Finite Cohomological Dimension Research Articles

Strongly locally homogeneous generalized continua of finite cohomological dimension

The notion of a Vn-continuum was introduced by Alexandroff [1] as a generalization of the concept of n-manifold. In this note we consider the cohomological analogue of Vn-continuum and prove that any strongly locally homogeneous generalized continuum X with cohomological dimension dimG⁡X=n is a generalized Vn-space with respect to the cohomological dimension dimG. In particular, every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in [12].An analog of the Mazurkiewicz theorem that no subset of covering dimension ≤n−2 cuts any region of the Euclidean n-space is also obtained for strongly locally homogeneous generalized continua X of cohomological dimension dimG⁡X=n.

Amenable groups of finite cohomological dimension and the zero divisor conjecture

We prove that every amenable group of cohomological dimension two whose integral group ring is a domain is solvable and investigate certain homological finiteness properties of groups that satisfy the analytic zero divisor conjecture and act on an acyclic CW-complex with amenable stabilisers.

Generalized Euler classes, differential forms and commutative DGAs

In the context of commutative differential graded algebras over [Formula: see text], we show that an iteration of “odd spherical fibration” creates a “total space” commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its “fibrations” with “fiber” of finite cohomological dimension, the induced map on cohomology is injective.

Dynamic characterizations of quasi-isometry and applications to cohomology

We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property $H_{FD}$. As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type $FP_n$ (over arbitrary rings) is a coarse invariant, and that being a (Poincar\'e) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every self coarse embedding of a Poincar\'e duality group over an arbitrary ring must be a coarse equivalence.

The Thom–Sebastiani theorem for the Euler characteristic of cyclic L∞-algebras

Let L be a cyclic L∞-algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L∞-algebra structure on the cohomology H⁎(L). The inner product plus the higher products of the cyclic L∞-algebra defines a superpotential function f on H1(L). We associate with an analytic Milnor fiber for the formal function f and define the Euler characteristic of L is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber.In this paper we prove a Thom–Sebastiani type formula for the Euler characteristic of cyclic L∞-algebras. As applications we prove the Joyce–Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi–Yau threefolds. A motivic Thom–Sebastiani type formula and a conjectural motivic Joyce–Song formula for the motivic Milnor fiber of cyclic L∞-algebras are also discussed.

The Infinitesimal Moduli Space of Heterotic G2 Systems

Heterotic string compactifications on integrable G2 structure manifolds Y with instanton bundles {(V,A), (TY,tilde{theta})} yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative {mathcal{D}} and show that it is equivalent to a heterotic G2 system encoding the geometry of the heterotic string compactifications. This operator {mathcal{D}} acts on a bundle {mathcal{Q}=T^*Y oplus {rm End}(V) oplus {rm End}(TY)} and satisfies a nilpotency condition {check{{mathcal{D}}}^2=0} , for an appropriate projection of {mathcal D}. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group {check H^1_{check{{mathcal{D}}}}(mathcal{Q})}. We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the {alpha'} expansion.

On homology of complements of compact sets in Hilbert cube

We introduce the notion of cohomologically weakly infinite-dimensional spaces and show the acyclicity of the complement Q∖X in the Hilbert cube Q of a cohomologically weakly infinite-dimensional compactum X. As a corollary we obtain the acyclicity of the complement results when(a)X is weakly infinite-dimensional;(b)X has finite cohomological dimension.

2-polyhedra for which every homotopy domination over itself is a homotopy equivalence

We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups.In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian.The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others.One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian).The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR's homotopy dominating each other have the same homotopy type?

On profinite groups of type FP∞

Suppose R is a profinite ring. We construct a large class of profinite groups L′HRˆF, including all soluble profinite groups and profinite groups of finite cohomological dimension over R. We show that, if G∈L′HRˆF is of type FP∞ over R, then there is some n such that HRn(G,R〚G〛)≠0, and deduce that torsion-free soluble pro-p groups of type FP∞ over Zp have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.

Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks

Laszlo and Olsson constructed Grothendieck’s six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. We give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed by Zheng (2014). As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and ℓ-adic coefficients.

Algebraic groups and compact generation of their derived categories of representations

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups.

Convergence of Voevodsky's slice tower

We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension, we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheavesa,bfnE is finite, exhaustive and separated at each stalk (after inverting the exponential characteristic of k). This partially verifies a conjecture of Voevodsky. 2010 Mathematics Subject Classification: Primary 14F42; Secondary 55P42

Cohomological obstruction theory for Brauer classes and the period-index problem

AbstractLet U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

The cohomological restriction map and FP-infinity groups

We ask, following Bartholdi, whether it is true that the kernel of the restriction map from the cohomology of a group G to the cohomology of a finite index subgroup H is finitely generated as an ideal. We show that in case the group has virtual finite cohomological dimension it is true, and we will show that if G does not have virtual finite cohomological dimension it might not be true, even in case G is an FP infinity group.

The acyclic group dichotomy

Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G in a binate (hence acyclic) group. In the other direction, there are no nontrivial homomorphisms from binate groups to groups of finite cohomological dimension. Binate groups are shown to be of significance in relation to a number of important K-theoretic isomorphism conjectures.

Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

In this paper, we study the growth with respect to dimension of quite general homotopy invariants Q applied to the CW skeleta of spaces. This leads to upper estimates analogous to the classical “dimension divided by connectivity” bound for Lusternik–Schnirelmann category. Our estimates apply, in particular, to the Clapp–Puppe theory of A–category. We use cat1(X) (which is A–category with A the collection of 1–dimensional CW complexes), to reinterpret in homotopy-theoretical terms some recent work of Dranishnikov on the Lusternik–Schnirelmann category of spaces with fundamental groups of finite cohomological dimension. Our main result is the inequality cat(X)≤ dim(Bπ1(X))+ cat1(X), which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].

Reflexivity in derived categories

An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works.

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call \({\mathcal{L}}\) this new class of pro-p groups. We show that the pro-p groups of \({\mathcal{L}}\) have finite cohomological dimension, type FP∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class \({\mathcal{L}}\) is either free pro-p or abelian.

Stable String Operations Are Trivial

We show that in closed string topology and in open-closed string topology with one D-brane, higher genus stable string operations are trivial. This is a consequence of Harer's stability theorem and related stability results on the homology of mapping class groups of surfaces with boundaries. In fact, this vanishing result is a special case of a general result that applies to all homological conformal field theories with the property that in the associated topological quantum field theories, the string operations associated to genus 1 cobordisms with one or two boundaries vanish. In closed string topology, the base manifold can be either finite-dimensional, or infinite-dimensional with finite-dimensional cohomology for its based loop space. The above vanishing result is based on the triviality of string operations associated to the homology classes of mapping class groups that are in the image of stabilizing maps.

Koszul differential graded algebras and BGG correspondence

The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra A is AS-regular, the Ext-algebra E of A is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left E-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG A-modules consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can be derived from the DG version.