El Mokhtar Mokkedem – From Cobordism to Stable homotopy

Abstract:In 1955, Pontrjagin exhibited a first link between cobordisms and stable homotopy in the case of framed cobordism. It was generalized later on by Thom to G-cobordism extending the link to stable homotopy groups of spectra. This connection provided stable homotopy with a strong geometrical motivation as cobordisms are well known for their applications in geometrical topology, notably through h-cobordism and Topological Quantum Field theories. Since then, stable homotopy has been a very active field of research and has met numerous applications in various fields of mathematics, including algebraic geometry and arithmetic through motivic homotopy, and complex K-theory and elliptic cohomology through chromatic homotopy. In this talk, we will make an attempt in presenting how the work of Thom and Pontrjagin highlighted the connection between cobordism and stable homotopy, and we will give a preview on how to realize computations in stable homotopy through the use of Adams spectral sequence and its generalizations.