This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if G is graph containing no 3- or 4-cycles, then the nth group A n (G) trivial all n?2. Second exhibit each n?1 natural homomorphism ?:A (G)?? (G), where ? cubical singular homology group, and an infinite family of which nontrivial ? surjective. It follows there ar...

It is proved that any countable topological group in which the filter of neighborhoods identity element not rapid contains a discrete set with precisely one nonisolated point. This gives negative answer to Protasov's question on existence ZFC nondiscrete all subsets are closed. also extremally disconnected implies ultrafilter and, hence, cannot be constructed ZFC.

We survey the cyclic cohomology associated with various algebras related to discrete groups. We then discuss the motivation and techniques of the cyclic theory approach to various problems in algebra and analysis.