The Århus integral of rational homology 3-spheres I: A highly non trivial flat connection on S ³

Abstract.

Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S ³ do not really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology spheres. We show that this invariant is equal (up to a normalization) to the LMO (Le-Murakami-Ohtsuki,[LMO]) invariant and that it recovers the Rozansky and Ohtsuki invariants.¶ This is part I of a 4-part series, containing the introductions and answers to some frequently asked questions. Theorems are stated but not proved in this part. Part II of this series is titled “Invariance and Universality”, part III “The Relation with the Le-Murakami-Ohtsuki Invariant”, and part IV “The Relation with the Rozansky and Ohtsuki Invariants”.