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Working group (Fall 2021) : Characteristic classes and exotic spheres

  • Abstract
    An exotic sphere is a differential manifold that is homeomorphic but not diffeomorphic to the standard n-sphere. We will be interested in these "exotic" manifolds, similar to a sphere, from the point of view of their topological properties, but with unusual differential structures. Are there exotic spheres? For which value of the dimension? Can we classify them?

    In general, we don't known for which dimensions exotic spheres exist (let alone how many they are). For example, there is no exotic sphere in dimension 12, the case of dimension 4 is still an open problem, etc. Historically, the first exotic spheres were built in 1956 by John Milnor in dimension 7. A few years later, Kervaire and Milnor established a more complete classification, showing that the set of diffeomorphism classes of oriented exotic spheres of dimension 7 form a cyclic group of order 28.

    The objective of this working group is to focus on the work of Milnor and Kervaire on exotic spheres. For this, we will introduce the notion of characteristic classes that are cohomological invariants which play a key role in the study of manifolds, vector bundles and the construction of exotic spheres.

    You can find the french version here .