arXiv research

Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.

Explains exceptional surgeries connecting maps and knot orbifolds.

Maps can transform non-homeomorphic manifolds into the same space.

Study the kernel of surgery map restricted to 1-loop part of homology cylinders.

Surgery method proves category inequality for specific manifolds.

For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In ad…

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $rel \partial$ case the surgery on…

Researchers identify graph components for unicellular collections.

The article explores the mapping class group using unicellular maps and provides filtrations.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

The study connects specific circle embeddings to 4-manifold diffeomorphisms.

Paper constructs fold maps with useful singular value sets.

Study exact surgery formula in involutive Heegaard Floer homology.

The paper explores various surgery equivalence relations on 3-manifolds.

Proves an equivariant version of Heegaard Floer link surgery formula.

Study embeddings of manifolds via acyclic maps and surgery.

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of …

Shows Anosov flows with genus one sections, supporting a conjecture.

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topolog…

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_…

Let $X$ be a connected compact 3-manifold with non-empty boundary. Consider the boundary $M$ of $X\times D^2$. $M$ is a 4-dimensional closed manifold and has the same fundamental group as $X$. Various examples of $X$ are known for which a certain assembly map $A:H_4(X;L)\to L_4(π_1(X))$ is injective. For such an $X$ an…

New mapping classes of knotted surfaces are computed via surgery.

New insights into cosmetic surgeries using Heegaard Floer homology.

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $σ^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds o…

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differentia…

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$ can be obtained from $N$ by surgery about a link in $N$ each of whose component…

Proves a formula for instanton Floer homology of knots.

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth-Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of su…

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…

Determines higher smooth surgery structure sets of complex projective spaces.

Computes extendable mapping classes for knotted surfaces in $S^4$.

We compare the star surgery operations introduced in [KS] to the generalized rational blow-down. We show that star surgery shares the properties that make rational blow-down useful for constructions of small exotic symplectic 4-manifolds. Then we show that star surgery operations provide a strictly more general class o…

We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different app…

Constructs infinite families of hyperbolic knots satisfying a volume conjecture.

Study on reducing surgeries on knots, developing thickness and genus bounds.

We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this tr…

A new surgery formula for knot lattice homology.

Geometrically represents L-homology classes using normal maps.

We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery o…

New cobordism maps for instanton knot homology help compute surgeries on knots.

Formula calculates knot Floer complexes for specific cable knots.

The paper introduces a group $LSP$ of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the $LSP$-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery ob…

Let $(M,ξ)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(Σ,Φ)$ supporting $(M,ξ)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,ξ)$ into a supporting open book decomposition. These constructions lead to a r…

Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.

New surgery exact triangles in Heegaard Floer homology for rational slopes.

In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating their source manifolds is fundamental, important and difficult. The author has int…

We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.

We compute the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in the three-sphere. As an application, we show that a knot concordance invariant of Hom can equiv…