arXiv subjects
Publications and source records attributed to Jin Miyazawa.
We establish a satellite formula for the real Seiberg-Witten Floer homotopy types of knots with odd patterns. Using this, we derive several applications to knot concordance theory. The satellite formula follows from a version of the excision theorem for real Floer homotopy types. Additionally, we show that the concordance invariants arising from real Seiberg-Witten theory depend only on the knot's zero-framed surgery.
We proved that the boundary Dehn twist on the Milnor fiber $M_c(2, q, r)$ is an exotic diffeomorphism relative to the boundary if $q, r$ are odd, coprime integers bigger than $3$ and $(q-1)(r-1)/4$ is an odd number. The proof is given by comparing the family relative Bauer--Furuta invariants of the mapping torus.
We give an infinite family of embeddings of $\mathbb{R} P^2$ to $S^4$ such that they are mutually topologically isotopic however are not smoothly isotopic to each other. Moreover, they are topologically isotopic to the standard $P^2$-knot. To prove that these $P^2$-knots are not smoothly isotopic to each other, we construct a gauge theoretic invariant of embedded surfaces in $4$-manifolds using a variant of the Seiberg--Witten theory, which is called the Real Seiberg--Witten theory.
We develop a version of Seiberg--Witten Floer cohomology/homotopy type for a spin$^c$ 4-manifold with boundary and with an involution which reverses the spin$^c$ structure, as well as a version of Floer cohomology/homotopy type for oriented links with non-zero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3-manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov-type inequalities which relate topological quantities of 4-manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, Nielsen realization problem for non-spin 4-manifolds, and non-smoothable unoriented surfaces in 4-manifolds.
Joyce, Tanaka, and Upmeier give an orientation of the $G$-instanton moduli spaces on a closed four manifolds which is canonically defined using the the $\mathrm{Spin}^c$ structure on the $4$-manifold. In this note, we describe the relation between the orientations given by other choices of the $\mathrm{Spin}^c$ structures, in a slightly more general setting. Furthermore, we give an alternative proof of the orientability of the instanton moduli spaces and an alternative construction of the orientation by Joyce, Tanaka and Upmeier, by using Witten localization of the index of a Dirac type operator.
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.
It is known that the Dirac index of a $Spin^c$ structure is localized to the characteristic submanifold. We introduce the notion of $G^{\pm}(n,s^+,s^-)$ structure on a manifold as a common generalization of the $Spin^c$ structure and the $H_n(s)$ structure defined by D.~Freed--M.~Hopkins, and formulate a version of characteristic submanifold for the $G^{\pm}(n,s^+,s^-)$ structure. We show that the $KO^*(pt)$-valued index associated with the $G^{\pm}(n,s^+,s^-)$ structure is localized to the characteristic submanifold. As an application, we give a topological sufficient condition for the moduli space of $Pin^-(2)$ monopoles to be orientable.