Families of ICIS with constant total Milnor number

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

Finite spaces and an axiomatization of the Lefschetz number

In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite {T_{0}} -spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite {T_{0}} -spaces. However, using the notion of the subdivision of a finite {T_{0}} -space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle {\mathbb{S}^{1}} .

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid {\mathrm{C}^{*}} -algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

On Essential Fixed Points of Compact Mappings on Arbitrary Absolute Neighborhood Retracts and Their Application to Multivalued Fractals

The existence of essential fixed points is proved for compact self-maps of arbitrary absolute neighborhood retracts, provided the generalized Lefschetz number is nontrivial and the topological dimension of a fixed point set is equal to zero. Furthermore, continuous self-maps of some special compact absolute neighborhood retracts, whose Lefschetz number is nontrivial, are shown to possess pseudo-essential fixed points even without the zero dimensionality assumption. Both results are applied to the existence of essential and pseudo-essential multivalued fractals. An illustrative example of this application is supplied.

Lefschetz Numbers for C*-Algebras

Using Poincaré duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear C*-algebra satisfying Poincaré duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with ℂ, as in the classical case.) We then examine endomorphisms of Cuntz–Krieger algebras OA. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism.