Trees, fundamental groups and homology groups Original Research Article

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