Formal group laws and genera
|the version used for publication as of 09:29, 1 April 2011 and the changes since publication.|
 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.
 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.
 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.
 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
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 .
 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).
 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 .
 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