Preprint: Directed combinatorial homology and noncommutative tori

Preprint: Directed combinatorial homology and noncommutative tori

[Marco Grandis]

The following preprint is available:

M. Grandis
Directed combinatorial homology and noncommutative tori
(The breaking of symmetries in algebraic topology),

Dip. Mat. Univ. Genova, Preprint 480 (May 2003), 29 p.

Abstract.

   This is a brief study of the homology of cubical sets, with two main
purposes.

   First, this combinatorial structure is viewed as representing *directed
spaces*, breaking the intrinsic symmetries of topological spaces. Cubical
sets have a *directed homology*, consisting of preordered abelian groups
where the positive cone comes from the structural cubes.

   But cubical sets can also express topological facts missed by ordinary
topology. This happens, for instance, in the study of group actions or
foliations, where a topologically-trivial quotient (the orbit set or the
set of leaves) can be enriched with a natural cubical structure whose
directed cohomology agrees with Connes' analysis in noncommutative
geometry. Thus, cubical sets can provide a sort of 'noncommutative
topology', without the metric information of C*-algebras.

MSC: 55U10, 55Nxx, 81R60.
Keywords: Cubical sets, combinatorial homology, noncommutative spaces,
directed algebraic topology.

Available in .pdf and .ps at:

http://www.dima.unige.it/~grandis/Bsy.pdf
http://www.dima.unige.it/~grandis/Bsy.ps

_____


Marco Grandis

Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/