We construct an extension (Stable Field Theory) of Cohomological Field Theory. The Stable Field Theory is a system of homomorphisms to some algebras generated by spheres and disks with punctures. It is described by a formal tensor series, satisfying to some system of ”differential equations”. In points of convergence the tensor series generate special noncommutative analogs of Frobenius algebra...

Let A L $A_L$ be the right-angled Artin group associated with a finite flag complex $L$ . We show that amenable category of equals virtual cohomological dimension Coxeter W $W_L$ In particular, groups satisfy question Capovilla–Löh–Moraschini proposing an inequality between and Farber's topological complexity.

Let R be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules (M, N) over R we study applications of rigidity of Tor(M, N), based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connecti...

We establish a Grothendieck–Lefschetz theorem for smooth ample subvarieties of projective varieties over an algebraically closed field characteristic zero and, more generally, whose complement has small cohomological dimension. A weaker statement is also proved in general context and all characteristics. Several applications are included.

Methods are developed to relate the action of a principal fibration relative Whitehead products in order determine homotopy type certain spaces. The methods applied thoroughly analyze based loops on cell attachments. Key examples (n-1)-connected Poincare Duality complexes dimension 2n or 2n+1 with minor cohomological conditions.

Let X = Γ\G/K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has Q-rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D0. We prove that D0 is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ-con...

The purpose of this note is to introduce a trick which relates the (non)-vanishing top-dimensional $\ell^2$-Betti numbers actions with that sub-actions. We provide three different types applications: we prove Aut($F_n$) and Out($F_n$) (and their Torelli subgroups) do not vanish in degree equal virtual cohomological dimension, subgroups 3-manifold groups have vanishing 3 2 for instance $F_2^d \t...

Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(F̄ /F ) and Gal(k̄/k) have the same 2-cohomological dimension and this dimension is finite. Then Hilbert’s Tenth Problem has a negative answer for any function field...