Cohomological Dimension Research Articles

Dimension invariants of outer automorphism groups

The geometric dimension for proper actions \underline{\mathrm{gd}}(G) of a group G is the minimal dimension of a classifying space for proper actions \underline{E}G . We construct for every integer r\geq 1 , an example of a virtually torsion-free Gromov-hyperbolic group G such that for every group \Gamma which contains G as a finite index normal subgroup, the virtual cohomological dimension vcd (\Gamma) of \Gamma equals \underline{\mathrm{gd}}(\Gamma) but such that the outer automorphism group Out (G) is virtually torsion-free, admits a cocompact model for \underline{E} Out (G) but nonetheless has vcd(Out (G))\le \underline{\mathrm{gd}} (Out (G))-r .

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.

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Let $X$ be a paracompact space, let $G$ be a finite group acting freely on $X$ and let $H$ a cyclic subgroup of $G$ of prime order $p$. Let $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that $\ind X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. In this work, we estimate the cohomological dimension of the set $A(f,H,G)$ of $(H,G)$-coincidence points of $f$. Also, we estimate the index of a $(H, G)$-coincidence set in the case that $H$ is a $p$-torus subgroup of a particular group $G$ and as application we prove a topological Tverberg type theorem for any natural number $r$. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently fail for all $r$ that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any integer $r$.

Stable finiteness properties of infinite discrete groups

Let $G$ be an infinite discrete group. A classifying space for proper actions of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are contractible for all finite subgroups $H$ of $G$. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper $G$-spectra and study its finiteness properties. We investigate when $G$ admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper $G$-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of $G$. Finally, if the group $G$ is virtually torsion-free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of $G$, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.

Presentations of rings with a chain of semidualizing modules

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

Examples of foliations with infinite dimensional special cohomology

We present two examples of foliations with infinite dimensional basic symplectic and complex cohomologies, along with a general sufficient condition for such phenomena. This puts restrictions on possible generalisations of several finiteness results from Riemannian foliations to any broader class. The examples are also noteworthy for the unusual behaviour of their basic de Rham cohomology.

Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

Abstract A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally FP 2 {\mathrm{FP}_{2}} ) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension 2 with respect to the family of parabolic subgroups. A class of groups where our result applies consists of C ′ ⁢ ( 1 / 6 ) {C^{\prime}(1/6)} small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness properties over Bredon modules and homological Isoperimetric inequalities.

Dimension of compact metric spaces

We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory.

About the cohomological dimension of certain stratified varieties

We determine an upper bound for the cohomological dimension of the complement of a closed subset in a projective variety which possesses an appropriate stratification. We apply the result to several particular cases, including the Bialynicki-Birula stratification; in this latter case, the bound is optimal.

Borsuk-Ulam Theorems and Their Parametrized Versions for $$\mathbb {F}P^m\times \mathbb {S}^3$$ F P m × S 3

Let $$G=\mathbb {Z}_p,$$ $$p>2$$ a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of $$\mathbb {F}P^m\times \mathbb {S}^3$$ , where $$m+1\not \equiv 0$$ mod p and $$\mathbb {F}=\mathbb {C}$$ or $$\mathbb {H}$$ . We wish to discuss the nonexistence of G-equivariant maps $$\mathbb {S}^{2q-1}\rightarrow X$$ and $$ X\rightarrow \mathbb {S}^{2q-1}$$ , where $$\mathbb {S}^{2q-1}$$ is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map $$f\!:\! X\rightarrow \mathbb {R}^n$$ , a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.

On the cohomological dimension of the moduli space of Riemann surfaces

The moduli space of Riemann surfaces of genus $g\geq 2$ is (up to a finite \'etale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$. This expectation is verified in low genus and supported by Harer's computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this paper we prove that such dimension is at most $2g-2$. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most $g$. In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: the construction of such a function relies on some basic geometric properties of translation surfaces.

Geometric dimension of lattices in classical simple Lie groups

We prove that if $\Gamma$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $\Gamma$-equivariantly homotopy equivalent to a proper cocompact $\Gamma$-CW complex of dimension the virtual cohomological dimension of $\Gamma$.

The paucity of universal compacta in cohomological dimension

Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim ⁡ X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G ⁡ X ≤ n .

Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Let R be a commutative Noetherian ring, I⊆J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q(J,M)≤q(I,M)+cd(J,M/IM). Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q(I,M)≤1, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. As a consequence of this result it is shown that, if q(I,R)≤1, then for any finitely generated R-module M, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. Finally, it is shown that the category of all I-cofinite R-modules C(R,I)cof is an Abelian subcategory of the category of all R-modules, whenever (R,m) is a complete Noetherian local ring and I is an ideal of R with q(I,R)≤1. These assertions answer affirmatively two questions raised by R. Hartshorne in [16], in the some special cases.

Sasakian structures a foliated approach

AbstractRecent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The paper presents a foliated approach to Sasakian manifolds on which the author gave several lectures. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and Kählerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds.

On dimensions of groups with cocompact classifying spaces for proper actions

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for E_G, the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for E_G. We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G-CW-complex.

A note on bounded-cohomological dimension of discrete groups

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

The codimension-one cohomology of SLnℤ

We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Steinberg module for SL_n Z whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL_2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.

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.

On generalized notions of amenability and operator homology of the Fourier algebra

Let G be a locally compact group, A (G) its Fourier algebra and Acb (G) the closure of A (G) in the space of completely bounded multipliers of A (G). We show that the Fourier algebras of weakly amenable, non-amenable groups are not approximately amenable. We also prove that A (G) is operator approximately biprojective if and only if G is discrete. Finally, we study various (operator) cohomological properties and its related (operator) homological properties of the algebra Acb (G).