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By Fox R.H., et al.

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Hence L(f) = L(g). These facts justify calling L(f) the sum of the indices of the fixed points off. It is a count of the 'algebraic number will ' of fixed points. 3. As stated schetz have Product spaces and intersections in the introduction, the techniques introduced by Lefhad a major influence on the development of algebraic topology. His proof of the first fixed-point theorem (for manifolds, 1 923) initiated the use of product spaces and intersections as basic tools. A map / of a manifold M into determines graph 0(f) in M x M consisting of the pairs (x,fx) for xzM.

Math. , 29: 429-462, 848. [37] Correspondences between algebraic curves. Ann. of Math. (2), 28: 342-354. The a complex on a manifold and related questions. Proc. , 13: 614-622, 805-807. [39] On the functional independence of ratios of theta functions. Proc. Nat. Acad. Sci. , 13: 657-659. [38] residual set of Nat. Acad. Sci. 1928 [40] Transcendental theory ; Singular correspondences between algebraic curves ; Hyperelliptic surfaces and Abelian varieties. Chap. 15-17, p. 310-395, vol. 1, of Selected Topics in Algebraic Geometry; Report of the Com- mittee on Rational Transformations of the National Research Council.

Should be remarked that Lefschetz handled the internal capproduct in a similar fashion. It 14. is Subsequent developments The moral of the preceding discussion is that the external product simple and geometrically intuitive; and the less obvious internal product can be obtained as an image of the external product under a homomorphism induced by a mapping. This lesson has been applied in several cases. An example is the J. H. C. Whitehead product in homotopy groups. - 1 ) -> (X, x ) be a map of a p-cell into which sends its the homotopy into # so a of that / represents an element boundary group TT^XJXQ).

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Algebraic geometry and topology. A symposium in honor of S. Lefschetz by Fox R.H., et al.


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