Algebraic Geometry and Topology by Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)

By Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)

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Whether this is so or not, the significance of the general theorem when applied to Abelian varieties is most illuminating, and one cannot but wonder whether we could not, with profit, use Abelian varieties to test out a number of the conjectures which are now current in the theory of algebraic varieties. Conclusion In the foregoing sections my purpose has been mainly to discuss the relation to current problems of mathematics of the more important contributions which Lefschetz has made to algebraic geometry.

STEENSOD complex, K* is a closed complex. As a geometric point set it is closed and contained in the open set L. In modern language, the correspondence er* defines isomor- K H phisms of homology with cohomology: q (KIL)&Hn -<*(K*). This is * hardly satisfactory as a result, since K is a new geometric configuraand a as set it on the tion, point depends triangulation. However, it contains all of K except for a neighborhood of L, and this can be made arbitrarily small by using a sufficiently fine triangulation.

He did define Betti numbers for a compact set L imbedded in a sphere and did he by the use of a decreasing sequence {Lj} of polyhedra converging to L\ and he did remark that these were topological invariants of L, and that the Alexander duality for Betti numbers holds in this general case. 9. The fixed -point theorem for a complex As stated earlier, Lefschetz extended the validity of his fixed-point formula from closed manifolds to manifolds with a regular boundary. In 1928, Hopf showed that it held for an arbitrary n-dimensional complex if the transformation isolated, is and are contained restricted so that the fixed points are in n-cells.

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Algebraic Geometry and Topology by Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)
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