Extension theory of infinite symmetric products

Abstract

Abstract. We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension ext-dim(X) was introduced by A. N. Dranishnikov [9] in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products SP(L). One of the main ideas of the paper is to treat ext-dim(X) ≤ SP(L) as the fundamental concept of cohomological dimension theory instead of dim G (X) ≤ n. In a subsequent paper [18] we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of the classical results on the cohomological dimension. The basic notion in [18] is that of homological dimension of a graded group which allows for simultaneous treatment of cohomological dimension of compacta and extension properties of CW complexes.We introduce cohomology of X with respect to L (defined as homotopy groups of the function space SP(L) X ). As an application of our results we characterize all countable groups G so that the Moore space M (G, n) is of the same extension type as the EilenbergMacLane space K(G, n). Another application is a characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex.

Algebra of dimension theory

“…The next result shows that our definition of the Bockstein basis coincides with the one in [13] and the only difference with the definitions in [16] or [8] is that the case of Z/p ∞ is treated differently. Proposition 6.11.…”

“…Let us give sufficient and necessary conditions for two pointed CW complexes in CW to be mapped to the same element of DGG. The result below was proved in [13] for countable CW complexes using different methods. 3.…”

“…The following result was proved in [9] (see Theorem 5.20) for countable CW complexes. The following result was proved in [13] using different methods.…”

“…Indeed, K(Z, n) ∼ X S n for all n ≥ 1 and all finite dimensional compacta X (this amounts to the Alexandroff Theorem stating that, in the class of finite-dimensional compacta, both covering dimension and the integral cohomological dimension coincide). However, for all n ≥ 2 and all m ≥ 0 there exist compacta X such that Σ m (K(Z, n)) ∼ X Σ m (S n ) fails (see Theorem 9.3 of [13]).…”

“…The main concept of the paper is the homological dimension dim G (A) of a graded group A with respect to an Abelian group G. In the case of A = H * (K) it is equal to the homological dimension dim G (K) introduced in [13] as the supremum of n such that H k (K; G) = 0 for all k < n. In the case of A = H − * (X ) it is equal to the negative of the cohomological dimension dim G (X) of X (it is the infimum of n such that K(G, n) is the absolute extensor of X).…”