abstracts-low

The three properties in the L-space Conjecture (not being an L-space, having left-orderable fundamental group, and supporting a co-orientable taut foliation) have been investigated for cyclic branched covers of several classes of knots and links. We will discuss some results in this direction and some patterns that emerge. This is joint work with Steve Boyer and Ying Hu.

The Borel conjecture predicts that closed, aspherical manifolds (i.e., those with contractible universal cover) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth version of this conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on closed aspherical 4-manifolds.

I’ll discuss an obstruction to a 3-manifold admitting an Anosov flow coming from (Heegaard) Floer homology and use it to produce infinite families of closed hyperbolic 3-manifolds which cannot admit such a flow. This is joint work with Jeremy Van Horn-Morris and Katherine Raoux.

Given a prime 3-manifold M, the L-space conjecture proposes a deep connection between the invariant left-orders on pi_1(M), the taut foliations on M, and its Heegaard Floer homology. In this talk, we’ll focus on the case where the 3-manifolds are toroidal. We will discuss how these three properties are related through the lens of slope detection for knot manifolds: the idea that certain slopes on the torus boundary of a knot manifold can be identified through orders, foliations, or Floer homology. We conjecture that the sets of slopes detected by each of these properties coincide, and explain why this conjecture should be equivalent to the L-space conjecture for toroidal manifolds. We establish the equivalence of the two conjectures for the properties given by foliations and Heegaard Floer homology. Finally, we will discuss what is known about the sets of detected slopes for each property and highlight some open problems. This is joint work with Steve Boyer and Cameron Gordon.

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An outstanding problem in three-dimensional topology is to understand the interplay between hyperbolic geometry and Floer theory, if any. In this talk I will discuss how one can completely describe the monopole Floer theory of certain torsion spin^c three-manifolds with b_1=1 in terms of data arising in hyperbolic geometry; even though these manifolds do not admit irreducible solutions to the Seiberg-Witten equations, they have non-trivial Floer homology arising from Dirac spectral flow. This is joint work with M. Lipnowski.

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I will discuss forthcoming work of myself, Daemi, and Scaduto giving a finite-dimensional formulation of the equivariant instanton complex of a rational homology spheres, and a connected sum theorem for this complex. This enables the computation of Kronheimer and Mrowka’s I#(Y) for more 3-manifolds Y and leads to new homology cobordism invariants, and I will discuss several applications.

Understanding when a manifold has a taut foliation has been a classical problem in 3-manifold theory, and has gained renewed interest due to its connection with the L-space conjecture. One approach to studying this conjecture involves analyzing Dehn surgeries on knots and links. Most existing techniques for constructing taut foliations on Dehn surgeries typically rely on properties of the link exterior, such as fiberedness. In this talk, I will present a different approach that uses purely diagrammatic properties of knots to ensure the existence of coorientable taut foliations on all their non-trivial surgeries. If time permits, I will also discuss further applications to specific families of knots and links.

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We will examine exotica for closed manifolds with cyclic
fundamental group, with special attention to definite examples,especially examples with the same rational homology as the complex projective plane (so called fake projective planes, FPP’s).

Ni and Ghiggini-Spano showed that if $K$ is a fibered knot in a closed oriented 3-manifold $Y$ then the rank of the second-to-top grading of $\widehat{HFK}(Y,K)$ essentially counts the number of fixed points of the monodromy of the fibering. Their proof uses ideas from the equivalence between Heegaard Floer homology and embedded contact homology. In this talk, we will explain a different approach to proving results of this flavor, based on the combinatorial machinery of veering triangulations. Our methods allow us to address periodic points with higher period, while having to restrict to pseudo-Anosov monodromy and specific framings. This is joint work with Antonio Alfieri.

For a link in a 3-manifold, finding and comparing which surgery slopes yield non-L-spaces, taut foliations, or left-orderable fundamental groups is an important topic motivated by the L-space conjecture. Since taut foliations are often closely related to certain associated pseudo-Anosov flows, it is also important to understand how degeneracy slopes relate to the filling slopes where taut foliations can be constructed. In this talk, I will discuss the construction of co-orientable taut foliations in the Dehn fillings of certain pseudo-Anosov mapping tori, where the stable foliation is co-orientable but the map reverses its co-orientation, and the filling slopes are constrained by bounds coming from the degeneracy slopes. In particular, for some Floer simple knot manifolds, we can construct taut foliations in all non-L-space Dehn fillings.

 If L is a link in a 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations? In this talk, I will discuss the ziggurat phenomenon: if one restricts to foliations transverse to a fixed flow on the link complement, the set of multislopes typically has a fractal staircase shape with rational corners. In work in progress with Thomas Massoni, we explain the ziggurat phenomenon in some contexts using tools from contact geometry.