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

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E. to the g-dimensional cohomology group. Lefschetz regarded the pseudo-cycle as a tool in the proof of the fixed-point theorem. He probably observed that two pseudo-cycles, ^ S, intersect in a pseudo-cycle; but there for the same imbedding K no indication that he considered the question of the topological invariance of the resulting ring. This was proved first in a paper 1935. e. without using also presented a definition of a cupAlexander imbeddings K<^S). under product. It was not satisfactory since it did not correspond intrinsic definitions of the NORMAN 36 E.

The 2 -cycle defined by L is sufficient to determine the spond mapping of the homology group of C into the homology group of /), and that of D into that of (7, and hence the usual equations of Hurwitz. Thus we have only to consider the algebraic cycles of C x D. If x is a point of (7, and y 1? , /^ C x D is homologous to A necessary and sufficient are integers. there should be a curve on condition that C x D homologous to this is that the double integrals of the first kind should have zero periods on this cycle.

And in question were products, intersections duality. was impelled by these considerations into in two fundamental contributions: the introduction of the concept of relative homology groups, and the discovery of the relative form of the Poincare duality theorem It is likely that Lefschetz an investigation which resulted [34,36,44]. It is an idle occupation but highly fascinating to speculate as to what might have happened if Lefschetz had suspected in 1924, say, that his fixed-point theorem was valid for an arbitrary complex, and not just for manifolds and relative manifolds.

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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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