On tertiary exotic characteristic classes

We prove the existence and nontriviality of tertiary exotic characteristic classes extending the results of Peterson and Ravenel for secondary exotic classes.

     

On exotic saturated fusion systems

In this paper,we prove that a product F_1×F_2 of saturated fusion systems is exotic if and only if at least one of the factors is exotic.This result provides a method to construct new exotic fusion systems by known exotic fusion systems.We also investigate central products of saturated fusion systems.     

On multiplicative maps

No abstract. November 15, 1999     

On countable-to-one maps

In this paper, it is proved that a space with a point-countable base is an open, countable-to-one image of a metric space, and a quotient, countable-to-one image of a metric space is characterized by a point-countable ℵ₀-weak base. Examples are provided in order to answer negatively questions posed by Gruenhage et al. [G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984) 303-332] and Tanaka [Y. Tanaka, Closed maps and symmetric spaces, Questions Answers Gen. Topology 11 (1993) 215-233].     

On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

On almost one-to-one maps

A continuous map of topological spaces is said to be almost -to- if the set of the points such that is dense in ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

     

On regularly branched maps

Let be a perfect map between finite-dimensional metrizable spaces and p1. It is shown that the space of all bounded maps from X into with the source limitation topology contains a dense G_δ-subset consisting of f-regularly branched maps. Here, a map is f-regularly branched if, for every n1, the dimension of the set is n(dimf+dimY)−(n−1)(p+dimY). This is a parametric version of the Hurewicz theorem on regularly branched maps.     

On f-bi-harmonic maps and bi-f-harmonic maps between Riemannian manifolds

Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects so as to broaden the notions of f-tension field and bi-tension field. We introduce a very large generalization of harmonic maps called f-bi-harmonic maps as the critical points of f-bi-energy functional, and then derive the Euler-Lagrange equation of f-bi-energy functional given by the vanishing of f-bi-tension field.Subsequently, we study some properties of f-bi-harmonic maps between the same dimensional manifolds and give a non-trivial example. Furthermore, we also study the basic properties of f-bi-harmonic maps on a warped product manifold so that we could find some interesting and complicated examples.     

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.     

On equivariant self-similar wave maps

By variational methods, we establish the existence of an infinite number of self-similar wave maps which evolve from smooth, compactly supported data and develop a singularity in finite time.     

On semistability of Albanese maps

In this note, we show that the Albanese map for a smooth projective variety X with numerically effective anticanonical bundle is surjective, equi-dimensional and semistable.