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Additional info for Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz
Of course, a mathematical theorem, rigorously proved, needs no such verification; but it is a common human instinct to see how a novel idea works in practice in a simple case, and Abelian where the topology is so simple, are ideal for applying Lefschetz 's general theory. 'But there is more to it than this; by seeing how this theory works out on an Abelian variety, where the results varieties, can be obtained by using the theory of ^-functions which is available, one gets a deeper understanding of the general theory, and so a greater mastery of it.
Much of their beauty lies in their completeness and finality. It is true that they have found numerous applications and have been extended in various directions. However, they have not inspired any large-scale new trends. In contrast, the theory of products has had an extensive development culminating in the cohomology ring and the reduced power operations associated with the ring structure. These appear as vital tools in all phases of algebraic topology beyond the most elementary. The fixed-point problem seems to have dominated nearly all of Lefschetz's work in topology.
In modern language, K* is a deformation retract ofK. Therefore the inclusion map K*<^K induces K isomorphisms K Hn ~ (K) w H n -<*(K*). q gives the final result Of course, the Combining the two isomorphisms ^ (KJL] ^ Hn_Q(R) original result was not stated in this form since the language of cohomology had not been developed. In the original form the statement reads: the q ih Betti number of equals the th Betti number of (n K, and the g-dimensional torsion numbers g) KmodL of K modi coincide with the (n q 1) -dimensional torsion numbers of K.