Nick Meyer
Assistant Professor of Mathematics at Winona State University
I am an Assistant Professor of Mathematics at Winona State University. I earned my PhD from University of Nebraska–Lincoln, where I was advised by Alex Zupan. My research lies broadly in geometric and algebraic topology, with a primary focus on the topology of 3- and 4-manifolds.
My research centers on knot theory in dimensions three and four, Heegaard splittings of 3-manifolds, and trisections of 4-manifolds. I am particularly interested in how these structures encode and reflect subtle geometric and algebraic information, and in how different decomposition techniques interact across dimensions. Alongside theoretical questions, I am drawn to computational approaches, especially computational algebraic topology and computational knot theory, as tools for experimentation, conjecture-building, and visualization.
At Winona State University, I teach a variety of undergraduate mathematics courses and contribute to the department’s mission of preparing students for careers, graduate study, and lifelong engagement with mathematics. I strive to create a learning environment that emphasizes conceptual understanding, clear communication, and mathematical reasoning. I view teaching and research as complementary pursuits, and I am committed to mentoring students as they develop mathematical maturity, engage in mathematical inquiry, and explore opportunities both within and beyond the classroom.
A copy of my CV is available at this link.
Education
2018-2024 PhD in Mathematics | ||
2018-2020 MS in Mathematics | ||
2015-2018 BS in Mathematics; Minor: Physics |
Publications
Pants distances of knotted surfaces in $4$-manifolds
We define a pants distance for knotted surfaces in 4-manifolds which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the (4-manifold, surface) pair has a simple topology. Furthermore, we calculate the exact values of our invariants for many new examples such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.
Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds
Suppose that $W$ and $W’$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S^1$ times the exterior $E_Y(K)$ of a fibered knot $K$ in a closed, connected, orientable $3$-manifold $Y$, or are diffeomorphic to $\Sigma_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f: \partial W’ \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W’$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W’ = T^2 \times D^2$ and $W = S^1 \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S^1 \times K$ is a $4$-manifold that fibers over $S^1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S^1$ times the unknot $\mathcal{U}$ in $S^1 \times S^3$ is diffeomorphic to $S^1$ times a lens space.
Torus Surgery, Fibrations, Multisections, and Spun $4$-Manifolds
A compact $n$-manifold $X$ is fibered if it is a fiber bundle where the fiber $F$ and base space $B$ are manifolds. Fibered manifolds are particularly nice, as they are essentially classified by their monodromy maps. Two common examples of 4-dimensional fibered manifolds are surface bundles over surfaces and 3-manifold bundles over the circle.
The main focus of this dissertation is to investigate fibered 4-manifolds whose boundaries are the 3-torus and how these manifolds glue together to give new closed, fibered 4-manifolds. In particular, suppose W is diffeomorphic to $S^1\times E_Y(K)$ where $Y$ is a closed, oriented 3-manifold and $K$ is a fibered knot in $Y$, or that $W$ is diffeomorphic to a $\Sigma_{g,1}$-bundle over the torus, and let $W’$ be defined similarly. If $f:\partial W’ \to \partial W$ is an orientation-preserving diffeomorphism of the $T^3$ -boundary, we have that $X = W \cup_f W’$ fibers over the circle. We also study spun 4-manifolds and construct 4-secting Morse 2-functions on these manifolds. Suppose that $Y$ is a compact, oriented, connected 3-manifold with connected boundary $F = \partial Y$ and that $f:F \times S^1 \to F \times S^1$ is an orientation-preserving diffeomorphism. Then, we show that the $f$-spin of $Y$ admits a $(2g - h; g)$ 4-section if $h \neq -1$ or if $h = 1$ and $f$ is isotopic to the identity, where $h$ is the genus of $F$ and $g$ is the Heegaard genus of $Y$. This generalizes the work of Meier on trisections of spun 4-manifolds and of Kegel and Schmäschke on trisections of $4$-dimensional open book decompositions.
Tri-plane diagrams for simple surfaces in $S^4$
Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.
Students Advised
Parker Smiley
Master’s Thesis: Vanishing Obstructions to Sliceness of Weaving Links Independent Study: Topology II (Algebraic Topology)
Bethany Harms
Master’s Project: Knot and Link Polynomials
Service
Reviewer for Algebraic & Geometric Topology
Serves as a peer reviewer for Algebraic & Geometric Topology, contributing to the evaluation and advancement of research in low-dimensional topology and related fields.
Member of MAA Committee for Minority Participation in Mathematics (CMPM)
Member of the Mathematical Association of America CMPM, supporting national efforts to advance diversity, equity, and inclusion in mathematics.
Vice Chair of Missouri Section of the MAA
Assisted the Chair in overseeing section activities.
Selected Talks
Telling the Warp from the Weft — Fibering and Surgery in Dimension 4
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.
Telling the Warp from the Weft — Fibering and Surgery in Dimension 4
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).
Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds
Morse Theory in Dimensions One and Two
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).
Torus Surgeries on the Hopf Surface
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 $S^3$.
Group Deficiencies from a $4$-Manifold Perspective
The Casson-Gordon Signature Invariant and Sliceness Obstructions
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.
Meier-Zupan Square Links and the Andrews-Curtis Conejecture
A Quest for Residual Finiteness: Geometrization and the Word Problem for 3-Manifolds
A Geometric Introduction to Heegaard Splittings and Trisections
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.”