Susumu Hirose
A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
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The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign of the spin numbers to the existence of a periodic diffeomorphism of odd prime order acting trivially on cohomology of homotopy K3 surfaces.
For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.
We consider two pairs: the standard unknotted $n$-sphere in $S^{n+2}$, and the product of two $p$-spheres trivially embedded in $S^{2p+2}$, and study orientation preserving diffeomorphisms of these pairs. Pseudo-isotopy classes of such diffeomorphisms form subgroups of the mapping class groups of $S^n$ and $S^p\times S^p$ respectively and we determine the algebraic structure of such subgroups when $n>4$ and $p>1$.
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples includ…
An orientation preserving diffeomorphism over a surface embedded in a 4-manifold is called extendable, if this diffeomorphism is a restriction of an orientation preserving diffeomorphism on this 4-manifold. In this paper, we investigate conditions for extendability of diffeomorphisms over surfaces in the complex projective plane.
We show that symplectically embedded $(-1)$-tori give rise to certain elements in the symplectic mapping class group of $4$-manifolds. An example is given where such elements are proved to be of infinite order.
In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range.
By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact connected subsurface of a closed surface can be extended to a diffeomorphism of the whole surface acting trivially on its homology. The special case of extensions by the identity diffeomorphism was considered by A. Putman, whose work was one of the main sources of inspiration for the present paper. It turns out that the answer depends on the position of the s…
Pseudo-Isotopy and Diffeomorphisms of the 4-Sphere I: Loops of Spheres
引入伪同伦和嵌入空间理论的新方法，构造不变量检测 $S^2 \ imes S^2$ 及其连通和中无法同伦为标准水平球面对偶球面环的非平凡2-球面嵌入环。
Let $K, K'$ be ribbon knottings of $n$-spheres with $1$-handles in $S^{n+2}$, $n\geq 2$. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and Nakagawa, after taking a finite number of connected sums with trivially embedded copies of $S^{n-1}\times S^{1}$.
We establish the existence of a pair of exotic surfaces in a punctured $K3$ which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer--Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the $S^{1}$-equivariant family Bauer--Furuta invariant of any orientation-preserving diffeomorphism on $S^{4}$ is trivial and that the $\mathrm{Pin}(2)$-equivaria…
Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins
本文通过构造从$S^2 \ imes S^2$中2-球面环路群到4-球面光滑映射类群的满同态，并利用Montesinos孪生对上的扭转给出显式生成元，推进了对4-球面光滑映射类群的理解。
Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold $M$, on the linking form associated to the homology groups of $M$. In particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theo…
We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds. In contrast, we also show that there are smoothly embedded 2-spheres that are both equivalent and topologically isotopic, but not smoothly isotopic.
A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in the 3-sphere to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.
An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds
本文提出一个伴随不等式障碍，用于判断4-流形中嵌入曲面在微分同胚下是否同痕，并据此构造了无限多个连续同痕但彼此不光顺同痕的嵌入曲面例子。
If a finite group of orientation-preserving diffeomorphisms of the 3-dimensional torus leaves invariant an oriented, closed, embedded surface of genus g>1 and preserves the orientation of the surface, then its order is bounded from above by 12(g-1). In the present paper we classify (up to conjugation) all such group actions and surfaces for which the maximal possible order 12(g-1) is achieved, and note that the unknotted surfaces can be realized by equivariant minimal surfaces in a 3-torus.
We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.
Exotic embedded surfaces and involutions from Real Seiberg-Witten theory
本文扩展了Miyazawa的构造，研究了4流形上的反演及非定向曲面的奇异嵌入，利用实Seiberg-Witten理论检测奇异性质，并证明了相关不变量的基本结果。
Torus knot filtered embedded contact homology of the tight contact 3-sphere
本文推广了结过滤嵌入接触同调以处理具有有理旋转数的退化结，并利用 Morse-Bott 方法和 Seiberg-Witten 理论计算了紧接触 3-球面上 T(2,q) 环面结的过滤嵌入接触同调，为后续正 T(p,q) 结计算及 Reeb 动力学应用奠定基础。
Stabilisation, scanning and handle cancellation
本文通过关联圆盘微分同胚群、球面余维1嵌入球面空间与余维2平凡纽结空间的同伦型，结合消去把柄、扫描映射及嵌入演算框架，证明Schoenflies球面幺半群关于连通和构成群，该结果在四维情形意义独特。
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