Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

2012-09-14T13:03:18Z

In Shipley's paper http://arxiv.org/abs/math/0209215 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.

Answer by Tyler Lawson for Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

2012-09-14T17:34:14Z

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.

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).

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.

Finally, we can define both homotopy groups and homology groups using the derived category: $$ H_i C = [\mathbb{Z}[n], C] $$ $$ \pi_i HC = [\Sigma^n H\mathbb{Z}, HC] $$ 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.

(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.)

Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence. - MathOverflow

Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

Answer by Tyler Lawson for Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.