Classical Differential Topology and Non-commutative Geometry

The message of this chapter is that very innocent looking, and actually quite useful, constructions in standard differential topology can, in the very precise situation of, what I call, BRUTAL VIOLATION of geometric simple connectivity, lead to non-commutative spaces. Geometric simple connectivity is a venerable, fundamental concept in geometric topology and its occurrence here has been a surprise to me. Geometric simple connectivity is also an important concept for geometric group theory (see [7, 11]). Also, my paper [9] contains a rather complete account of the whereabouts of this concept of “geometrically simply connected” (GSC). In slightly more detail, here is what this chapter does. In a purely general set-up, for a handlebody decomposition of a non-compact smooth four-manifold M, a certain auxiliary space is constructed, for the aim of investigating the asymptotic structure of M. In the GSC situation this turns out to be a smooth 3-manifold, endowed with a lamination. This is very useful for our Schoenflies paper, in this same volume. But, when GSC is BRUTALLY violated, then the auxiliary space is non-commutative, à la Alain Connes. We believe that this is a link to a conjectural new category, encompassing all finitely presented groups, which should occur there as the rationals amidst the irrationals. Not more is proved in this paper than the strict minimum necessary to show that the strange connection mentioned above is really there.