A survey of philosophical and explicit classification results for closed manifolds

Contact: Christian Kaiser (kaiser @ mpim-bonn.mpg.de)

By „philosophical“ results I mean a global picture of manifolds, which in general does not imply that one can decide whether two given manifolds are equal. The best answer in this direction is the Wall exact surgery sequence for manifolds of dimension >4, which I will explain. By „explicit“ results I mean that one gives a set of „computable“ invariants which classify a given class of manifolds. The prototype is the classification of surfaces and the Poincaré conjecture or, in complete analogy, the Borel conjecture. I will give a more or less complete survey of all explicit classification results, which indicates that to my knowledge there are rather few results. I will indicate which methods are used to obtain the different results.