TITLE: Are there applications of algebraic geometry into algebraic topology?
QUESTION [13 upvotes]: 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?

REPLY [6 votes]: A nice conceptual link between derived algebraic geometry and algebraic topology is established in
Bertrand Toen, Champs affines ,
http://arxiv.org/abs/math/0012219
An impression of the central conceptual picture developed there is given at
nLab: rational homotopy theory in an (oo,1)-topos.
The main point is, in slight paraphrase of what Toen writes, that he shows that the relation between dg-algebras and rational homotopy theory factors through derived algebraic stacks. This allows to closely connect algebraic topology with derived algebraic geometry and improve on ordinary rational homotopy theory by admitting algebraic models for spaces with arbitrary fundamental groups.
This is supposed to go back to and provide a solution to Grothendiecks "schematization problem" in terms of "schematic homotopy types".