zbMATH — the first resource for mathematics

Free compact groups. I: Free compact abelian groups. (English) Zbl 0589.22003
The structure of free compact abelian groups is investigated. A compact abelian group G is free compact abelian if and only if \(G\cong K\times {\hat {\mathbb{Q}}}^ a\times \prod_{p\in prime}{\mathbb{Z}}^ b_ p\), where \({\hat {\mathbb{Q}}}\) is the character group of the discrete group of rationals, K is a compact connected abelian group with dense torsion subgroup, and a,b are cardinal numbers such that \(a\geq \max \{2^{\aleph_ 0}\), b, dim \(K\}\) (Theorem 1.7.2). This theorem implies that if G is of the above form then it is the free compact abelian group on the underlying space of the compact abelian group \(X=K\times {\hat {\mathbb{Q}}}^ a\times F\), where F is a discrete abelian group of order b-1, if b is finite, and \({\mathbb{Z}}(2)^ b\), otherwise.
Compact Hausdorff spaces with isomorphic free compact abelian groups are completely characterized by means of their Alexander-Spanier cohomology group and the weight of them. This characterization gives a new information even for compact spaces with isomorphic free abelian topological groups. In particular, if A(X)\(\cong A(Y)\) and the weight of X, w(X), exceeds \(2^{\aleph_ 0}\), then \(w(X)=w(Y)\).
Reviewer: M.G.Tkachenko

22C05 Compact groups
20K99 Abelian groups
22B05 General properties and structure of LCA groups
55N05 Čech types
Full Text: DOI
[1] André, M., L’homotopies des groupes abéliens localement compacts, Comment. math. helv., 38, 1-5, (1963) · Zbl 0119.03302
[2] Arhangelskii, A.V., On the relations between invariants of topological groups and their subspaces, Uspehi mat. nauk, Russian math surveys, 35, 1-23, (1981)
[3] Arhangelskii, A.V., The principle of τ-approximation and a test for equality of dimension of compact Hausdorff spaces, Dokl. akad. nauk SSSR, Soviet math. dokl., 12, 805-809, (1980) · Zbl 0491.54028
[4] Bredon, G., A space for which \(H\^{}\{1\}(X, Z) ≠ [X, S\^{}\{1\}]\), Proc. amer. math. soc., 19, 396-398, (1968) · Zbl 0157.30102
[5] Bredon, G., Sheaf theory, (1968), McGraw Hill New York
[6] Enochs, E.E., Homotopy groups of compact abelian groups, Proc. amer. math. soc., 15, 878-881, (1964) · Zbl 0124.16201
[7] Graev, M.I., Free topological groups, Izv. akad. nauk SSSR ser. mat., Amer. math. soc. transl., Reprint amer. math. soc. transl., 8, 1, 305-364, (1962), English transl.
[8] Hofmann, K.H., An essay on free compact groups, (), Springer lecture notes, 915, 171-197, (1982)
[9] Hofmann, K.H.; Hofmann, K.H., Introduction to the theory of compact groups II, Tulane university lecture notes, Tulane university lecture notes, (1969)
[10] Hofmann, K.H.; Mostert, P.S., The cohomology of compact abelian groups, Bull. amer. math. soc., 74, 975-978, (1968) · Zbl 0165.34501
[11] Hofmann, K.H.; Mostert, P.S., Cohomology theories for compact abelian groups, (1973), Springer New York/Heidelberg, VEB Deutscher Verlag der Wissenschaften, Berlin
[12] Huber, P.J., Homotopical cohomology and C̆ech cohomology, Math. ann., 144, 73-76, (1961) · Zbl 0096.37504
[13] Joiner, C., Free topological groups and dimension, Trans. amer. math. soc., 220, 401-418, (1976) · Zbl 0331.54026
[14] Markov, A.A., On free topological groups, Bull. acad. sci. URSS ser. mat., Amer. math. soc. transl., Amer. math. soc. transl., 8, 1, 195-272, (1962), Reprint
[15] Morris, S.A., Free compact abelian groups, Mat. C̆asopsis sloven. akad. vied., 22, 141-147, (1972) · Zbl 0235.22012
[16] Morris, S.A., Free abelian topological groups, (), 375-391
[17] Scheerer, H.; Strambach, K., Idempotente multiplikationen, Math. Z., 182, 95-119, (1983) · Zbl 0505.55011
[18] Zambahidze, L.G., Relations between dimensions of free bases of free topological groups, Sooshch. akad. nauk gruzin SSSR, 97, 569-572, (1980) · Zbl 0452.54031
[19] Zambahidze, L.G., On relations between dimensional and cardinal functions of spaces imbedded in spaces of a special type, Soobshch. akad. nauk gruzin SSSR, 100, 557-560, (1980) · Zbl 0484.54007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.