We present a recent improvement of the homotopy Conley index, the so-called Conley index over a base. We apply it to endow the homology and cohomology Conley indices with the structure of module over the cohomology ring of the phase space, and indicate how that structure provides information on continuation of cohomology classes of isolated invariant sets.

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-going generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a reach combinatorial structure, e.g., it is a manifold with corners provided that the action is locally st...

We use bordism theory to construct periodic cohomology theories, which we call elliptic cohomology, for which the cohomology of a point is a ring of modular functions. These are complex-oriented multiplicative cohomology theories, with formal groups associated to the universal elliptic genus studied by a number of authors ([CC, LS, O, W1, Z]). We are unable to find a geometric description for t...

We describe the cohomology ring of the symplectic reductions by tori of coadjoint orbits, or weight varieties. Weight varieties arise from representation theory considerations, and are temed as such because they are the symplectic analogue of the T-isotypic components (weight spaces) of irreducible representations of G. Recently, Tolman and Weitsman expressed the (ordinary) cohomology ring for ...

Every homology or cohomology theory on a category ofE∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads. Mathematics Subject Classification (2000): 55P43, 55P48, 55U35

Let G be a virtually cyclic of the form (Za ⋊ Zb) ⋊ Z or [Za ⋊ (Zb × Q2i)]⋊Z. We compute the integral cohomology ring of G, and then obtain the periodicity of the Farell cohomology of these groups.

Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local ...

This paper primarily concerns Galois cohomology groups associated to Galois representations over a complete local ring R. The underlying Galois module and the corresponding cohomology groups which we consider are discrete R-modules. Under certain hypotheses, we prove that the first cohomology group is an almost divisible Rmodule. We also consider the subgroup of locally trivial elements in the ...

The goal of this work is to establish a new algorithm for computing the cohomology ring of cubical complexes. The cubical structure enables an explicit recurrence formula for the cup product. We derive this formula and, next, show how to extend the Mrozek and Batko [7] homology coreduction algorithm to the cohomology ring structure. The implementation of the algorithm is a work in progress. Thi...