For a tree T of its height equal to or less than ω1, we construct a space XT by attaching a circle to each node and connecting each node to its successors by intervals. H is the Hawaiian earring and H1 T(X) denotes a canonical factor of the first integral singular homology group. The following equivalences hold for an ω1-tree T: (1) π1(Xω1) is embeddable into π1(XT), if and only if H1 T(X)ω1≃Π ω1 σZ is embeddable into H1 T(XT), if and only if T is not an Aronzajn tree. (2) π1(XT) is embeddable into ××ωZ≃π1(H) if and only if H1 T(XT) is embeddable into Zω≃H1 T(H) if and only if T is a special Aronzajn tree. (3) π1(XT) has a retract isomorphic to an uncountable free group, if and only if H1 T(XT) has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain.
ASJC Scopus subject areas