Formal group laws and genera

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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


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

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

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

A formal group law $F$$F$ over $R$$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]]$$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$$R$ determines a formal group law over $R\otimes\mathbb Q$$R\otimes\mathbb Q$.

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

Proof. Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}$$\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))}$$\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))$$dg(u)=dg(F(u,v))$. This implies that $g(F(u,v))=g(u)+C$$g(F(u,v))=g(u)+C$. Since $F(0,v)=v$$F(0,v)=v$ and $g(0)=0$$g(0)=0$, we get $C=g(v)$$C=g(v)$. Thus, $g(F(u,v))=g(u)+g(v)$$g(F(u,v))=g(u)+g(v)$. $\square$$\square$

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

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

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

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

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

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

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

Proposition 3.1. The following relation holds in $U^2(X)$$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)}$$\alpha_{kl}\in\varOmega_U^{-2(k+l-1)}$ do not depend on $X$$X$. The series $F_U(u,v)$$F_U(u,v)$ is a formal group law over the complex bordism ring $\varOmega_U$$\varOmega_U$.

$\square$$\square$

The series $F_U(u,v)$$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)$$u\in U^2(X)$ is the first Conner-Floyd Chern class of the complex line bundle $\xi$$\xi$ over $X$$X$ obtained by pulling back the canonical bundle along the map $f_u\colon X\to\mathbb C P^\infty$$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)$$c_1^U(\xi\otimes\eta)\in U^2(X)$ of the tensor product of two complex line bundles over $X$$X$ in terms of the classes $u=c_1^U(\xi)$$u=c_1^U(\xi)$ and $v=c_1^U(\eta)$$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}$$H_{ij}$ ($0\le i\le j$$0\le i\le j$) are Milnor hypersurfaces and $H_{ji}=H_{ij}$$H_{ji}=H_{ij}$.

$\square$$\square$

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

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

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

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

$\square$$\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$$\mathbb Z/p$ as

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

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

4.1 Construction

Let $BU=\lim\limits_{n\to\infty}BU(n)$$BU=\lim\limits_{n\to\infty}BU(n)$. Then $H^*(BU)$$H^*(BU)$ is isomorphic to the graded ring of formal power series $\mathbb Z[[c_1,c_2,\ldots]]$$\mathbb Z[[c_1,c_2,\ldots]]$ in universal Chern classes, $\deg c_k=2k$$\deg c_k=2k$. The set of Chern characteristic numbers of a manifold $M$$M$ defines an element in $\Hom(H^*(BU),\mathbb Z)$$\Hom(H^*(BU),\mathbb Z)$, which in fact belongs to the subgroup $H_*(BU)$$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)$$H_*(BU)$ is obtained from the maps $BU_k\times BU_l\to BU_{k+l}$$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)$$\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)$$\varOmega_U\to H_*(BU)$ is a monomorphism, and Part 1 of the same theorem says that the corresponding $\mathbb Q$$\mathbb Q$-map $\varOmega_U\otimes\mathbb Q\to H_*(BU;\mathbb Q)$$\varOmega_U\otimes\mathbb Q\to H_*(BU;\mathbb Q)$ is an isomorphism. It follows that every homomorphism $\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$$\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\}$$\{K_i(c_1,\ldots,c_i),\;i\ge0\}$, $\deg K_i=2i$$\deg K_i=2i$. This sequence of polynomials cannot be chosen arbitrarily; the fact that $\varphi$$\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\}$$\mathcal K=\{K_i(c_1,\ldots,c_i),i\ge0\}$ with $K_0=1$$K_0=1$ satisfying these identities is called a multiplicative Hirzebruch sequence.

