Formal group laws and genera

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An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print.

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[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 R be a commutative ring with unit.

A formal power series F(u,v)\in R[[u,v]] is called a (commutative one-dimensional) formal group law over R if it satisfies the following equations:

  1. F(u,0)=u, F(0,v)=v;
  2. F(F(u,v),w)=F(u,F(v,w));
  3. F(u,v)=F(v,u).

The original example of a formal group law over a field \mathbf k is provided by the expansion near the unit of the multiplication map G\times G\to G in a one-dimensional algebraic group over \mathbf k. This also explains the terminology.

A formal group law F over R is called linearisable if there exists a coordinate change u\mapsto g_F(u)=u+\sum_{i>1}g_iu^i\in R[[u]] such that

\displaystyle    g_F(F(u,v))=g_F(u)+g_F(v).

Note that every formal group law over R determines a formal group law over R\otimes\mathbb Q.

Theorem 2.1. Every formal group law F is linearisable over R\otimes\mathbb Q.

Proof. Consider the series \omega(u)=\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}. 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).

We therefore have \frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}. Set

\displaystyle    g(u)=\int_0^u\frac{dv}{\omega(v)};

then dg(u)=dg(F(u,v)). This implies that g(F(u,v))=g(u)+C. Since F(0,v)=v and g(0)=0, we get C=g(v). Thus, g(F(u,v))=g(u)+g(v). \square

A series g(u)=u+\sum_{i>1}g_iu^i satisfying the equation g(F(u,v))=g(u)+g(v) is called a logarithm of the formal group law F; the above Theorem shows that a formal group law over R\otimes\mathbb Q always has a logarithm. Its functional inverse series f(t)\in R\otimes\mathbb Q[[t]] is called an exponential of the formal group law, so that we have F(u,v)=f(g(u)+g(v)) over R\otimes\mathbb Q. If R does not have torsion (i.e. R\to R\otimes\mathbb Q is monic), the latter formula shows that a formal group law (as a series with coefficients in R) is fully determined by its logarithm (which is a series with coefficients in R\otimes\mathbb Q).

Let F=\sum_{k,l}a_{kl}u^kv^l be a formal group law over a ring R and r\colon R\to R' a ring homomorphism. Denote by r(F) the formal series \sum_{k,l}r(a_{kl})u^kv^l\in R'[[u,v]]; then r(F) is a formal group law over R'.

A formal group law \mathcal F over a ring A is universal if for any formal group law F over any ring R there exists a unique homomorphism r\colon A\to R such that F=r(\mathcal F).

Proposition 2.2. Assume that a universal formal group law \mathcal F over A exists. Then

  1. The ring A is multiplicatively generated by the coefficients of the series \mathcal F;
  2. The universal formal group law is unique: if \mathcal F' is another universal formal group law over A', then there is an isomorphism r\colon A\to A' such that \mathcal F'=r(\mathcal F).

Proof. To prove the first statement, denote by A' the subring in A generated by the coefficients of \mathcal F. Then there is a monomorphism i\colon A'\to A satisfying i(\mathcal F)=\mathcal F. On the other hand, by universality there exists a homomorphism r\colon A\to A' satisfying r(\mathcal F)=\mathcal F. It follows that ir(\mathcal F)=\mathcal F. This implies that ir=\mathrm{id}\colon A\to A by the uniqueness requirement in the definition of \mathcal F. Thus A'=A. The second statement is proved similarly. \square

Theorem 2.3 ([Lazard1955]). The universal formal group law \mathcal F exists, and its coefficient ring A is isomorphic to the polynomial ring \mathbb Z[a_1,a_2,\ldots] 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 X be a cell complex and u,v\in U^2(X) two geometric cobordisms corresponding to elements x,y\in H^2(X) respectively. Denote by u+_{\!{}_H}\!v the geometric cobordism corresponding to the cohomology class x+y.

