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], which are important invariants of bordism classes of manifolds.

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.

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.

\square

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.

\square

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.

\square

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.

\square

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.

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.

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.

\begin{proposition}A multiplicative sequence \mathcal K is completely determined bythe series

Tex syntax error
 1+c_1+\cdots+c_n=(1+x_1)\cdots(1+x_n)
\displaystyle  we obtain that
 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.
Tex syntax error
 Q(x)=\frac x{f(x)}.
Tex syntax error
 \varphi[M]=\Bigl(\prod^n_{i=1}\frac{x_i}{f(x_i)},
 \langle M\rangle\Bigr)
Tex syntax error

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+f_1x+f_2x^2+\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.

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 4.1. 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.

\square

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).

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.

5 References

\le i\le j$) are [[Complex bordism#Multiplicative generators|Milnor hypersurfaces]] and $H_{ji}=H_{ij}$. {{endthm}} {{beginproof}} [[Media:proofs-fglgc32.pdf|Click here - opens a separate pdf file]]. {{endproof}} {{beginthm|Theorem|(Mishchenko, see \cite{Novikov1967})}} The logarithm of the formal group law of geometric cobordisms is given by the series $$ g_U(u)=u+\sum_{k\ge1}\frac{[\mathbb C P^k]}{k+1}u^{k+1} \in\varOmega_U\otimes\mathbb Q[[u]]. $$ {{endthm}} {{beginproof}} [[Media:proofs-fglgc33.pdf|Click here - opens a separate pdf file]]. {{endproof}} Using these calculations the following most important property of the formal group law $F_U$ can be easily established: {{beginthm|Theorem|(\cite{Quillen1969})}} The formal group law $F_U$ of geometric cobordisms is universal. {{endthm}} {{beginproof}} [[Media:proofs-fglgc34.pdf|Click here - opens a separate pdf file]]. {{endproof}} The earliest applications of formal group laws in cobordism concerned finite group actions on manifolds, or "differentiable periodic maps", see \cite{Novikov1967}, \cite{Buchstaber&Novikov1971}, \cite{Buchstaber&Mishchenko&Novikov1971}. For instance, a theorem of \cite{Novikov1967} describes the complex cobordism ring of the [[Wikipedia:Classifying_space|classifying space]] of the group $\mathbb Z/p$ as $$ U^*(B\mathbb Z/p)\cong\varOmega_U[[u]]/[u]_p, $$ 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 $p$th 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 \cite{Conner&Floyd1964}. The universality of the formal group law of geometric cobordisms has important consequences for the [[Wikipedia:Stable_homotopy_theory|stable homotopy theory]]: it implies that [[Complex bordism|complex bordism]] is the universal complex oriented [[Wikipedia:Homology_theory|cohomology theory]]. == 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 \cite{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. === 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 [[Wikipedia:Chern class|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 $$ \varOmega_U\to H_*(BU). $$ 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 [[Wikipedia:Vector_bundle#Operations_on_vector_bundles|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 [[Complex bordism#Structure results|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 $$ \Hom\nolimits_{\mathbb Q}(H_*(BU;\mathbb Q),R\otimes\mathbb Q)=H^*(BU;\mathbb Q)\otimes R, $$ 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 $$ 1+c_1+c_2+\cdots=(1+c'_1+c'_2+\cdots)\cdot(1+c''_1+c''_2+\cdots) $$ implies the identity $$ \sum_{n\ge0}K_n(c_1,\ldots,c_n)= \sum_{i\ge0}K_i(c'_1,\ldots,c'_i)\cdot \sum_{j\ge0}K_j(c''_1,\ldots,c''_j). $$ 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. \begin{proposition}A multiplicative sequence $\mathcal K$ is completely determined bythe series $$ Q(x)=1+q_1x+q_2x^2+\cdots\in R\otimes\Q[[x]], %% where $x=c_1$, and $q_i=K_i(1,0,\ldots,0)$; moreover, every series $Q(x)$ as above determines a multiplicative sequence. \end{proposition} \begin{proof} Indeed, by considering the identity considering the identity $$ 1+c_1+\cdots+c_n=(1+x_1)\cdots(1+x_n) $$ we obtain that $$ 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. $$ \end{proof} Along with the series $Q(x)$ it is convenient to consider the series $f(x)\in R\otimes\mathbb Q[[x]]$ with leading term $x$ given by the identity $$ Q(x)=\frac x{f(x)}. $$ It follows that the ring homomorphisms $\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$ are in one-to-one correspondence with the series $f(x)\in R\otimes\mathbb Q[[x]]$. Under this correspondence, the value of $\varphi$ on an n$-dimensional bordism class $[M]\in\varOmega_U$ is given by $$ \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+f_1x+f_2x^2+\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$. === 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 laws and genera#Formal group law of geometric cobordisms|formal group law of geometric cobordisms]]. {{beginthm|Theorem}} 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$. {{endthm}} {{beginproof}} [[Media:proofs-hirzgen421.pdf|Click here - opens a separate pdf file]]. {{endproof}} A parallel theory of genera exists for oriented manifolds. These genera are homomorphisms $\varOmega_{SO}\to R$ from the [[Oriented bordism|oriented bordism ring]], and the Hirzebruch construction expresses genera over $\mathbb Q$-algebras via certain [[Wikipedia:Pontryagin class|Pontrjagin characteristic classes]] (which replace the [[Wikipedia:Chern class|Chern classes]]). === Examples === ; We take $\,R=\mathbb Z$ in these examples: # 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 [[Complex bordism#Stably complex structures|stably complex manifold]] $(M,c_{\mathcal T})$ equals the Euler characteristic of $M$ if $c_{\mathcal T}$ is an almost complex structure. # The $L$-genus $\,L[M]$ corresponds to the series $\,f(x)=\mathop{\mathrm{tanh}}(x)$ (the hyperbolic tangent). It is equal to the [[Wikipedia:Signature_(topology)|signature]] of $M$ by the classical Hirzebruch formula \cite{Hirzebruch1966}. # 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 \cite{Landweber1976}) the functor $h^*_\varphi(\cdot)$ gives rise to a complex-oriented [[Wikipedia:Homology theory|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 [[Wikipedia:K-theory|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 \cite{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 $$ 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 [[Wikipedia:Dolbeault_cohomology|Dolbeault complexes]] on $M$. == References == {{#RefList:}} [[Category:Bordism]] [[Category:Theory]]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.

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.

\square

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.

\square

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.

\square

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.

\square

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.

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.

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.

\begin{proposition}A multiplicative sequence \mathcal K is completely determined bythe series

Tex syntax error
 1+c_1+\cdots+c_n=(1+x_1)\cdots(1+x_n)
\displaystyle  we obtain that
 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.
Tex syntax error
 Q(x)=\frac x{f(x)}.
Tex syntax error
 \varphi[M]=\Bigl(\prod^n_{i=1}\frac{x_i}{f(x_i)},
 \langle M\rangle\Bigr)
Tex syntax error

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+f_1x+f_2x^2+\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.

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 4.1. 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.

\square

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).

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.

5 References

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