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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 ) .

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 .

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.

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 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?

We specify a result of Yokoi \cite{yo} by proving that if $G$ is an abelian group and $X$ is a homogeneous metric $ANR$ compactum with $\dim_GX=n$ and $\check{H}^n(X;G)\neq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: $\check{H}^{n-1}(A;G)\neq 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D^{n-2}_G$ of all spaces of dimension $\dim_G\leq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $\check{H}^n(X;G)\neq 0$, then $\check{H}^{n-1}(C;G)\neq 0$ for any partition $C$ of $X$ such that $\dim_GC\leq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu \cite{kp}., Comment: 10 pages

We offer a short and elementary proof that, for a Z -set A in a finite-dimensional ANR Y , dim A dim Y . This result is relevant to the study of group boundaries. The original proof by Bestvina and Mess relied on cohomological dimension theory.

Suppose that K is a CW-complex. When we say that a space Y is an absolute co-extensor for K , we mean that K is an absolute extensor for Y , i.e., that for every closed subset A of Y and any map f : A → K , there exists a map F : Y → K that extends f . Our main theorem will provide several statements that are equivalent to the condition that whenever K is a CW-complex and X is a space which is the topological sum of a countable collection of compact metrizable spaces each of which is an absolute co-extensor for K , then the Stone-Cech compactification of X is an absolute co-extensor for K .

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $X\tau_h K\implies \beta(X)\tau K$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite. Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite. Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$. Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of $X\tau {\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept., Comment: 20 pages

In this paper a precompact shape theory is investigated. Necessary and sufficient conditions are found for which the precompact shapes of remainders are coinsided. An intrinsically characterization of Cech (co)homology groups of remainders is given. Border cohomological dimension, dim A ∞ X , and coefficient of border cyclicity, η A ∞ X , are defined and the inequality dim A ∞ X ⩽dim A ( cX ⧹ X ) and the equality η A ∞ X = η A ( cX ⧹ X ) are proved for a space X normally adjoined to its remainder.

We show that if the fundamental group π of a PD4-complex X has cohomological dimension 2 there is a 2-connected degree 1 map from X to a “minimal” such complex. If moreover π has one end we relate the cohomological linking pairing to the attaching map for the top cell via the Whitehead quadratic functor, and show that every w-hermitian form on a finitely generated projective Z[π]-module is realized by some PD4-complex with fundamental group π. If π is a PD2-group the minimal complex is the total space of an S2-bundle over K(π,1) and is determined by cohomological invariants of X.

We prove that for every compactum X and every integer n ⩾2 there are a compactum Z of dim⩽ n and a surjective UV n −1 -map r :Z→X having the property that: for every finitely generated Abelian group G and every integer k ⩾2 such that dim G X ⩽ k ⩽ n we have dim G Z ⩽ k and r is G -acyclic, or equivalently: for every simply connected CW-complex K with finitely generated homotopy groups such that e-dim X ⩽ K we have e-dim Z ⩽ K and r is K -acyclic. (A space is K -acyclic if every map from the space to K is null-homotopic. A map is K -acyclic if every fiber is K -acyclic.)

Let Sm be a m-sphere (or, more generally, a homology m-sphere), Y a k-dimensional CW-complex and f :S m →Y a continuous map. Let G be a finite group which acts freely on Sm. Suppose that H⊂G is a nontrivial normal cyclic subgroup of a prime order. We estimate the cohomological dimension of the set A(f,H,G) of (H,G)-coincidence points of f.

Let G be an Abelian group admitting a homomorphism α : Z →G such that the induced homomorphisms α⊗ id : Z ⊗G→G⊗G and α ∗ : Hom (G,G)→ Hom ( Z ,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards–Walsh resolution ω : EW G (L,n)→|L| . As applications of it we give several resolution theorems. In particular, we have Theorem. Let G be an arbitrary Abelian group. For every compactum X with c-dim G X ⩽ n there exists a G -acyclic map f :Z→X from a compactum Z with dim Z ⩽ n +2 and c-dim G Z ⩽ n +1. Our methods determine other results as well. If the group G is cyclic, then one can obtain Z with dim Z ⩽ n . In certain other cases, depending on G , we may resolve X in such a manner that dim Z ⩽ n +1 and c-dim G Z ⩽ n .

We prove the following theorem: Theorem 1. For a compactum X with c - dim Z /p X⩽n and c - dim Z (q) X⩽n for some distinct prime numbers p,q , and c - dim Z X⩽n+1 , where n>1 , there exists an (n+1) -dimensional compactum Z with c - dim Z /p Z⩽n , c - dim Z (q) Z⩽n and a cell-like map f :Z→X . Moreover, giving the following theorem, we note that Theorem 1 cannot be true in the case of n=1 . Theorem 2. For every pair p,q of distinct prime numbers there exists an infinite-dimensional compactum X such that c - dim Z /p X=1 , c - dim Z (q) X=1 and c - dim Z X=2 .

