Trees, fundamental groups and homology groups

✍ Scribed by Katsuya Eda; Masasi Higasikawa

• Book ID: 104307100
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 173 KB
• Volume: 111
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(01)00025-2

✦ Synopsis

For a tree T of its height equal to or less than !1, 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 H T 1 (X ) denotes a canonical factor of the ÿrst integral singular homology group. The following equivalences hold for an !1-tree T :

and only if T is a special Aronzajn tree. (3) 1(XT ) has a retract isomorphic to an uncountable free group, if and only if H T 1 (XT ) has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain.