What are the higher homotopy groups of Spec Z ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:14:39Z http://mathoverflow.net/feeds/question/3103 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3103/what-are-the-higher-homotopy-groups-of-spec-z What are the higher homotopy groups of Spec Z ? Jonathan Wise 2009-10-28T18:10:43Z 2009-11-15T19:28:14Z <p>The homotopy groups of the &eacute;tale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly &pi;<sub>1</sub> is trivial because Spec Z has no unramified &eacute;tale covers, but what is known about the higher homotopy groups?</p> http://mathoverflow.net/questions/3103/what-are-the-higher-homotopy-groups-of-spec-z/3970#3970 Answer by David Treumann for What are the higher homotopy groups of Spec Z ? David Treumann 2009-11-03T17:49:55Z 2009-11-04T04:11:55Z <p>If etale pi_1 classifies obstructions to trivializing finite flat unramified Z-algebras, it would be nice if the whole etale homotopy type classified obstructions to trivializing simplicial commutative Z-algebras that were finite, flat, and unramified in a homotopy sense. I think all of these notions make sense: "finite" means that the homotopy groups vanish in high degrees and are finitely generated, "flat" means that these homotopy groups have no torsion, and "unramified" means that the cotangent complex is zero. Is that right?</p> <p>Presumably algebraic topologists have thought about the sphere spectrum version of this question. Are there any connective E-infinity ring spectra that are finite, flat, and unramified over the sphere?</p> <p>After Tyler's comments, I see that this is a bad analogy. The dictionary between etale locally constant sheaves of sets and finite flat unramified algebras (which in one direction takes an algebra and associates the sheaf of sections of its spectrum over Spec Z) just doesn't extend to a dictionary between homotopy-style locally constant sheaves and homotopy-style finite flat unramified algebras.</p>