We present a technique for construction of infinite-dimensional compacta with given extensional dimension. We then apply this technique to construct some examples of compact metric spaces for which the equivalence XτM(G,n)⇔XτK(G,n) fails to be true for some torsion Abelian groups G and n⩾1 .

We prove that every compactum X , having rational dimension n , is an image under a rationally acyclic map of an (n+1) -dimensional compactum Y . If n>1 , then additionally dim Q Y=n . As a consequence we obtain existence of rational homology k -manifolds with covering dimension greater than k .

Let Γ be a discrete group of finite virtual cohomological dimension. Using a rational splitting for G -equivariant K -theory (when G is a finite group) due to T. tom Dieck, we find a rational splitting for the classifying space of such groups.

The main purpose of this paper is to present a unified treatment of the formula for dimension of the transversal intersection of compacta in Euclidean spaces. A new contribution is the proof of inequality dim( X ∩ Y ) ⩾ dim( X × Y ) − n for transversally intersecting compacta X , Y ⊂ R n , based on a correct interpretation of the classical Cogosvili theorem. Also included is a short summary of a new direction of dimension theory, called extension theory, which is needed for the proof.

Sullivan (1970, 1974) pointed out the availability and applicability of localization methods in homotopy theory. We shall apply the method to dimension theory and analyze covering dimension and cohomological dimension from the viewpoint. The notion of localized dimension with respect to prime numbers shall be introduced as follows: the P-localized dimension of a space X is at most n (denoted by dimp X ⩽ n) provided that every map f : A → Snp of a closed subset A of X into a P-localized n-dimensional sphere Snp admits a continuous extension over X. The main results are: 1. (1) Let P1 ⊆ P2 ⊆ P. Then dimp1, X ⩽ dimp2 X (Theorem 1.1). 2. (2) Let X be a compactum. Then the following conditions are equivalent: 2.1. (a) dimX < ∞; 2.2. (b) for some partition P1,…, Ps of P, max{dimpi X: i = 1,…, s} < ∞; 2.3. (c) for any partition P1,…, Ps of P, max{dimpi; X: i = 1,…, s} < ∞ (Theorem 1.2). 3. (3) Let X be a compactum, G an Abelian group. We have that sup{c-dimGp X: p ϵ P} = c-dimGX (Theorem 1.4).

Instead of Lusternik–Schnirelmann π 1 -category we define the π 1 -cohomological dimension by the cohomology with local coefficients and a gap theorem for closed manifolds is shown to hold for this invariant.

We prove that for each countable Abelian group G and for each natural number n there exists an absorbing set in the Hilbert cube for the class of strongly finite-dimensional (in the sense of covering dimension) spaces X with dim G X ⩽ n .

We introduce new classes of compact metric spaces: Cannon—Stan'ko, Cainian, and nonabelian compacta. In particular, we investigate compacta of cohomological dimension one with respect to certain classes of nonabelian groups, e.g., perfect groups. We also present a new method of constructing compacta with certain extension properties.

This paper introduces a notion of strongly hereditarily aspherical compacta and gives a sufficient condition for an inverse limit to have this property. The main result shows that cell-like maps defined on strongly hereditarily aspherical compact metric spaces cannot raise dimension. It suggests why 2-dimensional examples of this sort are plentiful and then sets forth 3-dimensional and 4-dimensional examples.

The setting is a proper surjection f : X → Y between metrizable spaces such that each Cech cohomology group H i ( f -1 ( y );Z) is finitely generated for each y ϵ Y . Estimates (upper bounds) are established of the dimension of Y in terms of the dimension of X , the groups H i ( f -1 ( y );Z) for y ϵ Y , and the dimensions in which these groups are nonzero. The estimates can be viewed as generalizations of the classical result that, for a proper surjection f : X → Y between metrizable spaces with cardinality of f -1 ( y ) at most k +1 for y ϵ Y , dim Y ⩽ dim X + k .

The paper sets forth several examples of decompositions of ( n + k )-manifolds ( k >2) such that the associated space B is not a generalized manifold. On the other hand, a fundamental technical result, derived from a spectral sequence argument, reveals that B always has cohomological dimension k ; the main result attests that if each g in G has trivial Cech homology in dimensions 1,2,…, k −1, where n ⩾ k ⩾2, and if B is finite dimensional, then it is a generalized k -manifold. By way of application, when each g ϵ G has the shape of S n , it follows that p : M → B is an approximate fibration.

