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

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When his second Colloquium Publication entitled Algebraic Topology appeared in 1942, the adjective 'combinatorial' fell into disuse. 2. The fixed -point formula The results on fixed points which preceded the work of Lefschetz were the theorems of Brouwer for continuous mappings of the n-cell and n-sphere, and a theorem of Alexander on topological mappings of a 2-dimensional manifold. The basic step toward a full-fledged result was Lefschetz's discovery in 1923 [25]f of a formula. Its description runs as follows.

He constructed a compact connected set X in Euclidean 3-space which was the intersection of a decreasing sequence of 3-cells. Its Cech homology is therefore the same as that of a 3-cell. 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].

E. 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.

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

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