Lifting gaps in the homology of spectra to skeleta and converse stable Hurewicz theorems

Hybrid talk. For zoom details contact Christian Kaiser (kaiser@mpim-bonn.mpg.de)
 

The stable Hurewicz theorem says that the singular homology of a connective spectrum E is concentrated in non-negative degrees. The converse is true if E is bounded below; yet it fails for general spectra since singular homology does not change if one extends E by an acyclic spectrum S (whose homology is zero; S may be non-trivial). I proved that examples of this type are the only possible ones. This statement is a consequence of the following one:  the singular homology H_1(E) vanishes and H_2(E) is a free abelian group if only if there exists a choice of a -1-skeleton $S=E^{(-1)}$ of E that is simultaneously its -2-skeleton. Moreover, in this case this S is canonical and its singular homology is the negative part of the homology of E. Another corollary: a choice of cellular tower for E (as defined by H.R. Margolis) is the same thing as a Postnikov tower for E with respect to the spherical weight structure on SH (defined by B.). To prove this, one studies weight structures (defined by B. and D. Pauksztello) and a morphism E\to E' that kills some weights, that is, "sends an m-skeleton of E into an n-skeleton of E'" for some n>m. My theory extends some of the results of the connective stable homotopy theory to unbounded objects; this includes the equivariant setting (that is, categories of the type SH(G)).

The talk should be accessible to anybody who heard just a little about triangulated categories and the stable homotopy category SH.