arXiv subjects
Publications and source records attributed to Hokuto Konno.
We give a generalized Thurston--Bennequin-type inequality for links in $S^3$ using a Bauer--Furuta-type invariant for 4-manifolds with contact boundary. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth 4-manifold whose non-equivariant Bauer--Furuta invariant is non-zero.
We establish a version of Seiberg--Witten Floer $K$-theory for knots, as well as a version of Seiberg-Witten Floer $K$-theory for 3-manifolds with involution. The main theorems are 10/8-type inequalities for knots and for involutions. The 10/8-inequality for knots yields numerous applications to knots, such as lower bounds on stabilizing numbers and relative genera. We also give obstructions to extending involutions on 3-manifolds to 4-manifolds, and detect non-smoothable involutions on 4-manifolds with boundary.
We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod $2$ relations between these invariants and the Chern classes of the spin$^c$ index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a $4$-manifold $X$: they obstruct the existence of certain families of $4$-manifolds with fibres diffeomorphic to $X$. As a concrete geometric application, we shall detect a non-smoothable family of $K3$ surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce $K$-theoretic Seiberg-Witten invariants and give a formula expressing the Chern character of the $K$-theoretic Seiberg-Witten invariants in terms of the cohomological Seiberg-Witten invariants. This leads to new divisibility properties of the families Seiberg-Witten invariants.
We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.
We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.
We show $ 10/8$-type inequalities for some end-periodic $4$-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed $4$-manifolds which do not admit positive scalar curvature metrics.
We determine the local equivalence class of the Seiberg-Witten Floer stable homotopy type of a spin rational homology 3-sphere $Y$ embedded into a spin rational homology $S^{1} \times S^{3}$ with a positive scalar curvature metric so that $Y$ generates the third homology. The main tool of the proof is a relative Bauer-Furuta-type invariant on a periodic-end 4-manifold. As a consequence, we give obstructions to positive scalar curvature metrics on spin rational homology $S^{1} \times S^{3}$, typically described as the coincidence of various Frøyshov-type invariants. This coincidence also yields alternative proofs of two known obstructions by Jianfeng Lin and by the authors for the same class of 4-manifolds.
We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold $X$ with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants $α, β, γ$ of $\partial X$, the inclusion map $\mathrm{Diff}(X,\partial) \hookrightarrow \mathrm{Homeo}(X,\partial)$ between the groups of diffeomorphisms and homeomorphisms which fix the boundary pointwise is not a weak homotopy equivalence. This combined with a classical result in dimension 3 implies that the inclusion map $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ is also not a weak homotopy equivalence under the same assumption on $\partial X$. Our constraints generalize both of constraints on smooth families of closed 4-manifolds proven by Baraglia and a Donaldson-type theorem for smooth 4-manifolds with boundary originally due to Frøyshov.
We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 4-manifolds.
Using the Seiberg-Witten monopole equations, Baraglia recently proved that for most of simply-connected closed smooth $4$-manifolds $X$, the inclusions $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ are not weak homotopy equivalences. In this paper, we generalize Baraglia's result using the $\mathrm{Pin}^{-}(2)$-monopole equations instead. We also give new examples of $4$-manifolds $X$ for which $π_{0}(\mathrm{Diff}(X)) \to π_{0}(\mathrm{Homeo}(X))$ are not surjections.
We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map $π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which there is no splitting of the map $Diff(K3) \to π_{0}(Diff(K3))$, but there is a splitting of $Homeo(K3) \to π_{0}(Homeo(K3))$ over the image of $G$ in $π_{0}(Homeo(K3))$, which is non-trivial. The third is that the map $π_{1}(Diff(K3)) \to π_{1}(Homeo(K3))$ is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for $K3$ surfaces.
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations. The other variant is an equivariant Seiberg-Witten invariant of smooth group actions. We consider several applications of the gluing formula including: obstructions to smooth isotopy of diffeomorpihsms, computation of the mod $2$ Seiberg-Witten invariants of spin structures, relations between mod $2$ Seiberg-Witten invariants of $4$-manifolds and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group actions on $4$-manifolds.
We construct characteristic classes of 4-manifold bundles using $SO(3)$-Yang-Mills theory and Seiberg-Witten theory for families.
For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction inequalities for surfaces embedded in $m\mathbb{CP}^2\# n(-\mathbb{CP}^2)$ ($m, n \geq 2$). The proofs of these results are given by studying a family of Seiberg-Witten equations.