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By the simplicial approximation theorem, there exists a subdivision K' of K and a simplicial map g: K' -*K such that, for each x gx lies on the closure of the simplex of K containing /#. Denote the subdivision chain-mapping K-+K' by Then g is a chain-mapping and Hq (K)-*Hq (K). it induces the homomorphisms f K-+K, q Because of the fineness of the initial triangulation K and the nearness 9 . : LEFSCHETZ AND ALGEBRAIC TOPOLOGY 35 of g to/, each cell of K fails to intersect the chain gfa. Therefore, in er the trace of the chain-map g q C (K)-*C (K) is q q Hence the alternating sum of these traces is zero.

If . 5. Consider now The dim N < dim M The graphs of/ and g are M x N has dimension m + n < 2m. The geometric the case w-cycles, but intersection of the still intersection ring . e. a a with Kronecker index. If the two cycles are in general 0-cycle the dimension of the intersection ism n. position, It is quite probable that this consideration was one of those which led Lefschetz to develop the theory of the intersection ring of a manifold [31,32,33]. Perhaps more important considerations were the parallelism between cycles on a manifold and subvarieties of an algebraic variety, and the applications real logy theory to algebraic geometry.

A number of definitions of generalized manifold were promulgated by Alexandroff, Cech, Lefschetz [55] and Wilder. In all cases, the more important conditions were local in nature. In 1931, Borsuk introduced the concepts of the absolute retract (AR) and the absolute neighborhood retract (ANR). These spaces are generalizations of cells and complexes respectively. Borsuk extended to them numerous theorems valid for the latter. The definition of an ANR space is not exactly local in nature, however it was clear that such spaces are locally smooth in some sense.

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