Coincidence theory for spaces which fiber over a nilmanifold
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Let Y be a finite connected complex and p : Y → N a fibration over a compact nilmanifold N. For any finite complex X and maps f ,g : X → Y , we show that the Nielsen coincidence number N( f ,g) vanishes if the Reidemeister coincidence number R(p f , pg) is infinite. If, in addition, Y is a compact manifold and g is the constant map at a point a ∈ Y , then f is deformable to a map fˆ : X → Y such that fˆ −1 (a) = ∅. 1. Introduction The celebrated Lefschetz-Hopf fixed point theorem states that if a selfmap f : X → X on a compact connected polyhedron X has nonvanishing Lefschetz number L( f ), then every map homotopic to f must have a fixed point. On the other hand, if L( f ) = 0, f need not be homotopic to a fixed point free map. A classical result of Wecken asserts that if X is a triangulable manifold of dimension at least three, then the Nielsen number N( f ) is the minimal number of fixed points of maps in the homotopy class of f . Thus, in this case, if N( f ) = 0, then f is deformable to be fixed point free. For coincidences of two maps f ,g : X → Y between closed oriented triangulable n-manifolds, there is an analogous Lefschetz coincidence number L( f ,g), and L( f ,g) = 0 implies {x ∈ X | f (x) = g (x)} = ∅ for all f ∼ f and g ∼ g. Schirmer [14] introduced a Nielsen coincidence number N( f ,g) and proved a Wecken-type theorem. While the theory of Nielsen fixed point (coincidence) classes is useful in obtaining multiplicity results in fixed point (coincidence) theory and in other applications, the computation of the Nielsen number remains one of the most difficult and central issues. One of the major advances in recent development in computing the Nielsen number is a theorem of Anosov who proved that for any selfmap f : N → N of a compact nilmanifold N, N( f ) = |L( f )|. By a nilmanifold, we mean a coset space of a nilpotent Lie group by a closed subgroup. Thus, the computation of N( f ) reduces to that of the homological trace L( f ). Anosov’s theorem does not hold in general for selfmaps of solvmanifolds or infranilmanifolds. Meanwhile, the theorem has been generalized to coincidences for Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 89–95 2000 Mathematics Subject Classification: 55M20 URL: http://dx.doi.org/10.1155/S1687182004308107
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Coincidences for spaces over a nilmanifold
maps between closed oriented triangulable manifolds of the same dimension. In particular, coincidences of maps from a manifold to a solvmanifold or an infrasolvmanifold have been studied (see, e.g., [8, 10, 15]). In [9], it was shown that if f ,g : X → Y are maps from a finite complex X to a compact nilmanifold Y , then R( f ,g) = ∞ implies N( f ,g) = 0. This result is false in general, for example, when Y is a solvmanifold (see, e.g., [8]). In this work, the main objective is to generalize this result for more general spaces, in particular, for finite connected complexes Y which fiber over a compact nilmanifold N. We should point out that such a space Y necessarily fib
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