Two prominent questions in low dimensional topology are: which knots are slice, and which $\mathbb{Q}$-homology $S^3$'s bound $\mathbb{Q}$-homology $B^4$'s? These questions are connected by a theorem that states if a knot $K$ in $S^3$ is slice, then the 2-fold branch cover of $S^3$ over $K$ bounds a $\mathbb{Q}$-homology $B^4$.
4-manifold topology is characterized by unexpected differences between the smooth and topological categories. For instance, it is the only dimension where there can exist infinitely many manifolds $Y_i$ which are homeomorphic to but not diffeomorphic to $X$. A natural question: how does one construct examples of this phenomenon? In this talk, we focus on the method of reverse engineering, which allows for the construction of “small” exotic 4-manifolds. Surprisingly, symplectic geometry is the main ingredient that makes this approach work!
This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:
Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)
Complex manifolds:
Stein manifolds
Stein compactifications
Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds
Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof.
This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:
Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)
Complex manifolds:
Stein manifolds
Stein compactifications
Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds
One of the first results on concordance was a condition on the Alexander polynomials of slice knots, now known as the Fox-Milnor condition. In this talk, we discuss a generalization of the Fox-Milnor condition to links and their multivariable Alexander polynomials. The main tool is an interpretation of the Alexander polynomials in terms of “Reidemeister torsion”, a notion defined for general manifolds. We will see that the Fox-Milnor condition is a reflection of a certain duality theorem for Reidemeister torsion.
This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:
Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)
Complex manifolds:
Stein manifolds
Stein compactifications
Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds
Taking the double branched cover of $S^3$ over a knot $K$ is natural way to associate $K$ with a 3-manifold, and to study the double branched cover, we often want a Dehn surgery description for it. The Montesinos trick gives a systematic way to get such a description. In this talk, we will go over the broad statement of this trick: that a rational tangle replacement on the knot corresponds to Dehn surgery on the double branched cover. This gives particularly nice descriptions for some satellites of $K$ as surgery on $K \mathrel\# K^r$.
This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:
Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)
Complex manifolds:
Stein manifolds
Stein compactifications
Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds
I'll report on an ongoing project, partly joint work with J. Hillman, aimed at finding criteria for the existence of sections on a given Lefschetz fibration over a surface. We will start by presenting a nice algebraic criterion for the existence of sections in a surface bundle and explain what goes wrong if we try to apply it to the more general Lefschetz fibration case. The question of when a nullhomotopic loop in the boundary of a Lefschetz fibration over the disk can be extended to a section over the whole disk is one such subtle issue.
