# Formal group laws and genera

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## Contents |

## [edit] 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], which are important invariants of bordism classes of manifolds.

## [edit] 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

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

We therefore have . Set

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.

## [edit] 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 :

where the coefficients do not depend on . The series is a formal group law over the complex bordism ring .

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:

The coefficients of the formal group law of geometric cobordisms and its logarithm may be described geometrically by the following results.

**Theorem 3.3** (Mishchenko, see [Novikov1967])**.**
The logarithm of the formal group law of geometric cobordisms is
given by the series

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.

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

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.

## [edit] 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.

### [edit] 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

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

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

implies the identity

A sequence of homogeneous polynomials
with satisfying these
identities is called a *multiplicative Hirzebruch sequence*.

**Proposition 4.1.**
A multiplicative sequence is completely determined by the series

where , and ; moreover, every series as above determines a multiplicative sequence.

**Proof.**
Indeed, by considering the identity

we obtain that

Along with the series it is convenient to consider the series with leading term given by the identity

It follows that ring homomorphisms are in one-to-one correspondence with series with leading term . Under this correspondence, the value of on an -dimensional bordism class is given by

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 .

### [edit] 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 4.2.**
For every genus , the
exponential of the formal group law is given
by the series corresponding to .

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).

### [edit] 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

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 .

## [edit] 5 References

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*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,
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*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
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*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