Proposition 4.1. A multiplicative sequence $\mathcal K$$\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$$x=c_1$, and $q_i=K_i(1,0,\ldots,0)$$q_i=K_i(1,0,\ldots,0)$; moreover, every series $Q(x)$$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

$\displaystyle Q(x_1)\cdots Q(x_n)=1+K_1(c_1)+K_2(c_1,c_2)+\cdots+ K_n(c_1,\ldots,c_n)+K_{n+1}(c_1,\ldots,c_n,0)+\cdots.$
$\square$$\square$

Along with the series $Q(x)$$Q(x)$ it is convenient to consider the series $f(x)\in R\otimes\mathbb Q[[x]]$$f(x)\in R\otimes\mathbb Q[[x]]$ with leading term $x$$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$$\varphi\colon\varOmega_U\otimes\mathbb Q\to R\otimes\mathbb Q$ are in one-to-one correspondence with series $f(x)\in R\otimes\mathbb Q[[x]]$$f(x)\in R\otimes\mathbb Q[[x]]$ with leading term $x$$x$. Under this correspondence, the value of $\varphi$$\varphi$ on an $2n$$2n$-dimensional bordism class $[M]\in\varOmega_U$$[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$$c_1,\ldots,c_n$ for the elementary symmetric functions in $x_1,\ldots,x_n$$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)$$\langle M\rangle\in H_{2n}(M)$.

The homomorphism $\varphi\colon\varOmega_U\to R\otimes\mathbb Q$$\varphi\colon\varOmega_U\to R\otimes\mathbb Q$ given by the formula above is called the Hirzebruch genus associated to the series $f(x)=x+\cdots\in R\otimes\mathbb Q[[x]]$$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]]$$f(x)\in R\otimes\mathbb Q[[x]]$ having leading term $x$$x$ and genera $\varphi\colon\varOmega_U\to R\otimes\mathbb Q$$\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)}$$\prod^n_{i=1}\frac{x_i}{f(x_i)}$ of a complex vector bundle $\xi$$\xi$ by $\varphi(\xi)$$\varphi(\xi)$; so that $\varphi[M]=\varphi({\mathcal T}\!M)\langle M\rangle$$\varphi[M]=\varphi({\mathcal T}\!M)\langle M\rangle$.

4.2 Connection to formal group laws

Every genus $\varphi\colon\varOmega_U\to R$$\varphi\colon\varOmega_U\to R$ gives rise to a formal group law $\varphi(F_U)$$\varphi(F_U)$ over $R$$R$, where $F_U$$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$$\varphi\colon\varOmega_U\to R\otimes\mathbb Q$, the exponential of the formal group law $\varphi(F_U)$$\varphi(F_U)$ is given by the series $f(x)\in R\otimes\mathbb Q[[x]]$$f(x)\in R\otimes\mathbb Q[[x]]$ corresponding to $\varphi$$\varphi$.

$\square$$\square$

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

4.3 Examples

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

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

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

As an example of this procedure, consider a formal indeterminate $\beta$$\beta$ of degree -2, and let $\,f(x)=1-e^{-\beta x}$$\,f(x)=1-e^{-\beta x}$. The corresponding genus, which is also called the Todd genus, takes values in the ring $\mathbb Z[\beta]$$\mathbb Z[\beta]$. By interpreting $\beta$$\beta$ as the Bott element in the complex K-group $\,\widetilde K^0(S^2)=K^{-2}(pt)$$\,\widetilde K^0(S^2)=K^{-2}(pt)$ we obtain a homomorphism $\mathop{\mathrm{td}}\colon \varOmega^*_U\to K^*(pt)$$\mathop{\mathrm{td}}\colon \varOmega^*_U\to K^*(pt)$. It gives rise to a multiplicative transformation $\,U^*\to K^*$$\,U^*\to K^*$ from complex cobordism to complex K-theory introduced by Conner and Floyd [Conner&Floyd1966]. In this paper Conner and Floyd proved that complex cobordism determines complex K-theory by means of the isomorphism $K^*(X)\cong U^*(X)\otimes_{\varOmega_U}\mathbb Z[\beta]$$K^*(X)\cong U^*(X)\otimes_{\varOmega_U}\mathbb Z[\beta]$, where the $\varOmega_U$$\varOmega_U$-module structure on $\mathbb Z[\beta]$$\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$$\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$$y\in\mathbb R$ is a parameter. Setting $y=-1$$y=-1$, $y=0$$y=0$ and $y=1$$y=1$ we get the top Chern number $c_n[M]$$c_n[M]$, the Todd genus $\mathop{\mathrm{td}}[M]$$\mathop{\mathrm{td}}[M]$ and the $L$$L$-genus $L[M]=\mathop{\mathrm{sign}}(M)$$L[M]=\mathop{\mathrm{sign}}(M)$ respectively.

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