Abstract This note corrects two errors in the paper Symplectomorphisms and spherical objects in the conifold smoothing [Compositio Math. 160 (2024), 2738–2773]. Neither affects the main results of the paper (in particular, neither affects any result stated in the Introduction of the paper). We apologise to our readers for these inaccuracies.

We discuss some examples in which symplectic monodromy (provably or conjecturally) splits off the symplectic mapping class group, hoping to illustrate different techniques and inputs to the arguments. Along the way we formulate several open questions and conjectures. Inter alia, we also construct a compact symplectic six-manifold which contains infinitely many pairwise disjoint Lagrangian 3-spheres.

. For a stably framed Liouville manifold X we [Bordism of flow modules and exact Lagrangians, Preprint] defined a Donaldson-Fukaya category "(X; S) over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from "(X; Z) to "(X; S). Here, we define a spectral Donaldson-Fukaya category for any 'graded tangential pair' Theta Phi of spaces living over BO BU, whose objects are Lagrangians L X for which the classifying maps of their tangent bundles lift to Theta Phi. The previous case corresponded to Theta = Phi = {pt}. We extend our obstruction theory to this setting. The flexibility to 'tune' the choice of Theta and Phi increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory Omega(Theta,Phi),degrees include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum R should exist, which may be of independent interest.

This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disk maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.

We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric products possess suitable idempotents. We re- late the existence of such idempotents to the manifold containing a sequence of Lagrangian links, whose number of components tends to infinity, satisfying a number of properties. Orbifold Floer cohomology for global quotient orbifolds is used axiomatically, and is constructed in a companion paper. We illustrate this strategy by giving a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere. The construction of suitable Lagrangian links in higher dimensions remains an intriguing open problem.

Let $X$ denote the 'conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T<^>*S<^>3$ or, equivalently, the plumbing of two copies of $T<^>*S<^>3$ along a Hopf link. Let $Y$ denote the 'conifold resolution', by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}<^>1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional 'affine $A_1$-case'). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the 'other side' of the mirror.

We construct bulk-deformed orbifold Hamiltonian Floer theory for a global quotient orbifold, that is the quotient of a smooth closed symplectic manifold by a finite group acting faithfully via symplectomorphisms. The moduli spaces define an `ordered marked Flow category', which we equip with a coherent presentation via derived orbifolds. The global charts for orbifold Floer cylinders are built from moduli spaces of holomorphic curves in a quotient of projective space by a free action of the given finite group.

Let X be a graded Liouville domain. Fix a pair of infinite loop spaces Ψ= (Θ→ Φ) living over (BO → BU). This determines a spectral Fukaya category ℱ(X;Ψ) whenever TX lifts to Φ, containing closed exact Lagrangians L for which TL lifts compatibly to Θ; and by Bott periodicity and index theory, a Thom spectrum R with bordism theory R_*. This paper has two main goals: we incorporate rank one spectral local systems ξ: L → BGL_1(R) into the spectral category; and we prove that the bordism class [(L,ξ)] defined by the open-closed map differs from the class [L] by a multiplicative two-torsion element in R^0(L)^× determined by an action of the stable homotopy class of the Hopf map η∈ π_1^st on ξ. Methods include a twisting construction associating flow categories to spectral local systems, and a model for the open-closed map incorporating Schlichtkrull's construction of the trace map BGL_1(R) ⊆ K(R) → R. The companion paper shows that (for Lagrangians which themselves admit spectral lifts) one can lift quasi-isomorphisms from ℤ to Ψ at the cost of introducing rank one local systems. Together with the open-closed computation given here, this gives an essentially complete picture of the bordism-theoretic consequences of quasi-isomorphism in the classical exact Fukaya category.

Let X be a graded Liouville domain. Fix a pair of infinite loop spaces Ψ= (Θ→ Φ) living over (BO → BU). This determines a spectral Fukaya category ℱ(X;Ψ) whenever TX lifts to Φ, containing closed exact Lagrangians L for which TL lifts compatibly to Θ; and by Bott periodicity and index theory, a Thom spectrum R with bordism theory R_*. Suppose that L and K are quasi-isomorphic in the Fukaya category over ℤ. We prove that: (a) if both lift to ℱ(X;Ψ), then there is a rank one R-local system ξ: L → BGL_1(R) over L so that (L,ξ) and K are quasi-isomorphic in the spectral Fukaya category; (b) when X is polarised and Ψ= (BO × F → BO), if only K lifts to ℱ(X;Ψ), then the composition L → B^2GL_1(R) of the stable Gauss map of L and the delooped J-homomorphism is nullhomotopic. Combined with the computation of the open-closed fundamental class associated to (L,ξ) in , these results have applications to bordism and stable homotopy types of quasi-isomorphic Lagrangians, to Hamiltonian monodromy groups, and to smooth structures on nearby Lagrangians. A key ingredient in the proofs is a new form of obstruction theory for flow categories `lying over' a manifold L, closely related to a `spectral Viterbo restriction functor' also introduced here.

For a stably framed Liouville manifold X , we construct a "Donaldson-Fukaya category over the sphere spectrum" F(X; S). The objects are closed exact Lagrangians whose Gauss maps are nullhomotopic compatibly with the ambient stable framing, and the morphisms are bordism classes of framed flow modules over Lagrangian Floer flow categories; this is enriched in modules over the framed bordism ring. We develop an obstruction theory for lifting quasi-isomorphisms in the usual Fukaya category to quasi-isomorphisms in appropriate truncations or quotients of F(X; S). Applications include constraints on the smooth structure of exact Lagrangians in certain plumbings, and the construction of non-trivial symplectic mapping classes which act trivially on the integral Fukaya category for a wide class of affine varieties.

Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory 𝕂 which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum 𝕂 -theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\Theta \to \Phi$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\Theta \to \Phi$. The previous case corresponded to $\Theta = \Phi = \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\Theta$ and $\Phi$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $\Omega^{(\Theta,\Phi),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest.

In , for a stably framed Liouville manifold X we defined a Donaldson-Fukaya category ℱ(X;𝕊) over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from ℱ(X;ℤ) to ℱ(X;𝕊). Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' Θ→ Φ of spaces living over BO → BU, whose objects are Lagrangians L→ X for which the classifying maps of their tangent bundles lift to Θ→ Φ. The previous case corresponded to Θ= Φ= {pt}. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of Θ and Φ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory Ω^(Θ,Φ),∘_*. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum R should exist, which may be of independent interest.

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.

We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi} \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $\pi$, Lalonde, McDuff and Polterovich proved that if $\phi$ lifts to the Hamiltonian group $\mathrm{Ham}(X,\omega)$, then the rational cohomology of $P_{\phi}$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_{\phi}$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. This class of examples includes all complex projective varieties equipped with a smooth morphism to $\mathbb{CP}^1$, in which case the analogous rational result was proved by Deligne using Hodge theory. The argument employs virtual fundamental cycles of moduli spaces of sections of $\pi$ in Morava $K$-theory and results from chromatic homotopy theory. Our proof involves a construction of independent interest: we build global Kuranishi charts for moduli spaces of pseudo-holomorphic spheres in $X$ in a class $\beta \in H_2(X;\mathbb{Z})$, depending on a choice of integral symplectic form $\Omega$ on $X$ and ample Hermitian line bundle over the moduli space of one-pointed degree $d = \langle \Omega,\beta\rangle$ stable genus zero curves in $\mathbb{CP}^d$.

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least 2 whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank 1 local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.