirzebruch genus
associated to the series
. Thus, there is a one-two-one correspondence
![f(x)\in R\otimes\mathbb Q[[x]]](/images/math/4/c/2/4c288a447d7068398ad73cece29a201d.png)
Tex syntax errorand genera
Tex syntax error.
We shall also denote the
characteristic class of a
complex vector bundle
by
; so that
.
5 References
- [Buchstaber&Mishchenko&Novikov1971] V. M. Buhštaber, A. S. Mišcenko and S. P. Novikov, Formal groups and their role in the apparatus of algebraic topology, Uspehi Mat. Nauk 26 (1971), no.2(158), 131–154. MR0445522 (56 #3862) Zbl 0226.55007
- [Buchstaber&Novikov1971] V. M. Buhštaber and S. P. Novikov, Formal groups, power systems and Adams operators, Mat. Sb. (N.S.) 84(126) (1971), 81–118. MR0291159 (45 #253) Zbl 0239.55005
- [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
- [Conner&Floyd1964] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Academic Press Inc., Publishers, New York, 1964. MR0176478 (31 #750) Zbl 0417.57019
- [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
- [Quillen1969] D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. MR0253350 (40 #6565) Zbl 0199.26705
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 bordism 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 complex bordism
ring
.
See the proof.
The series is called the formal group law of
geometric cobordisms; nowadays it is also usually referred to as "complex cobordism formal group law".
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 coefficients of the formal group law of geometric cobordisms and its logarithm may be described geometrically by the following results.
See the proof.
Theorem 3.3 (Mishchenko, see [Novikov1967]). The logarithm of the formal group law of geometric cobordisms is given by the series
![\displaystyle g_{\mathcal F}(u)=u+\sum_{k\ge1}\frac{[\mathbb C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\mathbb Q[[u]].](/images/math/d/1/d/d1db6dc3034e8a8e12ce289fee9dd6ee.png)
See the proof.
Using these calculations the following most important property of the formal group law can be easily established:
Theorem 3.4 ([Quillen1969]).
The formal group law of geometric cobordisms is universal.
See the proof.
The earliest applications of the formal group laws in cobordism concerned finite group actions on manifolds, or "differentiable periodic maps", see [Novikov1967], [Buchstaber&Novikov1971], [Buchstaber&Mishchenko&Novikov1971]. For instance, a theorem of [Novikov1967] describes the complex cobordism ring of the classifying space of the group as
![\displaystyle U^*(B\mathbb Z/p)\cong\varOmega_U[[u]]/[u]_p,](/images/math/b/d/7/bd7da689ff14861c5d795b6b619a1156.png)
where denotes the ring of power series in one generator
of degree 2 with coefficients in
, and
denotes the
th power in the formal group law of geometric cobordisms. This result extended and unified many earlier calculations of bordisms with
-actions from [Conner&Floyd1964].
The universality of the formal group law of geometric cobordism has important consequences for the stable homotopy theory: it implies that complex bordism is the universal complex oriented cohomology theory.
4 Hirzebruch genera
Every homomorphism from the
complex cobordism ring to a commutative ring
with unit can be
regarded as a multiplicative characteristic of manifolds which is
invariant on cobordism classes. Such a homomorphism is called
a (complex)
-genus.
Assume that the ring does not have additive torsion. Then
every
-genus
is fully determined by the corresponding
homomorphism
, which we shall
also denote by
. The following famous construction of [Hirzebruch1966] allows us to describe homomorphisms
by means of
universal
-valued characteristic classes of special type.
Construction
Let . Then
is
isomorphic to the graded ring of formal power series
in universal Chern classes,
.
The set of Chern characteristic numbers of a manifold
defines
an element in
, which in fact belongs to the
subgroup
in the latter group. We therefore obtain a
group homomorphism
![\displaystyle \varOmega_U\to H_*(BU).](/images/math/f/6/2/f624910c5bb97513b82195bff5d1eb54.png)
Since the multiplication in the ring is obtained from
the maps
corresponding to the
Whitney sum of vector bundles, and the Chern classes have the
appropriate multiplicative property,
is a
ring homomorphism.
