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

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In spite of this, however, Lefschetz's contributions call for some mention in the present setting. Apart from the actual results achieved, Lefschetz's work on Abelian varieties is well worth attention from those who derive inspiration from the study of the way in which a master works; for I have formed the impression that his Bordin Prize memoir, which contains an account of nearly all his work on Abelian varieties, contains many indications of the manner in which his ideas on the topology of varieties, integrals of the second kind, and on the characterization of algebraic cycles, developed.

It deviated from this by a numerical factor. Subsequently Alexander, Cech and Whitney found independently the definition which did correspond. 1 1 Singular homology theory As has already been remarked in 8, Lefschetz did define cycles and homologies which give the Vietoris-Cech homology groups for . a closed set in a sphere. He attempted to use these to extend the mean- ing and validity of his fixed-point formula to such spaces. It soon became clear however that the fixed-point theorem in this form is such general spaces.

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. Using the regularity conditions on cells of L, Lefschetz showed that any chain of could be deformed into *. 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.

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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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