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I have recently been reading about multiplicative sequences and genera from a couple of different sources, most notably "Spin Geometry" by Lawson and Michelsohn and "Characteristic Classes" by Milnor and Stasheff. The exposition in both of these are very good. There is, however, one point that seems to elude me, namely why the spesific functions used to define the various multiplicative sequences and genera are natural to consider.

For instance, take the Todd sequence associated to $$\textrm{td}(x)=\frac{x}{1-e^{-x}}$$ or the Hirzebruch L sequence associated to $$\ell(x) = \frac{\sqrt{x}}{\tanh \sqrt{x}}.$$

If the answer is something like "They crop up naturally when you calculate the topological index of (pseudo-)differential operators, and so they are useful in combination with the Atiyah-Singer index theorem.", that would of course be fine, but I would like to see a more "a priori"/intuitive/heuristic? reason why exactly these functions gives more useful invariants than other.

EDIT: In this question at MO, a part of my question is answered by OP, namely that the Todd sequence is unique in satisfying a certain condition coming from wanting the Todd class of complex projective spaces to be especially simple.

I guess this is the kind of answers I was looking for, but I realise my original intent might have been vague, at best.

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  • $\begingroup$ I suggest you have a look at the classic Habilitationsschrift of F. Hirzebruch, Topological methods in algebraic geometry, which starts with an abstract discussion of multiplicative sequences and shows how they arose in the study of cohomology operations (characteristic classes). This should at least give some indications as to how and why these sequences come up. $\endgroup$ – t.b. Apr 15 '11 at 17:05
  • $\begingroup$ @Theo - Thanks for the reference, I will have a look at that. $\endgroup$ – Raeder Apr 17 '11 at 19:19

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