Realization of fixed point sets

Letf:XX be a selfmap of a compact connected polyhedron, andA a nonempty closed subset ofX. In this paper, we shall deal with the question whether or not there is a mapg:XX homotopic tof such that the fixed point set Fixg ofg equalsA. We introduce a necessary condition for the existence of such a mapg. It is shown that this condition is easy to check, and hence some sufficient conditions are obtained.Partially supported by the Natural Science Foundation of Liaoning University.     

Compact groups and fixed point sets

Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.

     

Dimensions of fixed point sets of involutions

Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fⁿ and F² of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fⁿ is small if the normal bundle of F² does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that . In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F² in M; further, we determine in these cases the equivariant cobordism class of (M, T). Received: 25 August 2005     

The fixed point property for small sets

There exist exactly eleven (up to isomorphism and duality) ordered sets of size 10 with the fixed point property and containing no irreducible elements.The great part of the work presented here has been done when the author was visiting Ivan Rival at the University of Ottawa, Department of Computer Science.     

Absolute fixed point sets for continuum-valued maps

The notion of an absolute fixed point set in the setting of continuum-valued maps will be defined and characterized.

     

Absolute fixed point sets for multi-valued maps

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.

     

On fixed point sets of homeomorphisms of then-ball

Conditions are investigated under which a subsetA can be the fixed point set of a homeomorphism ofB ⁿ . If eitherA ∩ ?B ⁿ ≠ Ø andn arbitrary orA ∩ ?B ⁿ =Ø andn even it is necessary and sufficient thatA is non-empty and closed. IfA ∩ ?B ⁿ =Ø andn odd, conditions which are either necessary or sufficient (but not both) are given.     

Nielsen fixed point theory for partially ordered sets

In this paper, we introduce a Nielsen type number for any selfmap f of a partially ordered set of spaces. This Nielsen theory relates to various existing Nielsen type fixed point theories for different settings such as maps of pairs of spaces, maps of triads, fibre preserving maps, equivariant maps and iterates of maps, by exploring their underlying poset structures.     

The strong fixed point property for small sets

It is shown that every partially ordered set with the fixed point property and with ten or fewer elements actually has the strong fixed point property.Supported by NSERC Operating Grant A7884.Supported by NSERC Operating Grant 41702.     

On retractable sets and the fixed point property

Call a subsetA of an ordered setP retractable tob P iff the map that mapsA tob and leaves all other points fixed is a retraction. We prove fixed point theorems for sets that contain a retractable set and also use this tool to study the fixed point property for products. The results in this paper show that three classical approaches to the fixed point property: irreducible points, cutsets and lexicographic sums can be viewed as special cases of the situation described above.Presented by I. Rival.This research was funded in part by ONR grant nr. N00014-89-J-1824.     

Virtually stable maps and their fixed point sets

We introduce the concept of virtually stable selfmaps of Hausdorff spaces, which generalizes virtually nonexpansive selfmaps of metric spaces introduced in the previous work by the first author, and explore various properties of their convergence sets and fixed point sets. We also prove that the fixed point set of a virtually stable selfmap satisfying a certain kind of homogeneity is always star-convex.     

Cohomology of fixed point sets and deformation of algebras

Under certain conditions the cohomology algebra (with appropriate coefficients) of the fixed point set of a G-space (G=Z/pZ=Zp, p prime or G=S¹) can be considered a deformation - in a purely algebraic sense - of the cohomology algebra of the space itself. On one hand this gives restrictions on the isomorphism type of algebras that can occur as cohomology algebra of the fixed point set of G-spaces if the cohomology algebra of the space itself is given. On the other hand it leads to the definition of obstructions for the cohomology algebra of the fixed point set to be isomorphic (as an ungraded algebra) to the cohomology algebra of the space itself.     

Deformations of algebras and cohomology of fixed point sets

In [13] it is shown that under certain conditions the cohomology algebra of the fixed point set of a space with group action is in an algebraic sense a deformation of the cohomology algebra of the space itself. Here we attempt to prove a converse of the above statement, i.e. we try to realize geometrically a given algebraic deformation of a (commutative) graded algebras as the cohomology algebra of the fixed point set of a suitable space with group action. The first part of this note in a sense reduces this realization problem in equivariant topology to a non-equivariant problem while the second part uses Sullivan's theory of minimal models to actually obtain a converse for S¹-actions, where cohomology is taken with rational coefficients.     

Weak chain-completeness and fixed point property for pseudo-ordered sets

In this paper the notion of weak chain-completeness is introduced for pseudo-ordered sets as an extension of the notion of chain-completeness of posets (see [3]) and it is shown that every isotone map of a weakly chain-complete pseudo-ordered set into itself has a least fixed point.     

Strengthened fixed point property and products in ordered sets

Strengthened fixed point property for ordered sets is formulated. It is weaker than the strong fixed point property due to Duffus and Sauer and stronger than the product property meaning that A × Y has the fixed point property whenever A has the former and Y has the latter. In particular, doubly chain complete ordered sets with no infinite antichain have the strengthened fixed point property whenever they have the fixed point property, which yields a transparent proof of the well-known theorem saying that doubly chain complete ordered sets with no infinite antichain have the product property whenever they have the fixed point property. The new proof does not require the axiom of choice. Presented at the Summer School on General Algebra and Ordered Sets, Malá Morávka, 4–10 September 2005.