Trees, fundamental groups and homology groups

Abstract

For a tree T of its height equal to or less than ω 1 , we construct a space X T by attaching a circle to each node and connecting each node to its successors by intervals. H is the Hawaiian earring and H 1 T(X) denotes a canonical factor of the first integral singular homology group. The following equivalences hold for an ω 1 -tree T: (1) π 1(X ω 1 ) is embeddable into π 1(X T) , if and only if H 1 T(X) ω 1 ≃ Π ω 1 σ Z is embeddable into H 1 T(X T) , if and only if T is not an Aronzajn tree. (2) π 1(X T) is embeddable into ×× ω Z ≃ π 1( H) if and only if H 1 T(X T) is embeddable into Z ω ≃ H 1 T( H) if and only if T is a special Aronzajn tree. (3) π 1(X T) has a retract isomorphic to an uncountable free group, if and only if H 1 T(X T) has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain.