Formal group laws and genera
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Contents |
1 Introduction
The theory of formal group laws, which originally appeared in algebraic geometry, was brought into bordism theory in the pioneering work [Novikov1967]. The applications of formal group laws in algebraic topology are closely connected with 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
a 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 an 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 a 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 formal group laws in 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=F_U(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,](/images/math/a/4/e/a4eaa859e763440d4ad86e9c16c2068b.png)
where the coefficients do
not depend on
. The series
is a formal group law over the complex bordism
ring
.
See the proof here (opens a separate pdf).
The series is called the formal group law of
geometric cobordisms; nowadays it is also usually referred to as the "formal group law of complex cobordism".
The geometric cobordism is the
first Conner-Floyd Chern 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)=F_U(u,v).](/images/math/c/3/d/c3d341dca4020e7befa190ed0b5a55aa.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 here (opens a separate pdf).
Theorem 3.3 (Mishchenko, see [Novikov1967]). The logarithm of the formal group law of geometric cobordisms is given by the series
![\displaystyle g_{F_U}(u)=u+\sum_{k\ge1}\frac{[\mathbb C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\mathbb Q[[u]].](/images/math/2/d/8/2d8f4b2be853b616e6611ab88346f37f.png)
See the proof here (opens a separate pdf).
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 here (opens a separate pdf).
The earliest applications of 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 bordism with
-actions from [Conner&Floyd1964].
The universality of the formal group law of geometric cobordisms 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
an invariant of cobordism classes. Such a homomorphism is called
a (complex)
-genus. (The term "multiplicative genus" is also used, to emphasise that such a genus is a
ring homomorphism; in classical algebraic geometry, there are instances of genera which are not
multiplicative.)
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.
4.1 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 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+\cdots=(1+c'_1+c'_2+\cdots)\cdot(1+c''_1+c''_2+\cdots)](/images/math/c/8/5/c85ad5cef3074172ede0e5f32fa7a00b.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.
Such a multiplicative sequence is completely determined by the
series
where
, and
; moreover, every series
as above determines a multiplicative sequence. Indeed, by
considering the identity
![\displaystyle 1+c_1+\cdots+c_n=(1+x_1)\cdots(1+x_n)](/images/math/9/3/7/9371e14cdac3792b191f9e9f168c6183.png)
we obtain that
![\displaystyle Q(x_1)\cdots Q(x_n)=1+K_1(c_1)+K_2(c_1,c_2)+\cdots+ K_n(c_1,\ldots,c_n)+K_{n+1}(c_1,\ldots,c_n,0)+\cdots.](/images/math/e/d/7/ed75a3fdbcf719b2e8ea4da4580f2f09.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+\cdots.](/images/math/1/5/3/1534b620043b2b973931c95a53ffaec9.png)
It follows that the ring homomorphisms
are in
one-to-one correspondence with the series
.
Under 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
between series
having leading term
and
genera
.
We shall also denote the
characteristic class of a
complex vector bundle
by
; so that
.
4.2 Connection to formal group laws
Every genus gives rise to a formal
group law
over
, where
is
the formal group law of geometric cobordisms.
Theorem 4.1.
For every genus , the
exponential of the formal group law
is given
by the series
corresponding to
.
See the proof here (opens a separate pdf).
A parallel theory of genera exists for oriented manifolds. These
genera are homomorphisms from the oriented
bordism ring, and the Hirzebruch construction expresses genera
over
-algebras via certain Pontrjagin characteristic classes
(which replace the Chern classes).
4.3 Examples
We take in these examples:
- The top Chern number
is a Hirzebruch genus, and its corresponding
-series is
. The value of this genus on a stably complex manifold
equals the Euler characteristic of
if
is an almost complex structure.
- The
-genus
corresponds to the series
(the hyperbolic tangent). It is equal to the signature of
by the classical Hirzebruch formula [Hirzebruch1966].
- The Todd genus
corresponds to the series
. It takes value 1 on every complex projective space
.
The "trivial" genus
corresponding to the series
gives rise to the augmentation transformation
from complex cobordism to ordinary cohomology (also known as the Thom homomorphism).
More generally, for every genus
and a space
we
may set
. Under
certain conditions guaranteeing the exactness of the sequences of
pairs (known as the Landweber exact functor theorem [Landweber1976])
the functor
gives rise to
a complex-oriented cohomology theory with the coefficient ring
.
As an example of this procedure, consider a formal indeterminate of degree -2, and let
. The corresponding genus, which is also
called the Todd genus, takes values in the ring
.
By interpreting
as the Bott element in the complex
K-group
we obtain a
homomorphism
. It gives rise
to a multiplicative transformation
from complex
cobordism to complex K-theory introduced by Conner and
Floyd [Conner&Floyd1966]. In this paper Conner and Floyd proved that complex cobordism determines
complex K-theory by means of the isomorphism
, where the
-module structure on
is given by the Todd genus. Their proof makes use of the Conner-Floyd Chern
classes; several proofs were given subsequently, including one which follows directly from the Landweber exact functor theorem.
Another important example from the original work of Hirzebruch is
given by the -genus. It corresponds to the series
![\displaystyle f(x)=\frac{1-e^{-x(1+y)}}{1+ye^{-x(1+y)}},](/images/math/8/c/5/8c542759fc2a7659f6ae828a779ef65e.png)
where is a parameter. Setting
,
and
we get the top Chern number
, the Todd genus
and
the
-genus
respectively.
If is a complex manifold then the value
can be
calculated in terms of the Euler characteristics of
Dolbeault complexes on
.
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
- [Conner&Floyd1966] P. E. Conner and E. E. Floyd, The relation of cobordism to
-theories, Springer-Verlag, Berlin, 1966. MR0216511 (35 #7344) Zbl 0161.42802
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Landweber1976] P. S. Landweber, Homological properties of comodules over
and
, Amer. J. Math. 98 (1976), no.3, 591–610. MR0423332 (54 #11311) Zbl 0355.55007
- [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