Part 2 of the [Complex bordism#Structure results|structure theorem for complex bordism] says that
is a monomorphism, and Part 1 of the same theorem says that the corresponding
-map
is an isomorphism. It follows
that every homomorphism
can be interpreted as an element of
![\displaystyle \Hom_{\mathbb Q}(H_*(BU;\mathbb Q),R\otimes\mathbb Q)=H^*(BU;\mathbb Q)\otimes R,](/images/math/e/e/c/eecca17722f6fdc856a3cde67992b5ba.png)
or as a sequence of homogeneous polynomials
,
. This sequence of
polynomials cannot be chosen arbitrarily; the fact that
is a ring homomorphism imposes certain conditions. These
conditions may be described as follows: an identity
![\displaystyle 1+c_1+c_2+\ldots=(1+c'_1+c'_2+\ldots)\cdot(1+c''_1+c''_2+\ldots)](/images/math/4/4/f/44f74eb767fedd1c683390090a2cb4de.png)
implies the identity
![\displaystyle \sum_{n\ge0}K_n(c_1,\ldots,c_n)= \sum_{i\ge0}K_i(c'_1,\ldots,c'_i)\cdot \sum_{j\ge0}K_j(c''_1,\ldots,c''_j).](/images/math/c/d/d/cdde65a37ce330d554c546403e8c5dd9.png)
A sequence of homogeneous polynomials
with
satisfying these
identities is called a multiplicative
Hirzebruch sequence.
A multiplicative sequence is completely determined by the
series
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where , and
; moreover, every series
as above determines a multiplicative sequence. Indeed, by
considering the identity
![\displaystyle 1+c_1+\ldots+c_n=(1+x_1)\cdots(1+x_n)](/images/math/5/a/1/5a17609b12e31b22d8746e5bfd4ec596.png)
we obtain from that
![\displaystyle Q(x_1)\cdots Q(x_n)=1+K_1(c_1)+K_2(c_1,c_2)+\ldots+ K_n(c_1,\ldots,c_n)+K_{n+1}(c_1,\ldots,c_n,0)+\ldots.](/images/math/2/6/d/26db6dd1b255d5976cdefc99bc0b3b98.png)
Along with the series it is convenient to consider the
series
given by the identity
![\displaystyle Q(x)=\frac x{f(x)};\quad f(x)=x+f_1x+f_2x^2+\ldots.](/images/math/7/6/7/767432fe0fa9b0a0bae1f384dd09c645.png)
It follows that the ring homomorphisms
are in
one-to-one correspondence with the series
.
By this correspondence, the value of
on an
-dimensional bordism class
is given
by
![\displaystyle \varphi[M]=\Bigl(\prod^n_{i=1}\frac{x_i}{f(x_i)}, \langle M\rangle\Bigr)](/images/math/2/3/4/234d2f4052f422a1b5dd5ed93933832a.png)
where one needs to plug in the Chern classes for the elementary symmetric functions
in
and then calculate the value of the resulting characteristic class on the fundamental class
.
The homomorphism given by
the formula above is called the Hirzebruch genus
associated to the series
. Thus, there is a one-two-one correspondence
![f(x)\in R\otimes\mathbb Q[[x]]](/images/math/4/c/2/4c288a447d7068398ad73cece29a201d.png)
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We shall also denote the
characteristic class of a
complex vector bundle
by
; so that
.
5 References
- [Buchstaber&Mishchenko&Novikov1971] V. M. Buhštaber, A. S. Mišcenko and S. P. Novikov, Formal groups and their role in the apparatus of algebraic topology, Uspehi Mat. Nauk 26 (1971), no.2(158), 131–154. MR0445522 (56 #3862) Zbl 0226.55007
- [Buchstaber&Novikov1971] V. M. Buhštaber and S. P. Novikov, Formal groups, power systems and Adams operators, Mat. Sb. (N.S.) 84(126) (1971), 81–118. MR0291159 (45 #253) Zbl 0239.55005
- [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
- [Conner&Floyd1964] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Academic Press Inc., Publishers, New York, 1964. MR0176478 (31 #750) Zbl 0417.57019
- [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
- [Quillen1969] D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. MR0253350 (40 #6565) Zbl 0199.26705
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