Tuesday, 25th of November 2025, 13:15-14:00 in HG03.085.
Euan Aitken
A brief history of exotic structures in topology.
Manifold topology comes in various flavours, each coming with natural structure preserving maps and a resulting notion of isomorphism, such as: topological manifolds with continuous maps between them, piecewise linear manifolds with piecewise linear maps and smooth manifolds with smooth maps. In the early days of topology, it was assumed by many that these different flavours, and their corresponding notions of equivalence, were all essentially the same, and that the choice of which category to work in was just a matter of preference. In 1956, John Milnor disproved this assumption, constructing a smooth, seven-dimensional manifold which was homeomorphic, but not diffeomorphic, to the 7-sphere, with its standard smooth structure; a so-called exotic 7-sphere. In 1962. Milnor and Micheal Kervaire showed that the set of diffeomorphism classes of exotic n-spheres could be given a group structure, and computed the order of this group for many n. In this talk, we will discuss the history of these developments, some recent progress towards the classification of exotic spheres. We will also sketch some of the techniques involved, including some surprising interactions between differential topology and stable homotopy theory.