Let {\Lambda} be a {\mathbb{Q}}-algebra. A genus is a homomorphism

\displaystyle \phi: \Omega_{SO} \rightarrow \Lambda,

where {\Omega_{SO} } is the oriented cobordism ring. In other words, a genus {\phi} assigns to every compact, oriented manifold {M} an element {\phi(M) \in \Lambda}. This satisfies the conditions:

  1. {\phi(M \sqcup M') = \phi(M) + \phi(M')}.
  2. {\phi(M \times M') = \phi(M) \phi(M')}.
  3. {\phi(\partial N) =0 } for any manifold-with-boundary {N}.

A fundamental example of a genus is the signature {\sigma}, which assigns to every manifold {M} of dimension {4k} the signature of the quadratic form on {H^{2k}(M; \mathbb{R})}. (Also, {\sigma} is zero on manifolds whose dimension is not divisible by four.)

1. Genera and power series

I’d like next to describe the connection between genera and power series. This is very classical, but it’s going to be illuminated by some of the material on formal group laws discussed in the past.

A {\Lambda}-valued genus factors through the rationalization {\Omega_{SO} \otimes \mathbb{Q}}. There is an isomorphism

\displaystyle \Omega_{SO} \otimes \mathbb{Q} \simeq \mathbb{Q}[x_4, x_8, x_{12}, \dots ]

where we can take the even-dimensional complex projective spaces {\mathbb{CP}^{2i}} as polynomial generators. In particular, a genus {\phi} is uniquely determined by specifying { \phi(\mathbb{CP}^{2i}) \in \Lambda}. It follows that we can extract a power series

\displaystyle g(x) = \sum_{i = 0}^\infty \frac{\phi(\mathbb{CP}^{2i})}{2i+1} x^{2i+1}.

Moreover, this power series determines the genus.

Definition 1 The series {g} is called the logarithm of the genus {\phi}.

The word “logarithm” is adequately explained thanks to the discussion of formal groups earlier. Recall that there is a map

\displaystyle \Omega_U \rightarrow \Omega_{SO}

from the complex cobordism ring to the oriented one. After tensoring with {\mathbb{Q}}, we have a surjection

\displaystyle \Omega_U \otimes \mathbb{Q} \rightarrow \Omega_{SO} \otimes \mathbb{Q},

because the former is generated by the complex projective spaces and the latter by the even-dimensional complex projective spaces.

In any event, a genus gives by composition a map

\displaystyle \Omega_U \rightarrow \Lambda,

and since {\Omega_U} is the Lazard ring, this classifies a formal group law over the {\mathbb{Q}}-algebra {\Lambda}. However, since {\Lambda} is a {\mathbb{Q}}-algebra, we know that the formal group law classified has a logarithm. By naturality, the logarithm is just the push-forward of the logarithm in {\Omega_U \otimes \mathbb{Q}}; we’ve seen that said logarithm is

\displaystyle \sum_{i = 0}^\infty \frac{[\mathbb{CP}^i]}{i+1} x^i.

The push-forward of this to {\Lambda} is precisely what we called the logarithm above, since the odd-dimensional projective spaces are zero in {\Omega_{SO} \otimes \mathbb{Q}} (in fact, even in {\Omega_{SO}}).

The formal group law on {\Lambda} that the genus determines is given by

\displaystyle F(x, y) = g( g^{-1}(x) + g^{-1}(y)).

2. Power series again

Nonetheless, this is not the usual way in which one associates a genus to a power series. Given an 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

\displaystyle p(E) = \prod (1 + y_i^2),

where the {\left\{y_i\right\}} are the “Chern roots” of {E \otimes_{\mathbb{R}} \mathbb{C}}. Then, one defines

\displaystyle f(E) = \prod f(y_i).

This is symmetric in the {y_i} and invariant under the transformation {y_i \mapsto -y_i}, so it defines an honest stable, multiplicative characteristic class of real vector bundles.

Definition 2 Given the stable characteristic class {E \mapsto f(E)} as before, define a genus on manifolds by sending

\displaystyle M \mapsto \phi(M) = \phi(TM)[M] = \int_M \phi(TM).

In other words, one applies the stable characteristic class to the tangent bundle {TM}, and “integrates” over {M} (i.e., pairs with the fundamental class). One has to check that this is an honest genus; the key observation is that {\phi(M)} vanishes for a boundary {M = \partial N} as {TM \oplus \mathbf{1} = TN}. Thus,

\displaystyle \int_M \phi(TM) = \int_{\partial N} \phi(TN)|_M = 0

by Stokes’s theorem.

Proposition 3 Every genus arises from a unique even power series in this way.

In fact, given a genus {\phi}, we can explicitly compute the associated even power series (and in fact, express it in terms of the {g} earlier). We want to find an power series {f(x)} such that the associated genus, which I’ll write {\phi^f}, is equal to {\phi}: that is, we want

\displaystyle \phi^f(\mathbb{CP}^{2n}) = \phi(\mathbb{CP}^{2n})

for each {n}. Now, the tangent bundle of {\mathbb{CP}^{2n}} satisfies

\displaystyle T_{\mathbb{CP}^{2n}} \oplus \mathbf{1}= \mathcal{O}(1)^{2n+1},

where {p(\mathcal{O}(1)) = 1 + t^2} for {t \in H^2(\mathbb{CP}^{2n})} the hyperplane class. It follows that we can compute the stable characteristic class:

\displaystyle f(T_{\mathbb{CP}^{2n}}) = f(t)^{2n+1}.

In particular, we find that {\int_{\mathbb{CP}^{2n}} f(T_{\mathbb{CP}^{2n}})} is the coefficient of {t^{2n}} in {f(t)^{2n+1}}: that is, for any even power series {f}, we have

\displaystyle \phi^f(\mathbb{CP}^{2n}) = \mathrm{Res}_{x = 0} \left( \frac{f(x)}{x}\right)^{2n+1}.

So, if we want {\phi = \phi^f}, we need to find a power series {f(x)} such that

\displaystyle \frac{1}{2n+1}\mathrm{Res}_{x =0 } \left( \frac{f(x)}{x} \right)^{2n+1} = \frac{1}{2n+1}\phi(\mathbb{CP}^{2n}).

But this is just Lagrange inversion again: it tells us that we should have

\displaystyle g(x) = \left( \frac{x}{f(x)} \right)^{-1}, \quad \text{or} \frac{x}{f(x)} = g^{-1}(x) ,\ \ \ \ \ (1)

since {g(x) = \sum_{i = 0}^{\infty} \frac{\phi(\mathbb{CP}^{2i})}{2i+1} t^{2i+1}}. This is the basic relation between the two power series {f} and {g}, and it shows that any genus comes from an even power series in this way.

Example: The signature {\sigma} is a {\mathbb{Q}}-valued genus, which assigns to every {\mathbb{CP}^{2n}} the number one. What is the corresponding power series {f}? First, the logarithm is given by the series

\displaystyle \sum_{i=0}^\infty \frac{1}{2i+1}x^{2i+1} = \tanh^{-1}(x).

Consequently, we get

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

In particular, we find that the signature of a manifold is computed by the genus associated to the power series {\frac{x}{\tanh x}}; this is the Hirzebruch signature theorem.

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