Diffeomorphism Types Research Articles

On the diffeomorphic type of the complement to a line arrangement in a projective plane

Abstract We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011, 45(2), 137–148].

Classification of $Q$-trivial Bott manifolds

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $CP^1$-bundle starting with a point, where each $CP^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A $Q-trivial Bott manifold$ of dimension $2_n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(CP^1)^n$ with $Q$-coefficients. We find all diffeomorphism types of $Q$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $Z$-coefficients. As a consequence, the number of diffeomorphism classes of $Q$-trivial Bott manifolds of dimension $2_n$ is equal to the number of partitions of $n$. We even show that any cohomology ring isomorphism between two $Q$-trivial Bott manifolds is induced by a diffeomorphism.

DOUBLE KNOT SURGERIES TO S4 AND S2 × S2

It is known that S4 is a union of two fishtails, and S2 × S2 is a union of two cusps (glued along their boundaries). Here we prove that for any choice of knots K, L ⊂ S3 the knot surgery operations [Formula: see text] and S2 × S2 ⇝ (S2 × S2)K, L along both of these fishtails and cusps, respectively, do not change the diffeomorphism type of these manifolds.

Torus manifolds with non-abelian symmetries

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) with (G)-action then the action of a finite covering group of (G) factors through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action. We define invariants of torus manifolds with (G)-action which determine their (\tilde{G}')-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with (G)-action is determined by its admissible 5-tuple up to (\tilde{G})-equivariant diffeomorphism. Furthermore we prove that all admissible 5-tuples may be realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a finite covering group of (\tilde{G}').

M-theory, the signature theorem, and geometric invariants

The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections to the geometry of the underlying manifold, including positive scalar curvature. This results in a variation on the miraculous cancellation formula of Alvarez-Gaum\'e and Witten in 12 dimensions and leads naturally to the Kreck-Stolz $s$-invariant in 11 dimensions. Hence M-theory detects the diffeomorphism type of 11-dimensional (and seven-dimensional) manifolds and in the restriction to parallelizable manifolds classifies topological 11 spheres. Furthermore, requiring the phase of the partition function to be anomaly-free imposes restrictions on allowed values of the $s$-invariant. Relating to string theory in ten dimensions amounts to viewing the bounding theory as a disk bundle, for which we study the corresponding phase in this formulation.

On genus–1 simplified broken Lefschetz fibrations

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas' classification theorem of genus-1 Lefschetz fibrations, which states that the total space of a non-trivial genus-1 Lefschetz fibration over $S^2$ is diffeomorphic to an elliptic surface E(n), for some $n\geq 1$.

A note on the topology of the complements of fiber-type line arrangements in ℂℙ2

We prove that BDiff + (S 2 , {x 1 , ..., x n+1} ) is a K(π, 1) space, where π is the mapping class group of an (n+1)-punctured sphere. As a consequence we derive that the center-projecting braid monodromy of a fiber-type projective line arrangement determines the diffeomorphic type of its complement.

Examples of Einstein manifolds in odd dimensions

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler–Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano Kahler–Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.

Topological structure of candidates for positive curvature

In light of recent advances in the study of manifolds admitting Riemannian metrics of positive sectional curvature, the study of certain infinite families of seven dimensional manifolds has become a matter of interest. We determine the cohomology ring structures of manifolds belonging to these families. This particular ring structure indicates the existence of topological invariants distinguishing the corresponding homeomorphism and diffeomorphism type. We show that all families contain representatives of infinitely many homotopy types.

Diffeomorphism type of six-dimensional cohomogeneity one manifolds

In this article, we give a full diffeomorphism characterization of compact simply connected cohomogeneity one manifolds in dimension 6.

Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres

We construct open symplectic manifolds which are convex at infinity ("Liouville manifolds") and which are diffeomorphic, but not symplectically isomorphic, to cotangent bundles T^*S^{n+1}, for any n+1 \geq 3. These manifolds are constructed as total spaces of Lefschetz fibrations, where the fibre and all but one of the vanishing cycles are fixed. We show that almost any choice of the last vanishing cycle leads to a nonstandard symplectic structure (those choices which yield standard T^*S^{n+1} can be exactly determined). The Corrigendum changes the statement and proof of Lemma 1.1 in the original paper, which corrects our original description of the diffeomorphism type of the manifolds. We also fill a gap in the original proof of Lemma 1.2.

The making of Calabi‐Yau spaces: Beyond toric hypersurfaces

AbstractWhile Calabi‐Yau hypersurfaces in toric ambient spaces provide a huge number of examples, theoretical considerations as well as applications to string phenomenology often suggest a broader perspective. With even the question of finiteness of diffeomorphism types of CY 3‐folds unsettled, an important idea is Reid's conjecture that the moduli spaces are connected by certain singular transitions. We summarize the results of our recent construction of a large class of new CY spaces with small Picard numbers and of their mirrors via conifold transitions and discuss the benefits of other approaches to interesting locations in the web that have been or should be pursued.

Spectral and geometric bounds on 2-orbifold diffeomorphism type

We show that a Laplace isospectral family of two-dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.

The links of 3-dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5-dimensional links of 3-dimensional log-canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ℂ4 such that the order of H2(K) is 6k. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Chern numbers and diffeomorphism types of projective varieties

In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.

COMPACTNESS OF D-ISOSPECTRAL METRICS

We generalise the compactness of isospectral metrics in the C∞-topology modulo gauge equivalence in dimension three to the case for Schrödinger operators with bounded potentials, as well as the finiteness of diffeomorphism types of isospectral manifolds.

The Regeneration of a 5-point

The braid monodromy factorization of the branch curve of a surface of general type is known to be an invariant that completely determines the diffeomorphism type of the surface. Calculating this factorization is of high technical complexity; computing the braid monodromy factorization of branch curves of surfaces uncovers new facts and invariants of the surfaces. Since finding the branch curve of a surface is very difficult, we degenerate the surface into a union of planes. Thus, we can find the braid monodromy of the branch curve of the degenerated surface, which is a union of lines. The regeneration of the singularities of the branch curve, studied locally, leads us to find the global braid monodromy factorization of the branch curve of the original surface. So far, only the regeneration of the BMF of 3,4 and 6-point (a singular point which is the intersection of 3 / 4 / 6 planes) were done. In this paper, we fill the gap and find the braid monodromy of the regeneration of a 5-point. This is of great importance to the understanding of the BMT (braid monodromy type) of surfaces. This braid monodromy will be used to find the global braid monodromy factorization of different surfaces; in particular - the monodromy of the branch curve of the Hirzebruch surface $F_{2,(2,2)}$.

Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differen- tial equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the 2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

On symplectic fillings of lens spaces

Let ξ ¯ st \overline \xi _\textrm {st} be the contact structure naturally induced on the lens space L ( p , q ) = S 3 / Z / p Z L(p,q)=S^3/\mathbb {Z}/p\mathbb {Z} by the standard contact structure ξ st \xi _\textrm {st} on the three–sphere S 3 S^3 . We obtain a complete classification of the symplectic fillings of ( L ( p , q ) , ξ ¯ st ) (L(p,q),\overline \xi _\textrm {st}) up to orientation–preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two–dimensional cyclic quotient singularities.

On a Conformal Gap and Finiteness Theorem for a Class of Four-Manifolds

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and make a comparison of the solutions of the σk equations on a degenerating family of Bach-flat metrics.