Comparing Singular Homology with respect to Isometry

关于由非奇异 Morse-Smale 流定义的叶状结构的同调

On a homology of foliations defined by non-singular Morse-Smale flows

本文提出由非奇异 Morse-Smale 流定义的一维叶状结构的同调定义，并展示了计算与 Seifert 纤维化相关的此类叶状结构同调的方法。

奇异瞬子同调中的环结构

Ring structures in singular instanton homology

本文计算闭定向曲面上奇异瞬子 Floer 同调的环结构，并据此证明 n=1 时的切除公式，解决该理论切除公式的最后未知情形。

Instanton Floer homology for sutured manifolds with tangles

We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to define an instanton Floer homology theory for sutured manifolds with tangles. As applications, we prove that the annular Khovanov homology (1) detects the unlink, (2) detects the closure of the trivial braid, and (3) distinguishes braid closures from other links; we also prove that …

Khovanov同调与奇异瞬子同调中的配边映射 II

Cobordism maps in Khovanov homology and singular instanton homology II

本文将嵌入配边映射推广到浸入配边，证明其与奇异瞬子同调相容，并据此证明二桥环面结concordance诱导的Khovanov同调映射单射，以及相对奇异曲面在正扭转移动下的奇异性保持。

奇异瞬子同调中的关系

Relations in singular instanton homology

本文计算了 $S^1 \ imes S^2$ 中奇数 n-辫的带局部系数奇异瞬子同调，用代数曲线坐标环描述其结构，并预期等价于某 Fano 簇的量子上同调环。

奇异瞬子Floer同调的等变方面

Equivariant aspects of singular instanton Floer homology

本文利用$SU(2)$奇异瞬子规范理论为整数同调3-球面中的纽结构造等变不变量，包括类似纽结Floer同调的三元群和Frøyshov型配边不变量，并借助ADHM描述刻画球面纽结的模空间。

透镜空间中的奇异链环的网格同调与一个分解立方体

Grid homology for singular links in lens space and a resolution cube

本文定义透镜空间中奇异链环的网格同调，构造正则链环 knot Floer 同调的分解立方体，并证明其与经典同调一致，推广了奇异纽结理论至透镜空间。

On the coincidence of zeroth Milnor-Thurston homology with singular homology

In this paper we prove that the zeroth Milnor-Thurston homology group coincides with singular homology for Peano Continua. More- over, we show that the canonical homomorphism between these ho- mology theories may not be injective. However, it is proved that it is injective when a space has Borel path-components.

强可逆纽结的局部化与Floer同调

Localization and the Floer homology of strongly invertible knots

该研究在定向强可逆纽结K的纽结Floer同调中建立两个谱序列，分别关联K与三维球面的Heegaard Floer同调、K的奇异纽结Floer同调与K的商纽结Floer同调。

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the l^1-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology -- as already claimed by Thurst…

Mutation-invariance of Khovanov-Floer theories

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain strong Khovanov-Floer theories, including Szab{ó} homology and singular instanton homology, are invariant under Conway mutation. This confirms conjectures of Seed and Lambert-Cole. Along the way we prove two other conjectures about the structure of Szab{ó} homology.

The t-singular homology of orbifolds

For an orbifold M we define a homology group, called t-singular homology group t-H_q(M), which depends not only on the topological structure of the underlying space of M, but also on the orbifold structure of M. We prove that it is a b-homotopy invariant of orbifolds. If M is a manifold, t-H_q(M) coincides with the usual singular homology group. We calculate the t-singular homology groups with Z-coefficients of several orbifolds. The t-singular homology group with rational coefficients of an orbifold M is isomorphic to the singular homology group with rational coefficients of M (and to that of…

Link homology and equivariant gauge theory

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Flo…

On the homology of the space of singular knots

In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.

Singular chain intersection homology for traditional and super-perversities

We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superperversities (those satisfying p(2)>0). For the case p(2)=1, these latter perversitie were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptabilit…

Intersection homology duality and pairings: singular, PL, and sheaf-theoretic

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky-MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector, and Saralegi preserves product structures.

奇异圆盘与奇异瞬子Floer同调

Exotic disks and singular instanton Floer homology

利用带Chern-Simons滤波的奇异瞬子Floer同调构造奇异切片圆盘对，并证明存在强可逆Z-切片纽结，其任意对称Z-圆盘对均为奇异且稳定后仍奇异。

Relative measure homology and continuous bounded cohomology of topological pairs

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loeh's result plays a fundamental…

Equivalences for Morse homology

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudo-cycle and a well-defined integral homology class in singular homology.

Functorial semi-norms on singular homology and (in)flexible manifolds

A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classe…

Heegaard Floer homology and rational cuspidal curves

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.

Heegaard Floer同调与具有非尖点奇点的平面曲线

Heegaard Floer homology and plane curves with non-cuspidal singularities

本文利用Floer理论给出链环knotification的knot Floer复形上H₁-作用的一般公式，并借此研究复射影平面中一般代数曲线非尖点奇点的可能构型。

加权多面体的加权同调与上同调

Weighted Homology and Cohomology of Weighted Polyhedra

该研究定义了加权多面体的概念，构建了其加权奇异同调理论并证明其与加权单纯同调同构，还推广了相关上同调乘积并阐释了orbifold理论。

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.

Decomposition of the first Vassiliev derivative of Khovanov homology and its application

Khovanov homology extends to singular links via a categorified analogue of Vassiliev skein relation. In view of Vassiliev theory, the extended Khovanov homology can be seen as Vassiliev derivatives of Khovanov homology. In this paper, we develop a new method to compute the first derivative. Namely, we introduce a complex, called a crux complex, and prove that the Khovanov homologies of singular links with unique double points are homotopic to cofibers of endomorphisms on crux complexes. Since crux complexes are actually small for some links, the result enables a direct computation of the first…

A Duality Exact Sequence for Legendrian Contact Homology

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poin…

Singular crossings and Ozsváth-Szabó's Kauffman-states functor

Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings. Our bimodules are motivated by counting holomorphic disks in a bordered sutured version of a Heegaard diagram considered previously by Ozsváth, Stipsicz, and Szabó.

The pillowcase and perturbations of traceless representations of knot groups

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians…

论一种新的几何同调理论及其在范畴Gromov-Witten理论中的应用

On a new geometric homology theory and an application in categorical Gromov-Witten theory

本文证明带角光滑orbifold可三角剖分，并据此提出几何同调理论GHT以克服奇异同调的组合刚性，同时展示其在范畴Gromov-Witten理论中的应用。