arXiv subjects

Nickolas A. Castro

Publications and source records attributed to Nickolas A. Castro.

6 recordsLinked to original sources

arXiv2023

Slice obstructions from genus bounds in definite 4-manifolds

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.

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arXiv2020

The relative $\mathcal{L}$-invariant of a compact $4$-manifold

In this paper, we introduce the relative $\mathcal{L}$-invariant $r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\mathcal{L}$-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology ball, then $r\mathcal{L}(X)=0$ if and only if $X\cong B^4$. In order to better understand relative trisections, we also produce an algorithm to glue two relatively trisected 4-manifold by any Murasugi sum or plumbing in the boundary, and also prove that any two relative trisections of a given 4-manifold $X$ are related by interior stabilization, relative stabilization, and the relative double twist, which we introduce in this paper as a trisection version of one of Piergallini and Zuddas's moves on open book decompositions. Previously, it was only known (by Gay and Kirby) that relative trisections inducing equivalent open books on $X$ are related by interior stabilizations.

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arXiv2018

Trisections of 4-manifolds with Boundary

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.

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arXiv2017

Trisections of 4-manifolds via Lefschetz fibrations

We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold which admits a Lefschetz fibration over $S^2$ equipped with a section of square $-1$, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed $4$-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented $S^2$-bundle over any closed surface and in particular we draw the corresponding diagrams for $T^2 \times S^2$ and $T^2 \tilde{\times} S^2$ using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed $4$-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.

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arXiv2017

Diagrams for Relative Trisections

We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the $4$--dimensional analog of a sutured Heegaard diagram for a sutured $3$--manifold. We also give many foundational examples.

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arXiv2017

Trisecting Smooth 4-dimensional Cobordisms

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected $4$-manifolds can be glued to one another along diffeomorphic boundary components so as to induce a trisected manifold. These two results allow us to define a category $\textrm{Tri}$ whose objects are smooth, closed, oriented $3$-manifolds equipped with open book decompositions, and morphisms are relatively trisected cobordisms. Additionally, we extend the Hopf stabilization of open book decompositions to a relative stabilization of relative trisections.

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