We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka’s sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Matić in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on Giroux’s correspondence that Honda, Kazez and Matić’s contact class is well defined up to isomorphism.

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low-energy effective field theory approximations, and thereby produce a general formula for the group of symmetry protected topological (SPT) phases in terms of Thom’s bordism spectra; the only input is the dimension and symmetry type. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic $10$–fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.

We prove a structure theorem for pseudo-Anosov flows restricted to Seifert-fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration such that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A nonperiodic Seifert-fibered piece is called free. In a previous paper (Geom. Topol. 17 (2013) 1877–1954) we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. Here we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blowup of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. In the proof we introduce almost $k$–convergence groups and prove a convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we carefully define what is a hyperbolic blowup.

We study the generalization of quasipositive links from the $3$–sphere to arbitrary closed, orientable $3$–manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provides the first half of a topological characterization of links in $3$–manifolds which bound complex curves in a Stein filling. Our arguments replace pseudoholomorphic curve techniques with a study of characteristic and open book foliations on surfaces in $3$– and $4$–manifolds.

We show that if $K$ is a fibered ribbon knot in $S^3=\partial B^4$ bounding a ribbon disk $D$, then, given an extra transversality condition, the fibration on $S^3\setminus \nu (K)$ extends to a fibration of $B^4\setminus \nu (D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. More generally, we construct movies of singular fibrations on $4$–manifolds and describe a sufficient property of a movie to imply the underlying $4$–manifold is fibered over $S^1$.

We present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_{\varphi }$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $\varphi$. The symplectic manifold $T_{\varphi }$ is another Weinstein domain and its contact boundary is independent of $\varphi$. We distinguish $T_{\varphi }$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic cohomology groups are the same, as vector spaces, but that are different as Liouville domains. To our knowledge, this is the first example of such pairs that can be distinguished by their wrapped Fukaya category.

Previously, we have suggested a categorical model $M_{\varphi }$ for the wrapped Fukaya category $\mathcal{W}\left(T_{\varphi }\right)$, and we have distinguished $M_{\varphi }$ from the mapping torus category of the identity. We prove $\mathcal{W}\left(T_{\varphi }\right)$ and $M_{\varphi }$ are derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it preludes an algebraic description of wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor products.