Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:56:03Z http://mathoverflow.net/feeds/question/107175 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107175/homotopy-groups-and-homology-groups-for-the-h-mathbb-z-module-dg-module-corresp Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence. Justin Young 2012-09-14T13:03:18Z 2012-09-14T17:34:14Z <p>In Shipley's paper <a href="http://arxiv.org/abs/math/0209215" rel="nofollow">http://arxiv.org/abs/math/0209215</a> she proves a Quillen equivalence between the category of $H\mathbb Z$-modules and dg $\mathbb Z$-modules. So, to a chain complex $C$, she assigns a spectrum $HC$, but is it true in some sense that $\pi_i HC \cong H_i C$? I am aware there are some subtleties involved with homotopy groups of symmetric spectra, but I'm not sure how that plays out in this case.</p> http://mathoverflow.net/questions/107175/homotopy-groups-and-homology-groups-for-the-h-mathbb-z-module-dg-module-corresp/107197#107197 Answer by Tyler Lawson for Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence. Tyler Lawson 2012-09-14T17:34:14Z 2012-09-14T17:34:14Z <p>Shipley's result (of which an alternative version can be found in EKMM, IV.2) proves that there is a string of Quillen equivalences between the derived category of chain complexes over $\mathbb{Z}$ and the category of $H\mathbb{Z}$-modules. This, in particular, gives rise to a chain of natural equivalences between their homotopy categories. We can then prove the result you're asking for by defining both homotopy and homology in the same terms.</p> <p>Both homotopy categories have a suspension/shift functor, which is an autoequivalence. In both cases, this can be described as a homotopy pushout of the diagram $\ast \leftarrow X \rightarrow \ast$, where $\ast$ is the terminal object. Quillen equivalences preserves homotopy pushouts and so preserves shift (and hence, up to natural isomorphism, its inverse).</p> <p>In the category of chain complexes, there is an object $\mathbb{Z}$, concentrated in degree zero, and this is explicitly carried to $H\mathbb{Z}$ under the chain of Quillen equivalences.</p> <p>Finally, we can define both homotopy groups and homology groups using the derived category: <code>$$H_i C = [\mathbb{Z}[n], C]$$</code> <code>$$\pi_i HC = [\Sigma^n H\mathbb{Z}, HC]$$</code> Since all of the ingredients in this definition (shifting degree, the representing object, maps in the homotopy category) are preserved under Quillen equivalence, the two can be made naturally isomorphic.</p> <p>(Of course, this really manifests that one of Shipley's string of equivalences is Dold-Kan equivalence between simplicial abelian groups and nonnegatively graded chain complexes, which takes homology groups to homotopy groups.)</p>