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 multiplicative maps

No abstract. November 15, 1999     

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 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 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 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 certain K-equivalent birational maps

Mathematische Zeitschrift - A simple birational map is a K-equivalent birational map which is resolved by a single blowing-up. Examples of such maps include standard flops and twisted Mukai flops....     

On open real polynomial maps

We study open polynomial maps from ⁿ to ^p. For n = p we give a complete characterization, and for p = 2, n ≥ 3 we obtain some partial information.     

On exact atomless Milutin maps

We treat the notion of fiberwise measures as the natural expansion of measures from spaces onto continuous maps. The topological characterizations of maps admitting fiberwise measures with special properties—atomless and exact atomless Milutin maps—are given.     

On almost periodic multivalued maps

We consider multivalued maps with compact images in a complete metric spaceU. Conditions on the set of selections of a multivalued mapF are obtained which are necessary and sufficient for the multivalued mapF to be Stepanov almost periodic. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 82–90, July, 2000.     

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.