Werner Gysin

Werner Gysin (* 1915 ; † 1998 ) was a Swiss mathematician , after whom the gysin sequence and the gysin homomorphism are named.

Gysin wrote his dissertation at the Swiss Federal Institute of Technology in Zurich as a student of Heinz Hopf , the second reviewer was Eduard Stiefel .

His doctoral thesis made a significant contribution to understanding the relationship between the topological invariants of the base, total space and fibers of a bundle of spheres . By Hassler Whitney's work on spheres bundle many topological questions had been reduced to geometric why spheres bundles were considered fundamental objects. Hans Samelson had found the connection of the homology groups for compact groups and their homogeneous spaces, but this could only be applied to bundles of spheres in a few special cases. Gysin's new approach consisted of examining the relationship between the homology groups by means of the reverse homomorphism, later referred to as Gysin homomorphism: he assigns the simplicial cohomology by means of functoriality to the simplicial cohomology and the double application of the Poincaré duality to cycles through defined mapping . If the cycle is an edge , it is a cycle. This construction maps to . Gysin proved that the construction is well-defined and homotopy- invariant. If for a cycle , then it is not null homotop . This generalizes a construction of the Hopf invariant . ${\displaystyle S^{n}\to E\to B}$ ${\displaystyle f\colon E\to B}$ ${\displaystyle PD}$ ${\displaystyle f_{*}(z)=(f^{*}(z^{PD}))^{PD}}$ ${\displaystyle f_{*}\colon Z_{*}(B)\to Z_{*+n}(E)}$ ${\displaystyle f_{*}(z)}$ ${\displaystyle f_{*}(z)=\partial c}$ ${\displaystyle h(z):=f_{*}(c)\in C_{*+d+1}(B)}$ ${\displaystyle \left\{\left[z\right]\in H_{*}(B)\mid f^{*}(\left[z\right]^{PD})=0\right\}}$ ${\displaystyle H_{*+n+1}(B)/f_{*}(H_{*+n+1}(E))}$ ${\displaystyle h(z)\not =0}$ ${\displaystyle z}$ ${\displaystyle f}$

Gysin's work became central to new approaches to calculating homology groups and was one of the most cited topological works in the post-war period. Norman Steenrod gave a cohomological proof of the Gysin sequence and André Lichnerowicz an interpretation by means of integration along fibers in de Rham cohomology . Chern and Spanier generalized the gysin sequence to CW complexes .

Works

• On the homology theory of the mappings and grains of manifolds , dissertation, ETH Zurich, 1941
• On the homology theory of the mappings and grains of manifolds , Commentarii Mathematici Helvetici, 14: 61–122, 1942

literature

• J. McCleary: A History of Spectral Sequences: Origins to 1953 , in IM James (ed.): History of Topology , Elsevier, Amsterdam et al. 1999, ISBN 0-444-82375-1