Extreme point properties of fixed-point sets

Fixed Point Sets of Fiber-preserving Maps

Let F = (E, p, B;Y ) be a fiber bundle where E, B and Y are connected finite polyhedra. Let f : E → E be a fiber-preserving map and A ⊆ E a closed, locally contractible subset. We present necessary and sufficient conditions for A and its subsets to be the fixed point sets of maps fiber-homotopic to f . The necessary conditions correspond to those introduced by Schirmer in 1990 but, in the fiber...

Generalized Rochlin Invariants of Fixed Point Sets

The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree different...

Nielsen theory of transversal fixed point sets

Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps o...

Fixed Point Sets of Involutions on Spheres

Extreme Distances in Multicolored Point Sets

Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors. We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color. We do this by solving the more general prob...