Jerzy Dydak

Infinite dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture

IN [S], A. N. Dranishnikov produced examples of infinite dimensional metric compacta that have integral cohomological dimension equal to three. These examples established the distinct nature of these classical dimension theories, settling the problem, posed by P. S. Alexandroff in 1932 [l], of whether the integral cohomological dimension of a compact metric space is the same as its covering dimension. The techniques used in [S] shed little light on whether or not there could be such an example having integral cohomological dimension two. The infinite dimensionality of the examples in [S] is, ultimately, detected using the vanishing of the reduced complex K-theory with Z/p coefficients (p a prime) of an Eilenberg-MacLane complex K(Z, 3). While reduced complex K-theory with Z/p coefficients of an Eilenberg-MacLane complex K(Z, n) vanishes for n 2 3, the same is not true for K(Z, 2)). (The reader is referred to [2] and [4] for details of these K-theoretic assertions.) The specific nature of K-theory itself plays no direct role in the analyses in [S]. The essential feature is that it is a generalized cohomology theory for which K(Z, 3) behaves as a point; i.e., the reduced K-theory of K(Z, 3) is trivial. Actually, the latter is not true for complex K-theory itself but is valid when Z/p coefficients (p a prime) are used (see [2], [4]). The absence of readily available generalized cohomology theory for which K(Z, 2) is known to behave as a point requires an alternate approach to that in [S] in order to produce an infinite dimensional compact metric space having integral cohomological dimension equal to two. (In addition, the generalized cohomology theory would have to have the property that, in each dimension, its value on a finite complex is a finite group, the latter is true for K-theory with jinite coeficients.) An overview of the approach providing an alternate method to that in [S] and producing cohomological dimension two examples is:

Covering maps for locally path-connected spaces

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connected, then every Peano covering map is equivalent to the projection e X/H → X, where H is a subgroup of the fundamental group of X and e X equipped with the topology used in [2], [15] and introduced in [23, p.82]. The projection e X/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on e X for which one has a characterization of e X/H → X having the unique path lifting property if H is a normal subgroup of π1(X). Namely, H must be closed in π1(X). Such groups include π(U , x0) (U being an open cover of X) and the kernel of the natural homomorphism π1(X, x0) → π̌1(X, x0).

A Hurewicz theorem for the Assouad–Nagata dimension

Given a function

Extension theory of separable metrizable spaces with applications to dimension theory

f\colon X\to Y

Extension Theory: the interface between set-theoretic and algebraic topology

of metric spaces, its {\it asymptotic dimension}

Cohomological Dimension Theory

\asdim(f)

Strong shape for topological spaces

is the supremum of

COARSE AMENABILITY AND DISCRETENESS

\asdim(A)

Compactifications and cohomological dimension

such that

Large Scale Versus Small Scale

A\subset X