A metrizable space X is the cell-like image of a metrizable space Z of dimension ⩽ n iff the cohomological dimension X ⩽ n . If X is topologically complete and/or separable, then we may choose Z to be so. If X is a metrizable space with coh. dim. ⩽ n , then X can be embedded in a topologically complete metrizable space with coh. dim. ⩽ n .

The well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dim Z using a new characterization of dim Z In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with dim Z X ⩽ n to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with dim Z Z ⩽ n . These results are applied to the well-known open problem whether dim X = dim Z X . It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X.

Let M be a nilpotent CW-complex with finitely generated fundamental group. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X can be extended to a map X→M. This is a generalization of a result by Dranishnikov [Mat. Sb. 182 (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov forthcoming paper [Cannad. Bull. Math.] where such conditions were found for nilpotent CW-complexes M with finitely generated homotopy groups.

The following theorems follow from results proved in the paper: Theorem 1. For each Abelian group G≠0 there is a separable metric space X such that dimGX≤3 and all Hausdorff compactifications X′ of Sn× X, n≥0, have cohomological dimension dimGX′ strictly greater than dimG(Sn×X). Theorem 2. If G≠0 is either a torsion group or G is torsion free and l={p|p· G=G, p prime} is infinite, then there is a separable metric space X such that dimGX=2 and all Hausdorff compactifications X′ of Sn×X, n≥0, have cohomological dimension dimGX′; strictly greater than dimG(Sn×X).

The Lusternik–Schnirelmann π 1 -category, cat π 1 X , of a space X is the least integer k such that there is a covering of X by ( k + 1 ) open subsets, every loop in each of which is contractible in X . Let f : X → K ( π 1 ( X ) , 1 ) be a map inducing an isomorphism on π 1 . The π 1 -cohomological dimension, d π 1 ( X ) , of a space X is the largest integer k such that the homomorphism f ∗ : H k ( K ( π 1 ( X ) , 1 ) ; Λ ) → H k ( X ; Λ ) is non-trivial for some π 1 ( X ) -module Λ . The π 1 -cohomological dimension is a lower bound of the Lusternik–Schnirelmann π 1 -category. We determine the π 1 -cohomological dimension of all the non-simply connected compact simple Lie groups except for the projective orthogonal group PO ( 2 m ) with m ⩾ 5 . We also determine the Lusternik–Schnirelmann π 1 -category of SO ( m ) for 3 ⩽ m ⩽ 10 , Ss ( 2 r ) , PSp ( 2 r ) , PU ( p r ) and PO ( 8 ) .

A virtually torsion free, non-positively curved polygon of finite groups has virtual cohomological dimension two. In this paper, the assumption of non-positive curvature is dropped, and a simple construction is used to obtain a triangle of finite groups which is virtually Z k ,2 .

We give an alternative proof, based on Bokstein's theory, of the following result which was originally proved by the authors, jointly with E.V. Scepin (and independently, by S. Spiez): Let X and Y be 2-dimensional compact metric spaces such that dim( X × Y ) = 3. Then for every e > 0 and every pair of maps ƒ: X → R 4 and g : Y → R 4 there exist maps ƒ′ : X → R 4 and g ′ : Y → R 4 such that d (ƒ, ƒ′) e , d ( g, g ′) e and ƒ′( X ) ∪ g ′( Y ) = ∅.

A characterization of dim z (cohomological dimension with integer coefficients) is given for metrizable spaces. It generalizes one recently given by Mardesic for compact Hausdorff spaces and which has been useful in certain instances. The characterization states roughly that a metrizable space X has dim z X ⩽ n if and only if each map of X to a polyhedron P can be approximately factored through another polyhedron Q in such a manner that the ( n +1)-skeleton of Q maps into the n -skeleton of P by a slight adjustment. This result is then employed to show that if X = A ∪ B is a metric space then dim z X ⩽dim z B +1.

The concept of the cohomological dimension dim G X is defined for any Tychonov (i.e., completely regular and Hausdorff) space X and any countable abelian group G . We prove that dim G X = dim G νX , where νX denotes the Hewitt realcompactification of X . We also give a spectral characterization of the dimension dim G . These two results allow us to reduce several problems concerning the dimension dim G of general spaces to the corresponding problems for Polish (i.e., completely metrizable and separable) spaces. For instance, we show that the inequality dim Z ( A ∪ B ) ⩽ dim Z A + dim Z B + 1, validity of which was known for metrizable spaces (Rubin [1991]), remains true in general. We also establish the existence of universal spaces of a given cohomological dimension and of a given weight.

