I think it might be pretty hard to get an answer that says nothing about cohomology. I interpret/think of Brown Representability as saying that if you want to think about these types of invariants you should really look at spectra. That somehow cohomology is really about spectra. Once you see this you then interpret the axioms for a cohomology theory or rather what you know about singular cohomology and try to think of the structure you have on the representing object that comes from this and you get ring spectra. I dont know if non-topologists really think about these things or if i just think that they should. They bring up interesting questions about operads... that may have come up independently.
You also get cohomology operations for any cohomology theory from this, but only as a theoretical tool, that is i am not completely sure how to use Brown Representability to compute Adams operations, in fact i dont think that we know all of $K^*K$.
This is a pretty biased answer though, but it is a unifying idea of algebraic topology that we should look at ring spectra, or rather the representing objects of all "nice" cohomology theories.
anyone should feel free to appropriately fix this, especially if they have better historical or computational information
No comments:
Post a Comment