It is described in many sources that algebraic topology had been a major source of innovation for algebraic geometry. It is said that the uses of cohomology, sheaves, spectral sequences etc. in algebraic geometry were motivated by algebraic topology. Moreover it is said that Weil conjectures arose out of inspiration from algebraic topology.


So it seems a very clear thing that algebraic topology tremendously influenced algebraic geometry, at least historically.


But are there influences in the other way? Did it ever happen that the modern developments in algebraic geometry were ever taken back to algebraic topology and led to developments over there?


Edit: Topological K-theory is one application. Are there more?



Are there applications of algebraic geometry into algebraic topology?
elliptic cohomology, topological modular forms, stacks, formal groups, genera, to name but a few.


