Cohomology Spectral Sequence Research Articles

Cohomology of metacyclic groups

Let ${\mathbf {e}}:1 \to N \to G \to K \to 1$ be an extension of a finite cyclic group $N$ by a finite cyclic group $K$. Using homological perturbation theory, we introduce the beginning of a free resolution of the integers ${\mathbf {Z}}$ over the group ring ${\mathbf {Z}}G$ of $G$ in such a way that the resolution reflects the structure of $G$ as an extension of $N$ by $K$, and we use this resolution to compute the additive structure of the integral cohomology of $G$ in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension ${\mathbf {e}}$. The calculation is then completed by comparing the two bounds.

Change of rings and characteristic classes

AbstractWe construct characteristic classes for the change of rings spectral sequences. These have their values in appropriate Ext groups and provide descriptions of the first non-zero differentials in the spectral sequences. We use these classes to do some calculations in the cohomology spectral sequence of an extension of a finite cyclic group by a finite cyclic group.

Higher Divided Squares in Second-Quadrant Spectral Sequences

The geometric action of the Steenrod algebra on many mod 2 cohomology spectral sequences is complemented by the action of a completely different algebra.

Higher divided squares in second-quadrant spectral sequences

The geometric action of the Steenrod algebra on many mod 2 cohomology spectral sequences is complemented by the action of a completely different algebra.

�ber Konvergenz von Spektralfolgen der stabilen Homotopietheorie

We discuss the problem of convergence of spectral sequences that arise from a filtration of a spectrum in Boardman's stable homotopy category by applying a generalized homology, “homotopy” or cohomology theory. The criteria we get give e.g. the convergence of the Adams spectral sequence for a generalized homology theory in certain cases (using similar methods this equestion has been considered independently by J. F. Adams in his forthcoming Chicago lecture notes), and some results on the Adams cohomology spectral sequence including the well-known convergence properties in case of singular cohomology with Zp-coefficients and complex cobordism.