Colloquium: "Classification of 4-manifolds" by Dr. Guilherme Sadovski

Speaker: Dr. Sadovski is a member of this unit (visit his page)

Abstract. The situation in d>3 is very different from its lower-dimensional counterpart. In particular, there is no analog of neither a "geometrization" theorem nor a Rado/Moise theorem that could be used to give a full DIFF classification of higher-dimensional d-manifolds. The situation is particularly bad in d=4, where such classification has already been proved to be impossible due to severity of the exotic phenomena there. With these difficulties in mind, we proceed restricting ourselves to the classification of simply-connected and closed 4-manifolds in the TOP category alone. We introduce the concept of Kirby-Siebenmann (KS) invariant as an obstruction to the existence of a PL structure as well as the concept of an intersection form 'Q' and its classification by rank, parity, determinant, signature, etc. Finally, we state Friedmann's classification  as "the KS invariant and Q form a complete set of invariants classifying TOP 4-manifolds which are simply-connected and closed. Moreover, if the intersection form is unimodular, then all combinations of Q and the KS invariant is realized in some 4-manifold, except that when Q is even, the KS invariant is uniquely determined.". With this huge amount of non-homeomorphic 4-manifolds, we finish by given some important examples such as the \(\mathbb{S}^4,\; \mathbb{CP}^4,\; \mathbb{CP}^4_{\Theta}\) and the \(E_8\)-manifold.