Elliptic Genera and Hirzebruch's Signature theorem

An important object in algebraic topology is the oriented cobordism ring, a ring where elements are oriented cobordism classes of manifolds. The multiplication is given by the Cartesian product of manifolds, and the addition is given by the disjoint union of manifolds. The ring is graded by the dimensions of manifolds and is denoted by  
\[
\Omega _{*}^{SO}=\bigoplus _{n=0}^{\infty }\Omega _{n}^{SO}
\]
where \( \Omega _{n}^{SO} \) consists of oriented cobordism classes of manifolds of dimension \( n \).  

One can also define an unoriented cobordism ring, denoted by  
\[
\Omega _{*}^{O}.
\]
One can also restrict attention to complex/almost complex manifolds, in which case the one obtains the \emph{complex cobordism ring}, oriented or unoriented, which are denoted by replacing the $O$ in the real counterparts by $U$.

In any case, these rings are complicated, so to study them we consider homomorphisms to other algebras. Let $\Lambda$ be a $Q-$algebra. A genus is a homomorphism $\phi: \Omega _{*}^{SO} \to \Lambda$. Note that any such map factors through the rationalisation $\Omega _{*}^{SO} \otimes \mathbb{Q}$. It is a beautiful result that the rationalisation is a polynomial algebra in infinitely many generators, one for each $\mathbb{CP}^{2i}$

Power Series

Given an \emph{even} power series
\[
f(x) = 1 + a_1 x^2 + a_2 x^4 + \dots
\]
one can construct a genus in the following manner.

Given \( f \), one can define a stable characteristic class of real vector bundles. Let \( E \) be a real vector bundle, and write formally
\[
p(E) = \prod (1 + y_i^2),
\]
where the \( \{y_i\} \) are the Chern roots of \( E \otimes_{\mathbb{R}} \mathbb{C} \).

Define a characteristic class \( f(E) \) by
\[
f(E) = \prod f(y_i).
\]
Since \( f \) is an even power series, the function \( x \mapsto f(x) \) is symmetric in the \( y_i \) and is invariant under \( y_i \mapsto -y_i \). Thus, it defines a stable, multiplicative characteristic class of real vector bundles.


Definition Given the stable characteristic class \( E \mapsto f(E) \) as before, define a genus on manifolds by setting
\[
M \mapsto \varphi(M) = \varphi(TM)[M] = \int_M \varphi(TM).
\]


The hardest part about checking that this is a genus is showing it vanishes on manifolds $M$ with are a boundary $\partial N$. The appropriate collar neighbourhood theorem shows that $TM \bigoplus \R=TN|_M$, and we conclude using Stokes' theorem.

In fact, this construction is reversible. Given a genus $\varphi$, we want to find a power series $f(x)$ so that the associated genus $\varphi_f$ agrees with $\varphi$ on complex projective spaces.

$T_{\mathbb{CP}^{2n}} \oplus \mathbb{C}=\omega^{2n+1}$, the $2n+1$-fold tensor power of the tautological line bundle (or its inverse, depending on your convention).
Let $t$ be the hyperplane class generating second cohomology of $\mathbb{CP}^{2n}$. The Pontryagin class of the tautological line bundle is $1+t^2$, so $f(T_{\mathbb{CP}^{2n}})=f(t)^{2n+1}$. The upshot of following through the calculation is
that $\varphi_f(\mathbb{CP}^{2n})$ is the residue at $x=0$ of $(f(x)/x)^{2n+1}$.
The tool for this is called Lagrange inversion, and it tells us to set $f(x)=\frac{x}{g^{-1}(x)}$

Thus we have shown that

Proposition    Genera are in (naturally-defined) bijection with even power series

Hirzebruch's signature theorem

The earliest studied and most influential genus is the signature, defined for $4k$-dimensional manifolds as the signature of the bilinear form on cohomology in dimension $2k$ and 0 for manifolds of dimension not divisible by 4. One checks that the signature of $\mathbb{CP}^{2n}$ is always $1$, and this genus corresponds to the function/power series $f(x)=\frac{x}{\tanh{x}}$. This is the Hirzebruch signature theorem. We now show how to extract congruences on Pontryagin numbers using it.

Consider a sequence of polynomials
\[
K_1(X_1), \quad K_2(X_1X_2), \quad K_3(X_1X_2X_3), \dots
\]
with coefficients in \( A \), satisfying the homogeneity property:  
Each \( K_n(X_1X_2^2 X_3^3 \dots X_n^n) \) is homogeneous of degree \( n \).

