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Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

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

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.

For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

Let G be a simply connected Chevalley group over an infinite field K, and let $$ \tilde{w} $$ : Gn → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w1w2w3w4 is the product of four words in independent variables, then every noncentral element of G is contained in the image of $$ \tilde{w} $$. A similar result for a word w = w1w2w3, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B2 or G2. In the present paper, it is proved that for a group of type B2 or G2, all elements of the large Bruhat cell B nw0B are contained in the image of the word map $$ \tilde{w} $$, where w = w1w2w3 is the product of three independent words. For a group G of type Ar, Cr, or G2 (respectively, for a group of type Ar) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $$ \tilde{w} $$, where w = w1w2 is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $$ \tilde{w} $$ : G(K)n → G(K), where w = w1w2 is a product of two independent words, all unipotent elements are contained in Im $$ \tilde{w} $$.

We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I(R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.

Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont., Comment: 11 pages

Let (R, \(\mathfrak{m}\)) be a commutative Noetherian local ring, \(\mathfrak{a}\) be an ideal of R and M a finitely generated R-module such that \(\mathfrak{a}M \ne M\) and cd(\(\mathfrak{a}\), M) — grade(\(\mathfrak{a}\), M) ⩽ 1, where cd(\(\mathfrak{a}\), M) is the cohomological dimension of M with respect to \(\mathfrak{a}\) and grade(\(\mathfrak{a}\), M) is the M-grade of \(\mathfrak{a}\). Let D(−):= HomR(−, E) be the Matlis dual functor, where E:= E(R/\(\mathfrak{m}\)) is the injective hull of the residue field R/\(\mathfrak{m}\). We show that there exists the following long exact sequence $$\begin{array}{l}{0 \longrightarrow H_{\mathfrak{a}}^{n-2}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{\mathfrak{a}}^{n}(M))) \longrightarrow D(M)} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n-1}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n+1}(D(H_{\mathfrak{a}}^{n}(M)))} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(x_{1},\ldots, x_{n-1})}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(}^{n-1} M))) \longrightarrow \cdots},\end{array}$$ where n:= cd(\(\mathfrak{a}\), M) is a non-negative integer, x1,…, xn−1 is a regular sequence in \(\mathfrak{a}\) on M and, for an R-module L, \(H_{\mathfrak{a}}^{n}(L)\) is the ith local cohomology module of L with respect to \(\mathfrak{a}\).

Let R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the categor...

Weakly separable algebras are investigated in this article. For this purpose their cohomologies are studied. Conditions are found under which normed algebras have a zero cohomological dimension. Ab...

We give a proof without use of perfectoid geometry of the following vanishing theorem of Scholze: for $X\subset \mathbb{P}^n$ a projective scheme of dimension $d$ over an algebraically closed characteristic $0$ field, and $X_r$ the inverse image of $X$ via the map that assigns $(x_0^{p^r}: \dots : x_n^{p^r})$ to the homogeneous coordinates $(x_0:\ldots :x_n)$, the induced map $H^i(X, {{\mathbb{F}}}_p)\to H^i(X_r, {{\mathbb{F}}}_p)$ on étale cohomology dies for $i>d$ and $r$ large. Our proof holds in characteristic $\ell \neq p$ as well.

We construct free abelian subgroups of the group $U(A_\Gamma)$ of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group $U(A_\Gamma)$ was previously studied by Charney, Stambaugh and the second author, who constructed a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of the principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension., Comment: Improvements in exposition, Lemma 4.27 added to clarify proof of Theorem 4.28, remark 3.8 about the definition of "compatibility" added

We investigate the topological nilpotence degree, in the sense of Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$ is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian $p$-subgroups. By replacing centralizers of elementary abelian $p$-subgroups with components of Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn's result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn's result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our results apply much more generally, for example, we establish results for $p$-local compact groups in the sense of Broto-Levi-Oliver, for connected $H$-spaces with Noetherian mod $p$ cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson's depth conjecture in the case of a Noetherian unstable algebra of minimal depth., 42 pages; comments welcome v2; Significantly revised version, accepted for publication in Selecta Mathematica

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups., Comment: 15 pages, 2 figures. Revised background on RORGs, small changes elsewhere. Accepted to appear in Bulletin of the LMS

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$, or $\dim(R/\mathfrak{a}) \leq 1$, or $\dim(R) \leq 2$, then the local cohomology module $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i \in \mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3., Comment: It will appear in Kyoto Journal of Mathematics

In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in the literature., Comment: 24 pages

Throughout, let R be a commutative Noetherian ring. A ring R satisfies Serre’s condition (Sl) if for all p∈SpecR, depthRp≥min{l,dimRp}. Serre’s condition has been a topic of expanding interest. In ...

The cohomology $H^*(\Gamma, E) $ of a torsion-free arithmetic subgroup $\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be interpreted in terms of the automorphic spectrum of $\Gamma$. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes $\{\mathsf{P}\}$ of associate proper parabolic $\mathbb{Q}$-subgroups of $\mathsf{G}$. Each summand $H^*_{\mathrm{\{P\}}}(\Gamma, E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $\{\mathsf{P}\}$. The cohomology $H^*(\Gamma, E) $ vanishes above the degree given by the cohomological dimension $\mathrm{cd}(\Gamma) = \frac{n(n-1)}{2}$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $\{\mathsf{P}\}$ for which the corresponding summand $H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{P}\}}}(\Gamma, E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{Q}\}}}(\Gamma, \mathbb{C})$. Finally, in the case of a principal congruence subgroup $\Gamma(q)$, $q = p^{\nu} > 5$, $p\geq 3$ a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes $\{\mathsf{Q}\}$., Comment: 27 pages, no figures

We establish a “second vanishing theorem” for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

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., Comment: 29 pages; improved results and exposition; added and updated references

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B)., more detailed version - some corrections have been made -title has been changed

In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL 2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander–Milnor conjecture, (III) cohomology of SL 2 over number fields.

