@@ -132,7 +132,7 @@ This can be used to define a genus:

One obviously has to check, that this is in fact a genus, see \cite[14]{hirzebruch_modularforms}.

One can now define $f(x)\coloneqq x/Q(x)$ and obtains an odd power series with coefficients in $R$.

One can now define $f(x)\coloneqq x/Q(x)$ and by doing so obtain an odd power series with coefficients in $R$.

Let $g$ be the (formal) inverse of $f$, called the \Index{logarithm of the genus $\varphi_Q$}.

By resorting to the total Chern and Pontryagin classes of $\mathbb{C}\mathbb{P}^n$ in \cref{thm:axioms_chern,prop:axioms-pontryagin} and a tricky computation, one can show, that

\[

...

...

@@ -149,6 +149,16 @@ We therefore have

\[

\varphi(M)=\varphi(\Tang M)[M]

\]

\begin{example}

Letting $Q(x)=x/\tanh(x)$ yields an even power series (look at the Taylor expansion).

Following our earlier definitions, we get $f(x)=\tanh(x)$, $f'(x)=1-f(x)^2$ and hence

\[

g'(y)=\frac{1}{1-y^2}=1+ y^2+y^4+\ldots

\]

The genus obtained by this power series is called \Index[L-genus@$L$-genus]{$L$-genus}.

It was proven by \textcite{hirzebruch_Lgenus}, that this genus coincides with the signature.