TY - JOUR

T1 - Extensions of cantor minimal systems and dimension groups

AU - Glasner, Eli

AU - Host, Bernard

N1 - Funding Information:
The first author thanks the Israel Science Foundation (grant number 4699). The second author was partially founded by the Institut universitaire de France.

PY - 2013/9

Y1 - 2013/9

N2 - Given a factor map pW (X, T ) →(Y, S) of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups K 0(X)=K0(Y) in terms of intermediate extensions which are extensions of (Y, S) by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of (Y,S) embeds into a proper subgroup of the dimension group of (X, T), yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups H n(X / Y ) associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups Hn(X/ Y) are torsion groups. As a consequence we can now identify H0(X / Y ) as the torsion group of the quotient group K0(X)=K0(Y ).

AB - Given a factor map pW (X, T ) →(Y, S) of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups K 0(X)=K0(Y) in terms of intermediate extensions which are extensions of (Y, S) by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of (Y,S) embeds into a proper subgroup of the dimension group of (X, T), yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups H n(X / Y ) associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups Hn(X/ Y) are torsion groups. As a consequence we can now identify H0(X / Y ) as the torsion group of the quotient group K0(X)=K0(Y ).

UR - http://www.scopus.com/inward/record.url?scp=84888253436&partnerID=8YFLogxK

U2 - 10.1515/crelle-2012-0037

DO - 10.1515/crelle-2012-0037

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AN - SCOPUS:84888253436

SN - 0075-4102

SP - 207

EP - 243

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 682

ER -