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- W830665955 abstract "About 30 years ago, Robin Brooks [B1] undertook a study of the coincidence equation f(x) = g(x) for maps f, g:X → Y . The focus of the subject, as initiated ten years earlier by Schirmer [S], was the determination of the minimum cardinality of the set of solutions to the equation f (x) = g(x) among all maps f , g homotopic to f and g respectively. Brooks observed that there are two natural special cases of the coincidence equation. In addition to the well-known fixed point equation f(x) = x, for which X = Y and g is the identity map, there is what he called the root equation f(x) = a for which g is the constant map g(x) = a ∈ Y . Thus, given a map f :X → Y and a point a ∈ Y , he investigated MR[f, a], the minimum cardinality of the set f (a) for all maps f ′ homotopic to f . Brooks’ original approach, which arose in the context of coincidence theory, was published in [BB] and subsequent papers and given an exposition in [K]. However, starting with [B2] (see also [BO]), Brooks presented an equivalent, but in some ways more convenient, approach to root theory based on covering space theory. The goals of this paper are to give a more complete account of Brooks’ covering space approach to root theory than can be found in [B2] and [BO] and to present arguments for some of the main results of the theory that are more direct than those currently in the literature. These arguments arose as the result of taking a fresh look at the subject of root theory independent of its background of coincidence theory and finding that some technical apparatus that is essential in the more general theory is unneccessary if one is concerned only with roots. It was a problem to decide how to present this material. On the one hand, because the results are already known, a detailed, formal exposition of the material didn’t seem justified. On the other hand, a mere listing of definitions and results would fail to elucidate the novel features of the covering space approach and the non-traditional arguments. If we think of root theory as an unfamiliar village that I am trying to describe to the reader, the detailed exposition would be like listing every" @default.
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- W830665955 date "1999-01-01" @default.
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- W830665955 title "A middle-distance look at root theory" @default.
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- W830665955 doi "https://doi.org/10.4064/-49-1-29-41" @default.
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