Algebraic Geometry and Topology by Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)

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

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E. Steenrod Introduction Of the many contributions of Lefschetz to algebraic topology, the two which are best known and most often remarked are the fixedpoint formula and the duality theorem for manifolds with boundaries. Although these two results appear on the surface to be quite unis not the case. They possess a common underlying theme, namely, intersections and products. It was in the development of intersection theory and the algebraic techniques connected with related, this products that Lefschetz made contributions of the highest order.

An important role in these characterizations. It was natural to re- language of algebraic topology so as to obtain an automatic generalization to higher dimensions. Such a characterization generalization would surely be needed for a successful of the n-cell. influence was the thesis of van Kampen in 1928. Another formulate this property in the important He showed that the combinatorial conditions for a manifold could be NORMAN 38 E. STEENROD relaxed without losing the duality property. Instead of requiring that the transverse of a g-cell be an (w-g)-cell, he showed that it was sufficient for it to have the homology structure of a cell.

The result, however, is not easily expressed in invariant form in the original language of homology. The invariant form needs also the language of cohomology. Let JJL, v NORMAN 28 E. STEENROD denote the duality isomorphisms between homology and cohomology and with rational coefficients: in M N pq Let fq : Hq (M)-+Hq (N), : g*-* be the homomorphisms of homology and cohomology induced by / and g. 1) is Q = an endomorphism of Hq (M) K l H 9 -*"*fg for each q. 2) . 1), /i point of/. 2) reduces to the previous formula for L(f).

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