An exposition of the topological half of the Grothendieck-Hirzebruch-Riemann-Roch theorem in the fancy language of spectra

TL;DR

This paper provides an informal, spectrum-based exposition of pushforwards and orientations in generalized cohomology theories, clarifying the natural role of Todd classes in Grothendieck-Hirzebruch-Riemann-Roch formulas.

Contribution

It offers a conceptual, spectrum-oriented perspective on the topological aspects of the Grothendieck-Hirzebruch-Riemann-Roch theorem, emphasizing the naturality of Todd classes.

Findings

Todd classes are naturally integrated into the formulas.

Spectra provide a natural language for pushforwards and orientations.

The exposition clarifies conceptual foundations of the theorem.

Abstract

We give an informal exposition of pushforwards and orientations in generalized cohomology theories in the language of spectra. The whole note can be seen as an attempt at convincing the reader that Todd classes in Grothendieck-Hirzebruch-Riemann-Roch type formulas are not Devil's appearances but rather that things just go in the most natural possible way.