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type [Formula: see text]) counterexample based on Thompson’s group [Formula: see text]. We construct as well a finitely generated counterexample with finite decomposition complexity.

The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

Let X be a finitistic space with mod 2 cohomology algebra isomorphic to that of F P m × S 3 , where F = R , C or H . Let ( X , E , π , B ) be a fibre bundle and ( R k , E ′ , π ′ , B ) be a k -dimensional real vector bundle with fibre preserving G = Z 2 action such that G acts freely on E and E ′ − { 0 } , where {0} is the zero section of the vector bundle. We determine lower bounds for the cohomological dimension of the zero set f − 1 ( { 0 } ) of a fibre preserving G -equivariant map f : E → E ′ . As an application of this result, we determine a lower bound for the cohomological dimension of the coincidence sets of continuous maps f : X → R n . In particular, we estimate the size of the coincidence sets of continuous maps f : S i × S 3 → R k relative to any free involution on S i × S 3 , ( i = 1 , 2 , 4 ) .

Compacta [math] and [math] are said to admit a stable intersection in [math] if there are maps [math] and [math] such that for every sufficiently close continuous approximations [math] and [math] of [math] and [math] , we have [math] . The unstable intersection conjecture asserts that [math] and [math] do not admit a stable intersection in [math] if and only if [math] . This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case [math] , [math] and [math] , which still remains open.

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.

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of a finitely generated parafree augmented algebra of infinite cohomological dimension . Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and give a sufficient condition for an augmented algebra to be residually nilpotent in terms of its relations.

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 .

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 / I M ) . 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 H I i ( 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 H I i ( 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 ) c o f 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.

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 $FP_2$) 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 $C'(1/6)$ 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., Version accepted for publication in Journal of Group Theory

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.

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 $\Gamma$.

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.

We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one relator groups, the Hilbert modular group and $3$-manifold groups., Version accepted for publication in Glasgow Mathematical Journal. The title was changed, also some other minor corrections

For any right-angled Coxeter group $\Gamma$ on $k$ generators, we construct proper actions of $\Gamma$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some $\mathbb R^n$: this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension, etc. We also study some examples in cohomological dimension two and four, for which the dimension of the affine space may be substantially reduced., Comment: 41 pages, 4 figures

It is shown that not every metrizable compactum can be written as the inverse limit of an inverse sequence of finite triangulated polyhedra with simplicial bonding maps. This result came from a correspondence between Sibe Mardesic and Leonard R. Rubin, who has provided the Introduction below, and who with some suggestions from Sime Ungar and Vera Tonic, has edited the proof that was communicated to him by Sibe Mardesic. It will be found as Corollary 2.2 .

We show that a closed, connected and orientable Riemannian manifold of dimension $d$ that admits a quasiregular mapping from $\mathbb R^d$ must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom{d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a $d$-dimensional, simply connected manifold that does not admit a quasiregular map from $\mathbb R^d$. Our result gives an affirmative answer to this question.

We show that a countable group is locally virtually cyclic if and only if its Bredon cohomological dimension for the family of virtually cyclic subgroups is at most one., 14 pages, correction to Theorem 4.1, added countability assumption, typos corrected

Let R be a commutative Noetherian domain and let I be an ideal of R. In this paper it is shown that if cd(I,R) = t>0 and the R-module HomR(R∕I,HIt(R)) is finitely generated, then the R-module HIt(R) is faithful, i.e., AnnHIt(R)=0. This result provides a partially affirmative answer to Lynch’s conjecture in [10]. Moreover, in this paper we construct a counterexample to this conjecture.

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?

Let E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

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

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].

We study local-global questions for Galois cohomology over the function field of a curve defined over a p p -adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. Building upon the duality theorem, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third étale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction.

Let $R$ be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if $\mathfrak{a}$ is an ideal of $R$ which is linked by the ideal $I$, then $cd(\mathfrak{a},R) \in \{ grad \mathfrak{a}, cd(\fa, H^{grad \mathfrak{a}}_ {\mathfrak{c}} (R)) + grad \mathfrak{a}\}, $ where $\mathfrak{c} : = \bigcap_{\mathfrak{p} \in Ass \frac{R}{I}- V(\mathfrak{a})}\mathfrak{p}$. Also, it is shown that for every ideal $\mathfrak{b}$ which is geometrically linked with $\mathfrak{a},$ $cd(\mathfrak{a}, H^{grad \mathfrak{b}}_ {\mathfrak{b}} (R))$ does not depend on $\mathfrak{b}$, Comment: 12 pages Proposition 2.9, Remark 2.10 and Corollary 2.11 are added

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the cohomological dimension of groups admitting $P_k$-Anosov representations into $\mathsf{SL}(d,\mathbb R)$ and offer several characterizations of Benoist representations.

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.