The following preprint is available (in pdf and ps): Marco Grandis Equilogical spaces, homology and noncommutative geometry Dip. Mat. Univ. Genova, Preprint 491 (2003), 24 p. Abstract. After introducing singular homology for D. Scott's equilogical spaces, we show how these structures can express 'formal quotients' of topological spaces, which do not exist as ordinary spaces and are related with well-known noncommutative C*-algebras. This study also uses a wider notion of local maps between equilogical spaces, which might be of interest for the general theory of the latter. MSC: 18B30, 54A05, 55U10, 55Nxx, 46L80. Keywords: Equilogical spaces, cubical sets, singular homology, noncommutative C*-algebras. http://www.dima.unige.it/~grandis/Eql.pdf http://www.dima.unige.it/~grandis/Eql.ps Marco Grandis Dipartimento di Matematica Universita` di Genova Via Dodecaneso 35 16146-Genova, Italy e-mail: grandis@dima.unige.it home page: http://www.dima.unige.it/~grandis/