Deforming Calabi-Yau orbifolds

A Calabi-Yau 3-fold is a compact complex 3-manifold (X, J) equipped with a Ricci-flat Kähler metric g and a holomorphic volume form Ω which is constant under the Levi-Civita connection of g. Suppose X is a Calabi-Yau 3-fold and G a finite group that acts on X preserving J, g and Ω. Then X/

Nonstabilized Nielsen coincidence invariants and Hopf--Ganea homomorphisms

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs (f_1,f_2) of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC(f_1,f_2) (and MC(f_1,f_2), respectively) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1,f_2). Furthermore, we deduce finiteness conditions for MC(f_1,f_2). As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f_1,f_2) into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf--Ganea homomorphisms which turn out to yield finiteness obstructions for MC.Comment: This is the version published by Geometry & Topology on 24 May 200

On the topology of desingularizations of Calabi-Yau orbifolds

Let X/G be a 3-dimensional Calabi-Yau orbifold with codimension 2 singularities. The topology of crepant resolutions of X/G is described by the McKay correspondence (Reid, Ito). We study Calabi-Yau 3-folds Y that arise by deforming the complex structure of X/G. The McKay correspondence does not hold for such Y. We describe the topology of Y using the `Weyl group' of the singular set of X/G. Even in simple examples, this can give many different ways to desingularize X/G. It would be interesting to interpret these results in String Theory, which should lead to a generalization of the idea of orbifold CFT, similar to the idea of `discrete torsion' (Vafa, Witten).Comment: 25 pages, LaTeX, uses packages amstex and amssym

Automorphisms of generalized Thompson groups

We look at the automorphisms of Thompson type groups of piecewise linear homeomorphisms of the real line or circle that use slopes that are integral powers of a fixed integer n with n>2. We show that large numbers of "exotic" automorphisms appear---automorphisms that are represented as conjugation by non-PL homeomorphisms of the real line or circle. This is in contrast to the n=2 case where no such automorphisms appear.Comment: DVI and Post-Script files onl