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

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5. Consider now The dim N < dim M The graphs of/ and g are M x N has dimension m + n < 2m. The geometric the case w-cycles, but intersection of the still intersection ring . e. a a with Kronecker index. If the two cycles are in general 0-cycle the dimension of the intersection ism n. position, It is quite probable that this consideration was one of those which led Lefschetz to develop the theory of the intersection ring of a manifold [31,32,33]. Perhaps more important considerations were the parallelism between cycles on a manifold and subvarieties of an algebraic variety, and the applications real logy theory to algebraic geometry.

It is not necessary to enlarge here on the details of this theory, nor to discuss the application of the same arguments to the case in which D is a copy of C to obtain the Riemann relations between the periods of the integrals of the first kind on C. The brief sketch we have given should be sufficient to show that a most fruitful field of applica- tion of Lefschetz's result lies in the theory of correspondences. It is LEF8CHETZ AND ALGEBRAIC GEOMETRY 19 natural to apply similar methods to the theory of correspondences between irreducible algebraic varieties U and V of dimension r and s respectively.

Is under the map X Most of the properties of the internal product become entirely obvious with this definition. Another example is provided by the definition of addition in the Borsuk-Spanier cohomotopy groups. Here again product spaces are used liberally. A final example is provided by the cup-i-products which this reporter defined in 1946. They are a sequence of bilinear cochain such that u U v = u U v. These products u U v defined for i = 0, 1 . , t . . , cochain operations led to topologically invariant squaring operations which were used in the classification of mappings.

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