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 dimX≤n. The situation in cohomological dimension over an abelian group G, denoted dimG, 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 dimGX≤n.
We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant ...local coefficients, the simplicial version of Bredon–Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.
We prove a generalization of the Edwards–Walsh Resolution Theorem:
Theorem
Let G be an abelian group with
P
G
=
P
, where
P
G
=
{
p
∈
P
:
Z
(
p
)
∈
Bockstein basis
σ
(
G
)
}
. Let
n
∈
N
and let K be ...a connected CW
-complex with
π
n
(
K
)
≅
G
,
π
k
(
K
)
≅
0
for
0
⩽
k
<
n
. Then for every compact metrizable space X with XτK (
i.e., with K an absolute extensor for X)
, there exists a compact metrizable space Z and a surjective map
π
:
Z
→
X
such that
(a)
π is cell-like,
(b)
dim
Z
⩽
n
, and
(c)
ZτK.
We prove the following theorem.
Theorem. Let X be a nonempty compact metrizable space, let l1≤ l2≤ ⋅⋅⋅ be a sequence in N, and let X1 ⊂ X2⊂ ⋅⋅⋅ be a sequence of nonempty closed subspaces of X such ...that for each kN, dimZ/p Xk≤ lk. Then there exists a compact metrizable space Z, having closed subspaces Z1⊂ Z2⊂ ⋅⋅⋅, and a (surjective) cell-like map π:Z → X, such that for each kN,
(a) dim Zk≤ lk,
(b) π(Zk)=Xk, and
(c) π|Zk:Zk→ Xk is a Z/p-acyclic map.
Moreover, there is a sequence A1⊂ A2⊂⋅⋅⋅ of closed subspaces of Z such that for each k, dim Ak≤ lk, π|Ak:Ak → X is surjective, and for kN, Zk⊂ Ak and π|Ak:Ak→ X is a UVlk-1-map.
It is not required that X=∪∞k=1 Xk or that Z=∪∞k=1 Zk. This result generalizes the Z/p-resolution theorem of A. Dranishnikov and runs parallel to a similar theorem of S. Ageev, R. Jiménez, and the first author, who studied the situation where the group was Z.
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 introduce new classes of compact metric spaces: Cannon—Štan'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.
A characterization of dim
z (cohomological dimension with integer coefficients) is given for metrizable spaces. It generalizes one recently given by Mardešić 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.
This paper presents a translation of a theorem of Cartan into an equivariant setting. This work is largely based on the study of the homotopical algebra in the sense of Quillen of the category of ...simplicial objects over the category of rationalOg-vector spaces. The application is a solution to the equivariant commutative cochain problem. This solution is slightly better than the solution obtained earlier by Triantafillou in that the transformation groupG need not be finite.
A pair of maps
f:
X →
R
n
and
g:
Y →
R
n
of compacta
X and
Y into the Euclidean
n-space is said to have a
stable intersection if there exist ε>0 such that for any other pair of maps
f′:
X →
R
n
and
...g′:
Y →
R
n
, satisfying ρ(
f,
f′) <ε and ρ(
g,
g′)<ε, it follows that
f′(
X) ∩
g′(
Y) ≠ 0. The main result of this paper is the following theorem: Let
X and
Y be compacta and let
n = dim
X + dim
Y. Then there exists a pair of maps
f:
X →
R
n
and
g:
Y →
R
n
with stable intersection if and only if dim(
X ×
Y) =
n.
We give an alternative proof, based on Bokštein's theory, of the following result which was originally proved by the authors, jointly with E.V. Ščepin (and independently, by S. Spież): Let
X and
Y be ...2-dimensional compact metric spaces such that dim(
X ×
Y) = 3. Then for every
ε > 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′) <
ε,
d(
g, g′) <
ε and ƒ′(
X) ∪
g′(
Y) = ∅.
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