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

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It ization: (1) may groups not be characterized by numerical invariants, and coefficients to arbitrary coefficients where, again, from integer (2) numerical invariants are inadequate. Because of these language difficulties, it is not possible to give a simple answer to a question such as: Did Lefschetz use inverse limits, define the homology groups usually attributed to Cech? He did define Betti numbers for a compact set L imbedded in a sphere and did he by the use of a decreasing sequence {Lj} of polyhedra converging to L\ and he did remark that these were topological invariants of L, and that the Alexander duality for Betti numbers holds in this general case.

It is true that they have found numerous applications and have been extended in various directions. However, they have not inspired any large-scale new trends. In contrast, the theory of products has had an extensive development culminating in the cohomology ring and the reduced power operations associated with the ring structure. These appear as vital tools in all phases of algebraic topology beyond the most elementary. The fixed-point problem seems to have dominated nearly all of Lefschetz's work in topology.

4) Thus the external cross-products are easily and uniquely defined, and all properties are readily derivable. Now comes the essential feature of Lefschetz's method: the derivation of the cohomology cup- K K K ->K x be a complex, let d product from the cross-product. Let the be diagonal map, and let d* be the induced homomorphism of the cohomology of of K , define K x K into that of K. : If u, v are cohomology classes = d*(uxv). , A The properties of cup-products follow quickly from those of crossproducts using obvious properties of d.