arXiv research

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…

Study of 2-knots via trisection diagrams, showing unique descriptions and operations.

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

New surgery operation preserves monotonicity of Lagrangians.

Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.

We discuss some consequences Fintushel-Stern `knot surgery' operation on 4-manifolds coming from its handlebody description. We give some generalizations of this operation and give a counterexample to their conjecture.

Study curves on surfaces with surgeries connecting them.

Study shows flexibility of homology groups of Reeb spaces of fold maps through surgery operations.

Suspensions of manifolds by circle surgeries are key in free action constructions.

Study of unchaining surgery operation in symplectic 4-manifolds.

The paper introduces new contact structure modifications via round surgery.

Spaces of metrics with invertible Dirac operator are homotopy equivalent for cobordant manifolds.

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…

We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $…

Bi-contact surgery operations can be applied to Anosov flows.

We investigate the operation of torus surgery on tori embedded in $S^4$. Key questions include which 4-manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3-manifolds into 4-manifolds by viewing Dehn surgery as a cross section of a surgery on a su…

Band surgery affects knot signatures by 0 or 8.

Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formulæfor several geometric and spectral invariants associated to a Dir…

Study uses instanton Floer theory to obstruct knot unknotting operations.

Study bounds Urysohn width of manifolds under surgeries.

We give a simple criterion for a pointwise curvature condition to be stable under surgery. Namely, a curvature condition $C$, which is understood to be an open, convex, O(n)-invariant cone in the space of algebraic curvature operators, is stable under surgeries of codimension at least $c$ provided it contains the curva…

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary…

New findings on $L$-spaces and taut foliations in hyperbolic links.

Classifies $C_p$-surfaces using equivariant surgery methods.

Disk surgery on primitive disks of genus-3 Heegaard splittings of 3-sphere yields no primitive disks.

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…

Contact round surgery proves existence of contact structures on 3-manifolds.

We introduce the concept of `claspers,' which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called `C_k-equivalence,' which is generated by surgery operations of a certain kin…

We discuss the relation between Fintushel-Stern knot surgery operation on 4-manifolds and Scharlemann manifolds, and as a corollary show that they all are standard. Along the way we show that the fishtail can exotically knot in the 4-sphere infinitely many ways.

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

Khovanov homology remains unchanged by certain surface surgeries.

Kawauchi defined a group structure on the set of homology $S^1$\times$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injecti…

New 3-manifolds bound rational 4-balls through specific operations.

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for…

Paper constructs fold maps with useful singular value sets.

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We…

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.

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

Study shows bounds on symplectic fillings for spinal open book decompositions.

We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…

We review the work of the authors and their collaborators on the decomposition of the zeta-determinant of the Dirac operator into the contribution coming from different parts of a manifold.

Study of fold maps and Reeb spaces via surgery operations.

Paper explores constructing knots with identical traces using specific operations.

Proves cosmetic surgery conjecture for strongly invertible knots.

Researchers identify graph components for unicellular collections.

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

Geometric surgery theory applied to quantum statistics in spacetime.

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well known link projections. This gives a method of enumerating atoroidal graphs and he…