Obstructions to Homotopy Invariance in Parametrized Fixed Point Theory
نویسنده
چکیده
In “zero-parameter” or classical Nielsen fixed point theory one studies Fix(f) := {x ∈ X | f(x) = x} where f : X → X is a map. In case X is an oriented compact manifold and f is transverse to the identity map, idX , Fix(f) is a finite set each of whose elements carries a natural sign, ±1, the index of that fixed point. The set Fix(f) is partitioned into Nielsen classes. Adding the indices within a given Nielsen class yields the fixed point index of that class. The number of classes with nonzero fixed point index is called the Nielsen number of f . One-parameter fixed point theory is the study of the fixed point set, Fix(F ) := {(x, t) ∈ X × I | F (x, t) = x}, of a homotopy F : X × I → X. In the case where X is an oriented compact manifold and F is transverse to the projection map p : X × I → X, Fix(X) consists of naturally oriented circles in the interior of X × I and naturally oriented arcs whose endpoints lie in X ×{0, 1}. We give a summary of one-parameter fixed point theory, in analogy to the classical theory, as developed by the first two authors. While classical fixed point theory is a “homotopy invariant” theory, homotopy invariance in the one-parameter theory is obstructed by “torsion” type invariants. We discuss this phenomenon and give some new examples.
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