Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds *really*?

This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it *really* is.

In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually *exist*, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.

It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s *On piecewise linear cell decompositions*). (more…)