Related papers: Heegaard genus formula for Haken manifolds

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. We classify the mapping class groups of such Heegaard splittings in the case when the surface bundle has a sufficiently complicated monodromy map.

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold $M$ of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe…

The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the…

Let $Y_1$ be a closed, oriented 3-manifold and $\Sigma$ denote a non-separating closed, orientable surface in $Y_1$ which consists of two connected components of the same genus. By cutting $Y_1$ along $\Sigma$ and re-gluing it using an…

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible…

In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We…

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface…

We study solutions to the $\alpha$-SQG equations, which interpolate between the incompressible Euler and surface quasi-geostrophic equations. We extend prior results on existence of bounded patches, proving propagation of $H^k$-regularity…

A para-K\"ahler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $ K^2 = \mathrm{Id} $ or, equivalently,…

We define a Heegaard-Scharlemann-Thompson (HST) splitting of a 3-manifold M to be a sequence of pairwise-disjoint, embedded surfaces, {F_i}, such that for each odd value of i, F_i is a Heegaard splitting of the submanifold of M cobounded by…

Following work of Jaco and Rubinstein (2006), which (non-constructively) proved that any 3-manifold admits a one-vertex layered triangulation, we present an algorithm, with implementation using Regina, that uses a combinatorial presentation…

The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class…

We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with…

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…

We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove…

We show that if a closed hyperbolic 3-manifold has infinitely many finite covers of bounded Heegaard genus, then it is virtually fibered. This generalizes a theorem of Lackenby, removing restrictions needed about the regularity of the…

Idoneal genera are a generalization of Euler's idoneal numbers. We enumerate all idoneal genera by means of the Smith--Minkowski--Siegel mass formula. As an application, we classify transcendental lattices of K3 surfaces covering an…

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes…

When a Dehn filled link manifold contains a geometrically incompressible one-sided surface, it is shown there is a unique boundary incompressible position that the surface can take in the link space. The proof uses a version of the…

For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) = Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard splitting (M,F). Its path components are the isotopy classes of Heegaard splittings…