I am reading about complex-orientable cohomology theories for the first time, elliptic cohomology theories in particular.
An inconsistency(?) in the literature is the explicit reference to genera. Some references define and use the term, see for instance:
- Graeme Segal's survey, "Elliptic Cohomology."
- Charles Rezk's lecture notes for elliptic cohomology.
- Novikov's paper introducing the ANSS apparently uses genera (although I have trouble reading this paper.)
Yet several make no mention of it, for instance:
- Mike Hopkin's COCTALOS lecture notes.
- Haynes Miller's (Co)bordism lecture notes.
- Doug Ravenel's Complex cobordism and stable homotopy groups of spheres.
My understanding of even the topics foundational to this subject area is sparse (working on it!), which is making it hard to see why we ought to pick out and give a special name to these homomorphisms. Which brings me to my question:
What is the role of genera in the development of complex-oriented cohomology?
And why the hot-cold nature of genera's presence in literature? My impression is that genera are somehow related to the geometric/physical story of elliptic cohomology, and that perhaps(?) homotopy theorists just don't need all that. But that's speculation.
I'd especially appreciate an answer that provides historical context.