Bohdan Feshchenko
In this paper we give an algebraic description of fundamental groups of orbits of circle-valued smooth functions from some subspace of the space of smooth functions with isolated singularities on $2$-torus $T^2$ with respect to the action of the group of diffeomorphisms of $T^2$.
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Classification of torus bundles that bound rational homology circles
本文完全分类了界定具有圆的有理同调的4-流形的圆上环面丛，并探讨了链环整数手术及3-辫闭合双重分支覆盖与有理同调球的联系。
We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.
Non-singular extensions of circle-valued Morse functions
研究闭合定向曲面上的圈值Morse函数非奇点扩展条件，给出在 collar 邻域存在子浸入时的必要充分条件。
We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where $c$ is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S^1 with monodromy 2&1 1&1 ^n$. We also apply a recent result of Matveev and Pervova to show that c(M_n) \ge 2Cn with C\approx 0.598, where a compact manifold M_n is the total space of the torus bundle over S^1 with the same monodromy as N_n, and discuss an approach to the conjecture c(M_n)=2n+5 based on the equality c(N_n)=2n.
As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.
Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for knots in the full torus. Our approach deals with group theoretical techniques; it is completely combinatorial, and the groups are very easy to deal with.
Homotopy type of stabilizers of circle-valued functions with non-isolated singularities on surfaces
本文研究了有向曲面上光滑函数稳定子的同伦性质，完整描述了一类Morse-Bott函数推广的光滑函数稳定子恒等映射连通分量的同伦类型。
In this paper we classify maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. Furthermore we study certain controlled degenerations of such maps and describe how one can implement jumps from one homotopy class to another while allowing only "small" degenerations.
$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces
本文研究Sol流形中Z₂-Thurston规范的计算方法，并确定非定向闭曲面在其中的可嵌入性条件。
If M is a hyperbolic once-punctured torus bundle over the circle, then the trace field of M has no real places.
Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions
本文用特定 Morse-Bott 函数刻画表示为 Lens 空间、$S^2 \ imes S^1$ 和环面在圆上的丛的连通和的 $3$ 维流形，强化并明确了 Saeki 2006 年的相关结果。
Circle-valued angle structures and obstruction theory
该研究探讨三维流形理想三角剖分上的圆周值角结构空间，证明其连通分支由ℤ₂系数上同调群计数，并建立其与PSL(2,ℂ)表示提升障碍类的双射，为理解亚纯三维指标的状态积分域提供基础。
Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R.
Perfect circle-valued Morse functions on hyperbolic 6-manifolds
本文首次构造了具有完美圆值Morse函数的双曲6流形，并由此得到双曲群中一个类型为F2而非F3的子群新例子。
We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.
Hyperbolic 4-manifolds with perfect circle-valued Morse functions
本文构造了具有完美圆值Morse函数的有限体积双曲4-流形，证明了存在无限多个此类流形，其具有有界数量的1-和3-把手分解及有界的Betti数。
Given a torus bundle $Y$ over the circle and a cohomology class $[ω]\in H^2(Y;\mathbb{Z})$ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of $Y$ with twisted coefficients in the universal Novikov ring.
Free torus actions and twisted suspensions
本文将两流形连通和上的主圆丛全空间表示为各加项上的圆丛全空间，以此为球面乘积连通和上的自由圆与环面作用提供存在充分条件，并对具自由余齐次四环面作用的闭单连通流形进行拓扑分类，推得无限多具正 Ricci 曲率与等距环面作用的流形。
Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.
Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps
本文揭示了紧双曲流形的$\mathbb{Z}$-覆盖中距离极小叶状结构与球面轨道闭包的联系，构造了具有指定性质的$\mathbb{Z}$-覆盖并明确描述了所有horocycle轨道闭包，同时证明双曲度量的微小扰动会导致其拓扑发生剧烈变化。
The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round circles is always non-negative. We begin by proving a generalization of this lemma, to accommodate Jordan curves bounding domains which do not disconnect each other. We then apply this generalization to give a new proof of Schramm's Incompatibility Theorem, which was used by Sch…
Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.
Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. …
The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). Novikov homology provides lower bounds for MN(L). In the present paper we introduce the notion of twisted Novikov homology, which allows to obtain better lower bounds for MN(L) than the usual Novikov homology. Our twisted Novikov homology is a module over the Novikov ring Z((t)) bu…
The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a circle packing on the Riemann sphere, equivalently in the complex plane, all of whose tangency points, centers, and radii are algebraic, (2) that every flat conformal torus which admits a circle packing whose contact graph triangulates the torus has algebraic modulus, and (3) that if R is a compact Riemann surface of genus at least 2, having constant curvature …
The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian knots, knotoids, 3-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [6]. Long knots naturally appear in the study of classical knots [1, 8].
A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides an alternative route to the handleslide-invariance of Ozsvath-Szabo's Heegaard Floer homology.
We study the projective derivative as a cocycle of Möbius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the 3-torus for which we generalize the previous results.
Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author proved that under mild conditions on $f\in C^{\infty}(M,R^1)$ the corresponding stabilizers and orbits are homotopy equivalent: $S_{MR} \sim S_{M}$ and $O_{MR} \sim O_M$. These results are extended here to the actions on $C^{\infty}(M,S^1)$. It is prov…
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