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Cell is not a manifold. e. it is a relative manifold. If the techniques used in proving the fixed-point formula for manifolds could be extended to relative manifolds, then the Brouwer theorem might be The techniques included. and in question were products, intersections duality. was impelled by these considerations into in two fundamental contributions: the introduction of the concept of relative homology groups, and the discovery of the relative form of the Poincare duality theorem It is likely that Lefschetz an investigation which resulted [34,36,44].

When and in what connection would the basic concept of relative homology have been developed ? It is hardly necessary to describe in detail the definition of the is a chain comrelative homology groups. In modern language, if K L a subcomplex, then the factor groups Cq (K)ICq (L) with the plex, induced boundary operator give a chain complex KjL whose homology groups are the relative groups of modulo L. In the original form, the relative cycles are chains of K K whose boundaries are in L. Two relative homologous modi if their difference plus a chain of a boundary.

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. For each/ and each e > 0, there exists an e-approximation g to/ such that (i) the fixed points of g are finite in number, and (ii) for each fixed point, g maps some neighborhood topologically onto a neighborhood. ,&) of g, there is a local degree or index a t = I according as g preserves or reverses the orientation of the neighborhood.