Splittings and cross-sections in topological groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:435

Article

Bello, H. J.¹  Chasco, M. J.¹  Dominguez, X.²  Tkachenko, M.³ 

[1] Univ Navarra, Dept Fis & Matemat Aplicada, E-31080 Pamplona, Spain

[2] Univ A Coruna, Dept Metodos Matemat & Representac, La Coruna, Spain

[3] Univ Autonoma Metropolitana, Dept Matemat, Mexico City, DF, Mexico

关键词: Compact group;    Locally precompact group;    Free abelian topological group;    k(omega)-space;    Splitting extension;    Cross section;   

DOI  :  10.1016/j.jmaa.2015.11.040

来源: Elsevier

【 摘 要 】

This paper deals with the splitting of extensions of topological abelian groups. Given topological abelian groups G and H, we say that Ext(G, H) is trivial if every extension of topological abelian groups of the form 1 -> H -> X -> G -> 1 splits. We prove that Ext(A(Y), K) is trivial for any free abelian topological group A(Y) over a zero-dimensional k(omega)-space Y and every compact abelian group K. Moreover we show that if K is a compact subgroup of a topological abelia.n group X such that the quotient group X/K is a zero-dimensional k(omega)-space, then there exists a continuous cross section from X/K to X. In the second part of the article we prove that Ext(G, H) is trivial whenever G is a product of locally precompact abelian groups and H has the form T-alpha x R-beta for arbitrary cardinal numbers alpha and beta. An analogous result is true if G = Pi(i is an element of I) G(i) where each G(i) is a dense subgroup of a maximally almost periodic, Cech-complete group for which both Ext(G(i), R) and Ext(G(i), T) are trivial. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free