This paper studies properties of refinable maps and contains applications to dimension theory. It is proved that refinable maps between compact Hausdorff spaces preserve covering dimension exactly and do not raise small cohomological dimension with any coefficient group. The notion of a c -refinable map is introduced and is shown to play a comparable role in the setting of normal spaces. For example, c -refinable maps between normal spaces are shown to preserve covering dimension and S -weak infinite-dimensionality. These facts do not hold for refinable maps.

Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduced and investigated by fixed-point theorists, finitistic spaces found an application in cohomological dimension theory. In the paper, two characterizations of paracompact, finitistic spaces are proved. These characterizations allow to create a mechanism of generalizing results of finite dimension theory. As an application we obtain results on compact group actions on paracompact spaces which were previously known for compact Lie group actions.

For every countable CW complex K , we construct a universal separable metrizable space X with K ϵ AE ( X ). From this theorem we derive several corollaries devoted to the question of existence of universal spaces in cohomological dimension theory.

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(\pi,w)$) there are at most $2^\beta$ orbits of $k$-invariants determining "strongly minimal" complexes (i.e., those with homotopy intersection pairing $\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is determined by $\pi$, $w$, $\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabelian group is determined up to TOP isotopy and reflection by its group., Comment: 17 pages

For every prime p and each n =2,3,…,∞, we constructed in [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] an action of G =∏ ∞ i =1 ( Z / pZ ) on a two-dimensional compact metric space X with n -dimensional orbit space. The argument of [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] had a gap in Lemma 15 which affected the main lemma of the paper (Lemma 16). In this note we present corrected versions of Lemmas 15 and 16.

The hyperspaces of strongly countable dimensional compacta of positive dimension and of strongly countable dimensional continua of dimension greater than 1 in the Hilbert cube are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [ 0 , 1 ] . These facts hold true, in particular, for covering dimension dim and cohomological dimension dim G , where G is any Abelian group.

Let M be a nilpotent CW-complex. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X, can be extended to a map X→M.This is a generalization of a result by Dranishnikov [Mat. Sb. (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov [Canad. Bull. Math. (2001) and Topology Appl. (2002)] where a condition of finitely generatedness was imposed on the nilpotent CW-complex M.

Suppose that X is a nonempty compact metrizable space and X 1 ⊂ X 2 ⊂⋯ is a sequence of nonempty closed subspaces such that for each k∈ N , dim Z X k ⩽k . We show that there exists a compact metrizable space Z , having closed subspaces Z 1 ⊂Z 2 ⊂⋯ , and a surjective cell-like map π :Z→X , such that for each k∈ N , (a) dim Z k ⩽ k , (b) π ( Z k )= X k , and (c) π|Z k :Z k →X k is a cell-like map. Moreover, there is a sequence A 0 ⊂ A 1 ⊂⋯ of closed subspaces of Z such that for each k , Z k ⊂ A k , dim A k ⩽ k , π|A k :A k →X is surjective, and for k∈ N , π|A k :A k →X is a UV k −1 -map.

We prove that the following fundamental problems of geometric dimension theory are equivalent: (1) The Mapping Intersection Problem . Given compacta X and Y such that dim( X × Y ) n , can every pair of maps ƒ:X→ R n and g:Y→ R n be approximated arbitrarily closely by maps ƒ′:X→ R n and g′:Y→ R n such that ƒ′(X)∩g′(Y)=O ? (2) The Cohomological Dimension Approximation Problem . Given a compactum X of dimension ⩽ n −2, and an Abelian group G , can every map ƒ:X→ R n be approximated arbitrarily closely by a map ƒ′:X→ R n such that dim G> ƒ′(X)=dim G X ? (3) The Dimension Type Embedding Problem . Given a compactum X of dimension ⩽ n −2, does there exist a compactum X′⊂ R n of the same dimension type , i.e., dim G X ′= dim G X , for every Abelian group G ? By using this equivalence, we obtain the main result of the paper: Let X and Y be compacta such that dim ( X × Y ) n and ( codim X )( codim Y )⩾ n . Then every pair of maps ƒ:X→ R n and g:Y→ R n can be approximated arbitrarily closely by maps ƒ′:X→ R n and g′:Y→ R n such that ƒ′(X)∩g′(Y) =O , provided that dim X ⩽ n −3 and dim Y ⩽ n −3.

Instead of Lusternik–Schnirelmann π1-category we define the π1-cohomological dimension by the cohomology with local coefficients and a gap theorem for closed manifolds is shown to hold for this invariant.