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

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There is a point / has at least one fixed point xzX f References are to the Bibliography of the Publications of S. Lefschetz, pp. 44-9, this volume. NORMAN STEENROD The conclusion is valid whenever X is the space of a finite complex. 26 E. This result was proved by Lefschetz in 1928. His initial theorem in 1923 asserted the conclusion only when is a compact orientable manifold (without boundary). In this case he X was able to prove more by assigning a geometric significance to the numerical value of L(f) for an arbitrary/ in the following manner.

Hence L(f) = 1 for any map/. Borsuk showed that X admits arbitrarily false for small deformations without fixed points. The inadequacy of the Vietoris-Cech homology theory for the fixedpoint theorem probably led Lefschetz to investigate other methods of defining homology groups. In any case, he gave the first formal definition of the singular homology theory [45]. Singular cycles had been used by Veblen in his proof of the topological in variance of the homology group. The term singular was applied by Veblen to emphasize the fact that his cells and chains were continuous images of polyhedral cells and chains, and were not themselves polyhedral.

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. 4) implies the commutation law LEF8CHETZ AND ALGEBRAIC TOPOLOGY 41 In retrospect it is easy to see the source of the difficulties Alexander, Cech and Whitney had to overcome in their purely internal constructions.

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

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