Given an element \( a = 1 + a_1 + a_2 + \dots \in A \) with leading term 1,  
which is invertible in \( A \), define a new element \( K(a) \in A \), also with  
leading term 1, by the formula:
\[
K(a) = 1 + K_1(a_1) + K_2(a_1 a_2) + \dots
\]

The sequence \( \{ K_n \}_{n \geq 1} \) is a \emph{multiplicative sequence}, or briefly an  
\emph{m-sequence} of polynomials, if it satisfies the multiplicative property:
\[
K(ab) = K(a) K(b)
\]
for all \( A \)-algebras \( A \) and for all \( a, b \in A \) with leading term 1.

Lemma Given a formal power series:
\[
f (t) = 1 + \lambda_1 t + \lambda_2 t^2 + \dots
\]
with coefficients in \( A \), there is a unique m-sequence \( \{K_n\}_{n \geq 1} \)  
with coefficients in \( A \) satisfying the condition:
\[
K(1+t) = f(t).
\]
   
This condition is equivalent to requiring that the coefficient of \( X_n \) in each  
polynomial \( K_n(X_1, \dots, X_n) \) matches that given by the expansion of \( f(t) \).

Uniqueness: For any positive integer \( n \), set \( A = A[t_1, \dots, t_n] \). Let:
\[
\lambda = (1+t_1)(1+t_2) \cdots (1+t_n) = 1 + \lambda_1 + \lambda_2 + \dots + \lambda_n,
\]
where the polynomials \( \lambda_i \in A \) are elementary symmetric polynomials in  
\( t_1, \dots, t_n \). Then:
\[
K(\lambda) = K(1+t_1) K(1+t_2) \dots K(1+t_n) = f(t_1) f(t_2) \dots f(t_n)
\]
\[
= (1 + \lambda_1 t_1 + \lambda_2 t_1^2 + \dots) (1 + \lambda_1 t_n + \lambda_2 t_n^2 + \dots).
\]
Taking the homogeneous part of degree \( n \), it follows that \( K_n(\lambda_1, \dots, \lambda_n) \)  
is completely determined by the formal power series \( f (t) \).  
Furthermore, since the elementary symmetric polynomials are algebraically independent,  
each \( K_n \) is uniquely determined.
\\ 

Existence: We construct this fairly explicitly.
For any partition \( I = (i_1, \dots, i_r) \) of \( n \) with positive integers, let
\[
|I| = i_1 + \dots + i_r.
\]
Define the polynomial \( K_n \) by the formula:
\[
K_n(\lambda_1, \dots, \lambda_n) = \sum_{I} s_I(\lambda_1, \dots, \lambda_n),
\]
summing over all partitions \( I \) of \( n \), where \( s_I(\lambda_1, \dots, \lambda_n) \)  
is a homogeneous symmetric polynomial of degree \( n \) uniquely determined by  
the elementary symmetric polynomials.

By convention, we have:
\[
s_I(a) = s_I(1 + a_1 + a_2 + \dots) = s_I(a_1, \dots, a_n),
\]
for any partition \( I \) of \( n \). We also have the identity:
\[
s_I(ab) = \sum_{H, J=I} s_H(a) s_J(b),
\]
summing over all partitions \( H, J \) with juxtaposition \( HJ = I \).  
Thus, we obtain:
\[
K(ab) = \sum_I s_I(ab) = \sum_I \sum_{H,J=I} s_H(a) s_J(b)
\]
\[
= \sum_{H,J} s_H(a) s_J(b) = K(a) K(b),
\]
which holds for all \( a, b \in A \).

If \( I \) is not a trivial partition of \( n \), i.e., \( I \neq (n) \), then \( s_I(\lambda_1, \dots, \lambda_n) = 0 \).  
Since \( s_n(\lambda_1, \dots, \lambda_n) = \lambda_n \), we derive:
\[
K(1+t) = \sum_{I} s_I(t, 0, \dots, 0) = \sum_{I} \lambda_n t^n = f(t).
\]
For a partition \( I \) of 0, we have:
\[
\sum_{I} s_I(t, 0, \dots, 0) = 1.
\]
Thus, the construction is complete. $\blacksquare$

One then computes the multiplicative sequence, evaluates them on Pontryagin classes to define a cohomology class in top dimension, and pairs this with the fundamental class to get the signature. Since the signature must be an integer, but the polynomials have denominators (e.g. $L_1=\frac{x}{3}$), one obtains congruences between Pontryagin numbers.