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I take that you ask how in the world did Hirzebruch come up with the complicated expression of the L-genus?

The key fact behind this is that the signature is a genus, i.e. a ring morphism $\gamma:\Omega^\bullet_+\to\mathbb{R}$ from the oriented cobordism ring to the (ring of) reals. By Thom's work we deduce that a genus is determined by its values on $\mathbb{CP}^{2n}$ and thus by the generating series

$$r^\gamma(t)=1+\sum_{n\geq 1}\gamma(\mathbb{CP}^{2n})t^{2n}.$$

In the case of signature we have

$$r^\gamma(t)=\frac{1}{1-t^2}.$$

How does one go from this series to the the function $\frac{\xi}{\tanh \xi}$ that enters into the defintion of $L$? As a teaser, let me point out that

$$\xi =\log \left( \frac{1}{1-t^2}\right) \Leftrightarrow t=\tanh \xi.$$

The signature theorem follows from the above observations using a bit of algebraic combinatorics. For details see Chap. 7 of these lecture notes for a graduate course on this topic that I taught in 2008.

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I take that you ask how in the world did Hirzebruch come with the complicated expression of the L-genus?

The key fact behind this is that the signature is a genus, i.e. a ring morphism $\gamma:\Omega^\bullet_+\to\mathbb{R}$ from the oriented cobordism ring to the (ring of) reals. By Thom's work we deduce that a genus is determined by its values on $\mathbb{CP}^{2n}$ and thus by the generating series

$$r^\gamma(t)=1+\sum_{n\geq 1}\gamma(\mathbb{CP}^{2n})t^{2n}.$$

In the case of signature we have

$$r^\gamma(t)=\frac{1}{1-t^2}.$$

How does one go from this series to the the function $\frac{\xi}{\tanh \xi}$ that enters into the defintion of $L$? As a teaser, let me point out that

$$\xi =\log \left( \frac{1}{1-t^2}\right) \Leftrightarrow t=\tanh \xi.$$

The signature theorem follows from the above observations using a bit of algebraic combinatorics. For details see SectChap. 7.2-7.3 7 of these lecture notes for a graduate course on this topic that I taught in 2008.

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I take that you ask how in the world did Hirzebruch come with the complicated expression of the L-genus?

The key fact behind this is that the signature is a genus, i.e. a ring morphism $\gamma:\Omega^\bullet_+\to\mathbb{R}$ from the oriented cobordism ring to the (ring of) reals. By Thom's work we deduce that a genus is determined by its values on $\mathbb{CP}^{2n}$ and thus by the generating series

$$r^\gamma(t)=1+\sum_{n\geq 1}\gamma(\mathbb{CP}^{2n})t^{2n}$$.1}\gamma(\mathbb{CP}^{2n})t^{2n}.$$In the case of signature we have$$r^\gamma(t)=\frac{1}{1-t^2}.$$How does one go from this series to the the function \frac{\xi}{\tanh \xi} that enters into the defintion of L? As a teaser, let me point out that$$ \xi =\log \left( \frac{1}{1-t^2}\right) \Leftrightarrow t=\tanh \xi.

The signature theorem follows from the above observations using a bit of algebraic combinatorics. For details see Sect. 7.2-7.3 of these lecture notes for a graduate course on this topic that I taught in 2008.

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