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

Abstract

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: