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38
Variations Of Moduli Of Parabolic Bundles
 Math. Annalen
, 1994
"... 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 ..."
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Cited by 20 (6 self)
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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
ON THE SCALAR CURVATURE OF EINSTEIN MANIFOLDS
 MATHEMATICAL RESEARCH LETTERS 4, 843–854 (1997)
, 1997
"... We show that there are highdimensional smooth compact manifolds which admit pairs ofEinstein metrics for which the scalar curvatures have opposite signs. These are counterexamples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deforma ..."
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Cited by 12 (3 self)
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We show that there are highdimensional smooth compact manifolds which admit pairs ofEinstein metrics for which the scalar curvatures have opposite signs. These are counterexamples 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.
Einstein metrics on exotic spheres in dimensions 7
"... 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 SasakianEinstein. They also conjectured that all such homotopy spheres in dimension ..."
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Cited by 9 (7 self)
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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 SasakianEinstein. They also conjectured that all such homotopy spheres in dimension 4m − 1, m ≥ 2 admit SasakianEinstein 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 11spheres and 15spheres. Moreover, a program is given that determines the partition of the 8610 deformation classes of SasakianEinstein metrics into the 28 distinct oriented diffomorphism types in dimension 7. 1.
POINCARÉ DUALITY SPACES
"... At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R) ..."
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Cited by 3 (1 self)
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At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R)
Highly connected manifolds with positive Ricci curvature
"... 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, ..."
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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
Alexander polynomials of nonlocallyflat knots
, 2002
"... We generalize the classical study of Alexander polynomials of smooth or PL locallyflat knots to PL knots that are not necessarily locallyflat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification ..."
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Cited by 2 (2 self)
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We generalize the classical study of Alexander polynomials of smooth or PL locallyflat knots to PL knots that are not necessarily locallyflat. 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 lowdimensional case. This classification extends existing classifications for PL locallyflat knots. For knots with higherdimensional 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
ON THE CLASSIFICATION OF CERTAIN PIECEWISE LINEAR AND DIFFERENTIABLE MANIFOLDS IN DIMENSION EIGHT AND AUTOMORPHISMS OF # b i=1 (S2 × S 5) †
, 2002
"... ABSTRACT. In this paper, we will be concerned with the explicit classification of closed, oriented, simplyconnected 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 thes ..."
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Cited by 2 (0 self)
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ABSTRACT. In this paper, we will be concerned with the explicit classification of closed, oriented, simplyconnected 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.
Morse theory for manifolds with boundary, eprint arXiv:1207.3066
, 2012
"... Abstract. We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be m ..."
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Abstract. We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms. 1.
On the asymptotics of Morse numbers of finite covers of manifold , Eprint: math.DG/9810136
 Topology
, 1998
"... Abstract. Let M be a closed connected manifold. We denote by M(M) the Morse number of M, that is, the minimal possible number of critical points of a Morse function f on M. M.Gromov posed the following question: Let Nk, k ∈ N be a sequence of manifolds, such that each Nk is an akfold cover of M whe ..."
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Abstract. Let M be a closed connected manifold. We denote by M(M) the Morse number of M, that is, the minimal possible number of critical points of a Morse function f on M. M.Gromov posed the following question: Let Nk, k ∈ N be a sequence of manifolds, such that each Nk is an akfold cover of M where ak → ∞ as k → ∞. What are the asymptotic properties of the sequence M(Nk) as k → ∞? In this paper we study the case π1(M) ≈ Z m, dimM � 6. Let ξ ∈ H 1 (M,Z), ξ ̸= 0. Let M(ξ) be the infinite cyclic cover corresponding to ξ, with generating covering translation t: M(ξ) → M(ξ). Let M(ξ, k) be the quotient M(ξ)/t k. We prove that limk→ ∞ M(M(ξ, k))/k exists. For ξ outside a subset M ⊂ H 1 (M) which is the union of a finite family of hyperplanes, we obtain the asymptotics of M(M(ξ, k)) as k → ∞ in terms of homotopy invariants of M related to the Novikov homology of M. It turns out that the limit above does not depend on ξ (if ξ / ∈ M). Similar results hold for the stable Morse numbers. Generalizations for the case of noncyclic coverings are obtained. Introduction and the statement of the result Let M be a closed connected smooth manifold. Denote by M(M) the Morse number of