Proposition 3.1. The following relation holds in U^2(X):

\displaystyle    u+_{\!{}_H}\!v=F_U(u,v)=u+v+\sum_{k\ge1,\,l\ge1}\alpha_{kl}\,u^kv^l,

where the coefficients \alpha_{kl}\in\varOmega_U^{-2(k+l-1)} do not depend on X. The series F_U(u,v) is a formal group law over the complex bordism ring \varOmega_U.

Proof. Click here - opens a separate pdf file.


The series F_U(u,v) 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 u\in U^2(X) is the first Conner-Floyd Chern class of the complex line bundle \xi over X obtained by pulling back the canonical bundle along the map f_u\colon X\to\mathbb C P^\infty. It follows that the formal group law of geometric cobordisms gives an expression of the first class c_1^U(\xi\otimes\eta)\in U^2(X) of the tensor product of two complex line bundles over X in terms of the classes u=c_1^U(\xi) and v=c_1^U(\eta) of the factors:

\displaystyle    c_1^U(\xi\otimes\eta)=F_U(u,v).

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

Theorem 3.2 ([Buchstaber1970]).

\displaystyle    F_U(u,v)=\frac{\sum_{i,j\ge0}[H_{ij}]u^iv^j}   {\bigl(\sum_{r\ge0}[\mathbb C P^r]u^r\bigr)\bigl(\sum_{s\ge0}[\mathbb C   P^s]v^s\bigr)},

where H_{ij} (0\le i\le j) are Milnor hypersurfaces and H_{ji}=H_{ij}.

Proof. Click here - opens a separate pdf file.


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

\displaystyle    g_U(u)=u+\sum_{k\ge1}\frac{[\mathbb C P^k]}{k+1}u^{k+1}   \in\varOmega_U\otimes\mathbb Q[[u]].

Proof. Click here - opens a separate pdf file.


Using these calculations the following most important property of the formal group law F_U can be easily established:

Theorem 3.4 ([Quillen1969]). The formal group law F_U of geometric cobordisms is universal.

Proof. Click here - opens a separate pdf file.


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 \mathbb Z/p as

\displaystyle    U^*(B\mathbb Z/p)\cong\varOmega_U[[u]]/[u]_p,

where \varOmega_U[[u]] denotes the ring of power series in one generator u of degree 2 with coefficients in \varOmega_U, and [u]_p denotes the pth power in the formal group law of geometric cobordisms. This result extended and unified many earlier calculations of bordism with \mathbb Z/p-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 \varphi\colon\varOmega_U\to R from the complex cobordism ring to a commutative ring R 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) R-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 R does not have additive torsion. Then every R-genus \varphi is fully determined by the corresponding homomorphism \varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q, which we shall also denote by \varphi. The following famous construction of [Hirzebruch1966] allows us to describe homomorphisms \varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q by means of universal R-valued characteristic classes of special type.

[edit] 4.1 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

\displaystyle    \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 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

\displaystyle    \Hom\nolimits_{\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

\displaystyle    1+c_1+c_2+\cdots=(1+c'_1+c'_2+\cdots)\cdot(1+c''_1+c''_2+\cdots)

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).

A sequence of homogeneous polynomials \mathcal K=\{K_i(c_1,\ldots,c_i),i\ge0\} with K_0=1 satisfying these identities is called a multiplicative Hirzebruch sequence.

Proposition 4.1. A multiplicative sequence \mathcal K is completely determined by the series

\displaystyle    Q(x)=1+q_1x+q_2x^2+\cdots\in R\otimes\Qq[[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.

Proof. Indeed, by considering the identity

\displaystyle    1+c_1+\cdots+c_n=(1+x_1)\cdots(1+x_n)

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.

Along with the series Q(x) it is convenient to consider the series f(x)\in R\otimes\mathbb Q[[x]] with leading term x given by the identity

\displaystyle    Q(x)=\frac x{f(x)}.

It follows that ring homomorphisms \varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q are in one-to-one correspondence with series f(x)\in R\otimes\mathbb Q[[x]] with leading term x. Under this correspondence, the value of \varphi on an 2n-dimensional bordism class [M]\in\varOmega_U is given by

\displaystyle    \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\in H_{2n}(M).

The homomorphism \varphi\colon\varOmega_U\to R\otimes\mathbb Q given by the formula above is called the Hirzebruch genus associated to the series f(x)=x+\cdots\in R\otimes\mathbb Q[[x]]. Thus, there is a one-two-one correspondence between series f(x)\in R\otimes\mathbb Q[[x]] having leading term x and genera \varphi\colon\varOmega_U\to R\otimes\mathbb 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.

[edit] 4.2 Connection to formal group laws

Every genus \varphi\colon\varOmega_U\to R gives rise to a formal group law \varphi(F_U) over R, where F_U is the formal group law of geometric cobordisms.

Theorem 4.2. For every genus \varphi\colon\varOmega_U\to R\otimes\mathbb Q, the exponential of the formal group law \varphi(F_U) is given by the series f(x)\in R\otimes\mathbb Q[[x]] corresponding to \varphi.

Proof. Click here - opens a separate pdf file.


A parallel theory of genera exists for oriented manifolds. These genera are homomorphisms \varOmega_{SO}\to R from the oriented bordism ring, and the Hirzebruch construction expresses genera over \mathbb Q-algebras via certain Pontrjagin characteristic classes (which replace the Chern classes).

[edit] 4.3 Examples

We take \,R=\mathbb Z in these examples:

  1. The top Chern number \,c_n(\xi)[M] is a Hirzebruch genus, and its corresponding f-series is f(x)=\frac x{1+x}. The value of this genus on a stably complex manifold (M,c_{\mathcal T}) equals the Euler characteristic of M if c_{\mathcal T} is an almost complex structure.
  2. The L-genus \,L[M] corresponds to the series \,f(x)=\mathop{\mathrm{tanh}}(x) (the hyperbolic tangent). It is equal to the signature of M by the classical Hirzebruch formula [Hirzebruch1966].
  3. The Todd genus \mathop{\mathrm{td}}[M] corresponds to the series f(x)=1-e^{-x}. It takes value 1 on every complex projective space \,\mathbb C P^k.

The "trivial" genus \varepsilon\colon\varOmega_U\to\mathbb Z corresponding to the series f(x)=x gives rise to the augmentation transformation \,U^*\to H^* from complex cobordism to ordinary cohomology (also known as the Thom homomorphism). More generally, for every genus \varphi\colon\varOmega_U\to R and a space X we may set h^*_\varphi(X)=U^*(X)\otimes_{\varOmega_U}R. Under certain conditions guaranteeing the exactness of the sequences of pairs (known as the Landweber exact functor theorem [Landweber1976]) the functor h^*_\varphi(\cdot) gives rise to a complex-oriented cohomology theory with the coefficient ring R.

As an example of this procedure, consider a formal indeterminate \beta of degree -2, and let \,f(x)=1-e^{-\beta x}. The corresponding genus, which is also called the Todd genus, takes values in the ring \mathbb Z[\beta]. By interpreting \beta as the Bott element in the complex K-group \,\widetilde K^0(S^2)=K^{-2}(pt) we obtain a homomorphism \mathop{\mathrm{td}}\colon \varOmega^*_U\to K^*(pt). It gives rise to a multiplicative transformation \,U^*\to K^* 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 K^*(X)\cong U^*(X)\otimes_{\varOmega_U}\mathbb Z[\beta], where the \varOmega_U-module structure on \mathbb Z[\beta] 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 \chi_y-genus. It corresponds to the series

\displaystyle    f(x)=\frac{1-e^{-x(1+y)}}{1+ye^{-x(1+y)}},

where y\in\mathbb R is a parameter. Setting y=-1, y=0 and y=1 we get the top Chern number c_n[M], the Todd genus \mathop{\mathrm{td}}[M] and the L-genus L[M]=\mathop{\mathrm{sign}}(M) respectively.

If M is a complex manifold then the value \chi_y[M] can be calculated in terms of the Euler characteristics of Dolbeault complexes on M.

[edit] 5 References

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