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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?

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The short answer is "yes". –  Tyler Lawson May 19 '10 at 21:33
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The use of sheaves came not directly from algebraic topology, but rather from several complex variables. Lefschetz's famous book on algebraic geometry and analysis situs (from 1924) shows that the developments of algebraic geometry and algebraic topology were interwined. Some more recent developments, giving a slightly longer version of Tyler's answer: (a) Etale techniques played a key role in the Sullivan's proof of the Adams conjecture. (b) Moduli of elliptic curves (over the integers) is at the heart of the theory of topological modular forms. –  Emerton May 19 '10 at 21:48
    
@Emerton: I thought Leray invented sheaves during WWII. Am I (or the references on Wikipedia) mistaken? –  Steve Huntsman May 19 '10 at 22:15
    
I don't think you're mistaken, but the introduction of sheaves to algebraic geometry was made by Serre, following their use in SCV by Cartan and others (as part of their interpretation of Oka's fundamental work). –  Emerton May 20 '10 at 0:44
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3 Answers

up vote 13 down vote accepted

elliptic cohomology, topological modular forms, stacks, formal groups, genera, to name but a few.

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The Riemann-Roch (et al.) theorems were generalized by the Atiyah-Singer theorem.

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Atiyah-Singer is in differential topology, not algebraic topology? –  Akela May 19 '10 at 21:36
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The most common examples of AS give algebraic-topological results: en.wikipedia.org/wiki/… –  Steve Huntsman May 19 '10 at 23:17
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isn't it a statement about relating something analytic to K-theory? surely that is AT. –  Sean Tilson May 23 '10 at 9:08
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If you take the original approach...but even if you do the heat equation or SUSY approach the results are still ATish. –  Steve Huntsman May 23 '10 at 10:01
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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".

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