this paper, we study the moduli spaces of semistable parabolic bundles of arbitrary rank over a smooth curve X with marked points in a finite subset P of X. A parabolic bundle

We show that there are high-dimensional smooth compact manifolds which admit pairs ofEinstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.

Abstract. In a recent article the first three authors proved that in dimension 4m + 1 all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension 4m − 1, m ≥ 2 admit Sasakian-Einstein metrics [BGK03], and proved this for the simplest case, namely dimension 7. In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8610 deformation classes of Sasakian-Einstein metrics into the 28 distinct oriented diffomorphism types in dimension 7. 1.

At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological n-manifold X n satisfy the relation bi(X): = dimRHi(X; R)

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomials that is complete except for one special low-dimensional case. This classification extends existing classifications for PL locally-flat knots. For knots with higher-dimensional singularities, we further extend the necessary conditions on the invariants. We also construct several varieties of singular knots to demonstrate realizability of certain families of polynomials as generalized Alexander polynomials. These constructions, of independent interest, generalize known knot constructions such as

We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k–fold connected sum of S 2n 1 S 2n admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for computing the diffeomorphism types is given and tables are presented for dimensions 7 and 11. 53C25; 57R55

ABSTRACT. In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan Müller. In order to understand the structure of these manifolds, we will analyze their minimal handle presentations and describe explicitly to what extent these handle presentations are determined by the cohomology ring and the characteristic classes. It turns out that the cohomology ring and the characteristic classes do not suffice to reconstruct a manifold of the above type completely. In fact, the group Aut0 # b i=1 (S2 ×S5) ) ( /Aut0 # b i=1 (S2 ×D6) ) of automorphisms of # b i=1 (S2 ×S5) which induce the identity on cohomology modulo those which extend to # b i=1 (S2 × D6) acts on the set of oriented homeomorphy classes of manifolds with fixed cohomology ring and characteristic classes, and we will be also concerned with describing this group and some facts about the above action. 1.

Abstract not found

Let f: S p × S q × S r smooth embedding. → Sp+q+r+1, 2 ≤ p ≤ q ≤ r, be a In this paper we show that the closure of one of the two components of Sp+q+r+1 − f(S p × Sq × Sr), denoted by C1, is diffeomorphic to Sp × Sq × Dr+1 or Sp × Dq+1 × Sr or Dp+1 × Sq × Sr, provided that p + q ̸ = r or p + q = r with r even. We also show that when p + q = r with r odd, there exist infinitely many embeddings which do not satisfy the above property. We also define standard embeddings of S p × S q × S r into S p+q+r+1 and, using the above result, we prove that if C1 has the homology of S p ×S q, then f is standard, provided that q<r. 1. Introduction. In [A], Alexander has shown that a piecewise linearly embedded torus in the

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental