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Robion Kirby

Publications and source records attributed to Robion Kirby.

12 recordsLinked to original sources
arXiv2023

A group-theoretic framework for low-dimensional topology

A correspondence, by way of Heegaard splittings, between closed oriented 3-manifolds and pairs of surjections from a surface group to a free group has been studied by Stallings, Jaco, and Hempel. This correspondence, by way of trisections, was recently extended by Abrams, Gay, and Kirby to the case of smooth, closed, connected, oriented 4-manifolds. We unify these perspectives and generalize this correspondence to the case of links in closed oriented 3-manifolds and links of knotted surfaces in smooth, closed, connected, oriented 4-manifolds. The algebraic manifestations of these four subfields of low-dimensional topology (3-manifolds, 4-manifolds, knot theory, and knotted surface theory) are all strikingly similar, and this correspondence perhaps elucidates some unique character of low-dimensional topology.

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arXiv2019

Trisections and link surgeries

We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby. These links lie on Heegaard surfaces in $\#^j S^1 \times S^2$ and have surgeries yielding $\#^k S^1 \times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.

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arXiv2016

Group trisections and smooth 4-manifolds

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group theoretic. For example, the smooth 4-dimensional Poincaré conjecture can be reformulated as a purely group theoretic statement.

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arXiv2014

Indefinite Morse 2-functions; broken fibrations and generalizations

A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions mapping to arbitrary compact, oriented surfaces. "Uniqueness" means there is a set of moves which are sufficient to go between two homotopic indefinite Morse 2-functions while remaining indefinite throughout. We extend the existence and uniqueness results to indefinite, Morse 2-functions with connected fibers.

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arXiv2013

Trisecting 4-manifolds

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold $X$ arises from a Morse 2--function $G:X \to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3--manifold $Y$ arises from a Morse function $g : Y \to B^1$ and the obvious bisection of $B^1$.

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arXiv2012

Cohomotopy sets of 4-manifolds

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.

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arXiv2012

Reconstructing 4-manifolds from Morse 2-functions

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.

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arXiv2011

Fiber connected, indefinite Morse 2-functions on connected n-manifolds

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite". This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.

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arXiv2007

Constructing Lefschetz-type fibrations on four-manifolds

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4-manifolds without 3- and 4-handles, such as the homotopy 4-spheres arising from nontrivial balanced presentations of the trivial group.

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arXiv2004

Constructing symplectic forms on 4-manifolds which vanish on circles

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.

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arXiv1999

The E_8-manifold, singular fibers and handlebody decompositions

The E_8-manifold has several natural framed link descriptions, and we give an efficient method (via `grapes') for showing that they are indeed the same 4-manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we show how the degeneration of a regular fiber to a singular fiber in an elliptic surface provides rich examples of Gromov's compactness theorem.

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