Andrea Pantaleoni, Riccardo Piergallini
We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_g$, for any genus $g$. A complete classification of such involutions is given in terms of their fixed points.
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Applying Morse theory, we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies. We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form. Our algorithm is practical. Several examples are given to test the algorithm.
A note on some high-dimensional handlebodies
本文研究高维 $k$-环柄体的乘积结构，并引入 $(n,k)$-Kirby 图以推广四维 $2$-环柄体的经典 Kirby 图表示。
This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deformations (handle slides and creations/cancellations of handles of index <= 2). We factor this bijective correspondence through a map onto the closed morphisms in a universal braided category freely generated by a Hopf algebra object H. In this way we obtain a complete algebraic desc…
For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ``hyperelliptic'' three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the {\sl Weierstrass lines} of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierst…
We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini's category $4\mathrm{HB}$ of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathcal{C}$, which is not required to be semisimple. The main example of target category is provided by $H$-mod, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\mathrm{HB}$ is freely generated, as a braided monoidal category, by a BPH algebra (short for Bobtcheva-Piergallini Hopf algebra), and this is sent by the Kerler-Lyubashenko functo…
On diffeomorphisms of 4-dimensional 1-handlebodies
本研究给出四维1-柄体边界微分同胚可延拓至整个柄体的Laudenbach-Poénaru定理的新证明，将其推广到特定四维压缩体，并应用于紧致四维流形相对三分图研究。
It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove…
Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are similar choices with X closed and hyperbolic.
We define an action of Artin's braid group on a finite dimensional algebra.
Knotted handlebodies in the 4-sphere and 5-ball
本文构造了$S^4$中具有相同边界与切割系统但不同痕的3维1-柄体，证明了Budney-Gabai关于亏格至少2的猜想。
Extending families of homeomorphisms over 4-dimensional handlebodies
本文证明了对于所有亏格g≥0，由限制到边界诱导的映射B Homeo(H_g) → B Homeo(U_g)存在截面。
Invariants of 4-Dimensional 2-Handlebodies from the Temperley-Lieb Category in Positive Characteristic
本文研究了在正特征下，基于Temperley-Lieb范畴的四维2柄体不变量，探讨其在特定条件下的消失性质及对exotic平滑结构的检测潜力。
Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies
本文通过单模ribbon范畴构建4维2柄体中ribbon表面的量子不变量，并恢复多个已有不变量，同时推广Reshetikhin-Turaevfunctor到标记Kirby图类，产生带框架链接的不变量。
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Ap…
On the mapping class group of 4-dimensional 1-handlebodies via Budney-Gabai invariants
本文通过Budney-Gabai不变量研究四维1-柄体的映射类群，揭示了微分同胚群中线性无关元素的存在及无限生成子群的结构。
In this paper we consider all orientation-preserving $\mathbb{Z}_{4}$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(Γ($v$),\mathbf{G(v)})$, which determines a handlebody orbifold $V(Γ($v$),{\mathbf{G(v)}})\simeq{V_g}/\mathbb{Z}_{4}$. This algebraic characterization is used to enumerate the total number of $\mathbb{Z}_{4}$ group actions on such handlebodies, up to equivalence.
Algebraic Presentation of $4$-Dimensional $2$-Handlebodies and $3$-Dimensional Cobordisms
本文通过构造反函数，给出了3维cobordism和4维2柄体的有限代数表示，并证明其等价于Habiro的猜想。
A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies
本文提出了一种构造表面 skein TQFT 的方法，并扩展其到 4 维 2-柄体，通过归纳状态和构造和 Kirby 颜色来计算不变量。
Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies
本文细化4维2柄体的不变量，定义配对$(W,\omega)$的精细化不变量，并研究受限量子群及其辫子扩张情形，推导分解公式并建立与小量子群不变量的联系。
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for solvable groups and for the groups PSL(2,3^p)…
We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near a $2n$-dimensional disk embedded in the boundary, $\partial(D^{n+1}\times S^n)^{\natural g}$. We construct a map $colim_{g\to\infty}BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \longrightarrow Q_{0}BO(2n+1)\langle n \rangle_{+}$ and prove that it induces an isomorphism on integral homology in the case that $2n+1 \geq 9$. Above, $BO(2n+1)\langle n \rangle…
A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a…
Multisections of higher-dimensional manifolds
研究高维光滑或PL闭可定向流形的多剖分，通过将其分解为1-柄体并关联图，证明多剖分图在特定维度确定唯一流形，还表明任何闭可定向光滑5-流形都有一个多剖分。
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, in terms of local moves. This result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly non-simple coverings of S^3 br…
In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| < \min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359 ), we identify the homology of $colim_{g\to \infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ with that of the infinite loopspace $Q_{0}BO(2n+1)\langle n\rangle_{+}$, in the case that $n \geq 4$. Combining this "stable homology" calculation with this paper's homologic…
Given a (genus 2) cube-with-holes M, i.e. the complement in S^3 of a handlebody H, we relate intrinsic properties of M (like its cut number) with extrinsic features depending on the way the handlebody H is knotted in S^3. Starting from a first level of knotting that requires the non-existence of a planar spine for H, we define several instances of knotting of H in terms of the non-existence of spines with special properties. Some of these instances are implied by an intrinsic counterpart in terms of the non-existence of special cut-systems for M. We study a natural partial order on these insta…
Multisections of $(m+3)$-dimensional $m$-spun $3$-manifolds
本文将4维旋转流形概念推广至任意维度，为任意$m$构造了$m+3$维$m$-旋转3流形的多截面及多截面图，提供了所有维度下多截面流形的实例，并给出无穷多非微分同胚多截面流形的实例。
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $Σ$, guiding the gluing of the handlebodies. Any morphism $φ$ from $π_1(X)$ to a finitely generated free abelian group induces a morphism on $π_1(Σ)$. We express the twisted homology and Reidemeister torsion of $(X;φ)$ in terms of the first homology of $(Σ;φ)$ and the three subspaces generated by the collections of curves. We also express the intersec…
Involutions on S^4
本文研究了S^4上的局部线性对易变换，证明了在特定条件下对易变换必为线性，并给出了相关应用和结论。
In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.
The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.
We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulated in terms of the existence of free involutions on a certain class of 4-manifolds. We consider this question and analyze its relation to the A,B-slice problem.
In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on $S^6$ with fixed-point set an embedded $S^3$.
The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on …
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.
Three-Dimensional Small Covers and Links
本文研究了三维小覆盖在二阶映射下的商空间，并揭示了当商空间为3球面时，小覆盖是沿特定链接的2折叠覆盖。
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