Immersions of manifolds and the Hopf invariant

Prof. P. Eccles (University of Manchester)

Monday 12th November, 2012 15:30-16:30 417


The Hopf invariant was first defined by Heinz Hopf in 1931 in proving that the third homotopy group of the 2-sphere is non-trivial. In due course this was generalized in various ways and proved a key tool in determining the homotopy groups and the stable homotopy groups of spheres The Hopf invariant is essentially an obstruction to desuspension. An immersion of a manifold determines an element in a stable homotopy group of a Thom complex whereas an embedding determines an element of the corresponding unstable homotopy group. Thus the self-intersections intersections of the immersion provide an obstruction to the immersion being an embedding and this turns out to be a version of the Hopf invariant. This connection will be explained and some examples of applications given.

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