Hi, I'm Nick Meyer

Torus surgery provides a flexible way to construct new 4-manifolds, but it generally destroys geometric structures present before surgery. In this talk, we study when fibrations over the circle survive torus gluings. We show that if two compact oriented 4-manifolds are diffeomorphic to $S^1\times E_Y(K)$ for a fibered knot $K$, or more generally are $\Sigma_{g,1}$-bundles over $T^2$, then any orientation-preserving gluing of their $T^3$ boundaries produces a 4-manifold that fibers over $S^1$. The key ingredient is the flexibility of fibrations of $T^3$. We discuss applications to torus surgery, including generalized Fintushel–Stern knot surgery.

Slides.

Torus surgery is a fundamental operation in $4$-manifold topology, generalizing $3$-dimensional Dehn surgery to dimension $4$ and underpinning constructions such as Fintushel-Stern knot surgery. A natural question is how this operation interacts with fibered structures. In particular, we partially answer the question “When does drilling-and-filling a $4$-manifold along an embedded $2$-torus preserve the existence of a fibration?”

In this talk, we develop a framework for extending circle fibrations of a $3$-torus boundary across $4$-manifolds of the form $S^1 \times E_Y(K)$, where $K$ is a fibered knot in a closed, oriented $3$-manifold $Y$, as well as across oriented $\Sigma_{g,1}$-bundles over $T^2$. The key mechanism is the rigidity of essential tori in $T^3$, which forces boundary fibrations to extend uniquely under natural geometric conditions.

As applications, we show that broad families of $4$-manifolds obtained by torus surgery admit fibrations over the circle. We also discuss consequences for torus surgeries in $S^1 \times S^3$ and structural constraints on $4$-manifolds containing “torus fibered torus knots” (a generalization of a fibered knot in a $3$-manifold).

Geometric topology is a subfield of topology that studies manifolds and maps between manifolds. Originating in roughly 1935, the past eighty-nine years have seen unprecedented growth and development, especially in low-dimensional topology, which studies nice shapes, called manifolds, in dimensions one through four.

In this talk, I will give an overview of one tool used to study topology in low dimensions, Morse theory. Particular focus will be given to Morse theory in dimensions one and two, which is essentially the study of critical points of functions of the form $y=f(x)$ and $z=f(x,y)$.

No background higher than Calculus I will be assumed, but this talk will be most accessible to those who have seen some Multivariable Calculus (Calculus III).

The Hopf surface is diffeomorphic to $S^1\times S^3$, and fibers over $S^2$ with fiber $T^2$. The result of doing torus surgery along an embedded $2$-torus $T$ of self-intersection zero in a closed $4$-manifold $X$ is the $4$-manifold $X_\varphi(T)$ obtained by removing a regular neighborhood of $T$ and replacing it with a copy of $T^2\times D^2$ along some diffeomorphism $\varphi$ of the resulting $2$-torus boundary.

Zentner showed that if the result of doing torus surgery along two fibers of the Hopf surface results in a homology Hopf surface, then the result is diffeomorphic to the standard Hopf surface. We extend these results to show that every manifold obtained by performing torus surgery along two fibers of the Hopf surface is diffeomorphic to $Y \times S^1$ where $Y$ is a $3$-manifold obtained by Dehn surgery along the unknot in $S^3$.

In this talk, I construct the Casson-Gordon signature invariants of a knot K. In doing so, we will take a voyage through an ocean of abstract nonsense, touching on twisted homology, cobordism groups, representation theory, and more!

Time permitting, we will also see how a particularly interesting subset of these 3-dimensional invariants were triviallized by the groundbreaking work in 4-dimensions of Micheal Freedman.

In this talk, I will discuss Heegaard splittings of 3-manifolds and trisections of 4-manifolds. These decompositions cut the manifold into 1-handlebodies of appropriate dimension whose common intersection is a closed, orientable surface. By studying these decompositions, we obtain lots of information about the manifolds that they determine. This talk is based in part on David Gay’s “From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4.”

YouTube.