Formal group laws and genera
Taras Panov (Talk | contribs) (→Connection to the formal group laws) |
Taras Panov (Talk | contribs) (→Construction) |
||
Line 217: | Line 217: | ||
Let $BU=\lim\limits_{n\to\infty}BU(n)$. Then $H^*(BU)$ is | Let $BU=\lim\limits_{n\to\infty}BU(n)$. Then $H^*(BU)$ is | ||
isomorphic to the graded ring of formal power series | isomorphic to the graded ring of formal power series | ||
− | $\mathbb Z[[c_1,c_2,\ldots]]$ in universal Chern classes, $\deg c_k=2k$. | + | $\mathbb Z[[c_1,c_2,\ldots]]$ in universal [[Wikipedia:Chern class|Chern classes]], $\deg c_k=2k$. |
The set of Chern characteristic numbers of a manifold $M$ defines | 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 | an element in $\Hom(H^*(BU),\mathbb Z)$, which in fact belongs to the |
Revision as of 16:41, 17 April 2010
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 09:29, 1 April 2011 and the changes since publication. |
The user responsible for this page is Taras Panov. No other user may edit this page at present. |
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
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 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 :
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:
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
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
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.
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.
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
we obtain that
Along with the series it is convenient to consider the series given by the identity
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
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 starting from and genera .
We shall also denote the characteristic class of a complex vector bundle by ; so that .
4.2 Connection to the formal group laws
Every genus gives rise to a formal group law over , where is the formal group 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.
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
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
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 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 :
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:
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
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
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.
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.
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
we obtain that
Along with the series it is convenient to consider the series given by the identity
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
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 starting from and genera .
We shall also denote the characteristic class of a complex vector bundle by ; so that .
4.2 Connection to the formal group laws
Every genus gives rise to a formal group law over , where is the formal group 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.
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
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. |