Finite type invariants of 3-manifolds

A note on quantum 3-manifold invariants and hyperbolic volume

We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and different volume.

A Hennings type invariant of $3$-manifolds from a topological Hopf superalgebra

We prove the unrolled superalgebra $\mathcal{U}_ξ^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $ξ$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.

Right integrals and invariants of three-manifolds

This paper gives a summary of our approach to invariants of three manifolds via right integrals on finite dimensional Hopf algebras and their relation to the Kirby calculus.

Quantum SU(2)-invariants for three-manifolds associated with non-trivial cohomology classes modulo two

We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.

Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula

We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expressed in terms of the S- and T-matrices of the modular category. In another direction we derive a rational surgery formula, which states how the RT-invariants behave under rational surgery along framed links in arbitrary closed oriented 3-manifolds with embedded colored ribbon graphs…

Generalized Kashaev invariants for knots in three manifolds

Kashaev's invariants for a knot in a three sphere are generalized to invariants of a knot in a three manifold. A relation between the newly constructed invariants and the hyperbolic volume of the knot complement is observed for some knots in lens spaces.

Fusion 2-categories and a state-sum invariant for 4-manifolds

We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.

关于某些高亏格带框架的3-流形的Kuperberg不变量的规范不变性

On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds

本文研究了某些高亏格带框架的3-流形的Kuperberg不变量的规范不变性，并通过双曲3-流形提供了有限维Hopf代数规范不变量的第一个例子。

Values of the t-invariant for small Seifert manifolds

The t-invariant can be considered as the Turaev-Viro invariant of order 5 computed for integer colors only. We compute all values of the t-invariant for Seifert manifolds with base sphere and three singular fibers. As a result we show that the manifolds parameters modulo five define the value of the t-invariant. Partially we show that there are 12 distinct values of the t-invariant for these manifolds. Some examples show that the t-invariant for these manifolds is not defined by the first homology group.

Invariants of 3-manifolds from representations of the framed-tangle category

A new class of 3-manifold invariants is constructed from representations of the category of framed tangles.

群与3维流形的类quandle不变量

Groups versus quandle-like invariants of 3-manifolds

本文证明群范畴与非空忠实risandle范畴等价，定义3维流形的基本risandle并范畴化既有着色不变量，从而在无穷多流形上实现基本群与基本risandle相互恢复。

Bordered invariants of pairs of sutured manifolds with torus boundary

We establish a framework for extending invariants of sutured manifolds to invariants of pairs of sutured manifolds who differ by attaching a basic slice along a torus boundary component. In the particular case of (bordered-)sutured Floer homology we show that this recovers the bordered-sutured Floer homology of a corresponding bordered-sutured three-manifold - though the framework is broad enough that it should be applicable in the setting of other invariants of sutured manifolds.

A generalization of the Dijkgraaf-Witten invariants for cusped 3-manifolds

We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same homology groups, whereas with distinct generalized DW invariants.

Spherical Tetrahedra and Invariants of 3-manifolds

Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version of a formal invariant of Ponzano-Regge and Korepanov defined via Euclidean tetrahedra.

Open manifolds, Ozsvath-Szabo invariants and Exotic R^4's

We construct an invariant of open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold X homeomorphic to R^4 for which the invariant is non-trivial, showing that X is an exotic R^4.

Hyperbolic three-manifolds with trivial finite type invariants

We construct a hyperbolic three-manifold with trivial finite type invariants up to a given degree.

Survey on L^2-invariants and 3-manifolds

In this paper give a survey about L^2-invariants focusing on 3-manifolds.

G-monopole invariants on some connected sums of 4-manifolds

On a smooth closed oriented $4$-manifold $M$ with a smooth action of a finite group $G$ on a Spin$^c$ structure, $G$-monopole invariant is defined by "counting" $G$-invariant solutions of Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We compute $G$-monopole invariants on some $G$-manifolds. For example, the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant has nonzero $\Bbb Z_k$-monopole invariant mod 2, where the $\Bbb Z_k$-action is given by cyclic permutations of $k$ summands.

Four-manifold invariants from higher-rank bundles

Generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N). Some values of the invariants are calculated for the case that the 4-manifold arises by the knot-complement construction of Fintushel and Stern. The results are consistent with predictions of Marino and Moore.

Torsion Invariants of Combed 3-Manifolds with Boundary Pattern and Legendrian Links

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in contact 3-manifolds. Our approach uses a combinatorial encoding of vector fields, based on standard spines. In this paper we extend this encoding from closed manifolds to manifolds with boundary.

4-Manifold Invariants From Hopf Algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invar…

Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic 4-manifolds contain L-spaces as hypersurfaces.

Immersed projective planes, Arf invariants and even 4-manifolds

In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with $b_1=1$

In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.

闭3-流形的斜配边上循环不变量

Skew-rack cocycle invariants of closed 3-manifolds

该研究提出通过Dehn手术构造3-流形不变量的新方法，引入带良好对合与FR性质的斜配边并定义其上循环不变量，同时给出3-球面中在链环同伦下不变的链环不变量。

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants

In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.

3维流形的对称 Dijkgraaf-Witten 型不变量

Symmetric Dijkgraaf-Witten type invariants for 3-manifolds

该研究借鉴经典 Dijkgraaf-Witten 不变量思想，定义特殊 $G$-系统以构造适用于闭及带边界3维流形的不变量，并利用群上同调给出一维示例。

Realizing arithmetic invariants of hyperbolic 3-manifolds

These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results are included.

具有可定向基底的Seifert 3-流形的Dijkgraaf-Witten不变量

The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases

本文针对具有可定向基底的可定向Seifert 3-流形，推导出其Dijkgraaf-Witten不变量的计算公式，为相关拓扑不变量研究提供了显式结果。

An invariant of smooth 4-manifolds

We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein structures.) Other applications of this invariant are given.

Exotic Mazur manifolds and knot trace invariants

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $ν$ is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants $τ$ and $ε$ are not in…

闭异质4-流形的新构造与不变量

New constructions and invariants of closed exotic 4-manifolds

该研究借助Heegaard Floer同调提出可在覆盖中保持区分性的新型闭四维流形不变量，并构造多类闭异质4-流形。

Product Formulae for Ozsvath-Szabo 4-manifold Invariants

We give formulae for the Ozsvath-Szabo invariants of 4-manifolds X obtained by fiber sum of two manifolds M_1, M_2 along surfaces S_1, S_2 having trivial normal bundle and genus g>0. The formulae follow from a general theorem on the Ozsvath-Szabo invariants of the result of gluing two 4-manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this "perturbed" version o…

The Casson-Walker-Lescop invariant of periodic three-manifolds

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson-Walker-Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points.

Quantum invariants and free Z_{p^2} -actions on 3-manifolds

We give a congruence for the quantum invariant of a Z_p-quotient of a 3$-manifold with a Z_{p^2} action. We show the congruence does not hold for quotients of 3--manifolds with a Z_{5}xZ_{5} action.

Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces

We propose a modification of the three-manifold invariant based on the use of Euclidean metric values ascribed to the elements of manifold triangulation. We thus obtain a nontrivial invariant that can, in particular, distinguish non-homeomorphic lens spaces.

Every quasitoric manifold admits an invariant metric of positive scalar curvature

We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.

Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chern's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP^n; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique invariant of this type.

Floer theoretic invariants for 3- and 4-manifolds

Seiberg-Witten (Floer) theory, Ozsvath-Szabo's Heegaard Floer theory, Hutchings's embedded contact homology, in different stages of development, define (or are expected to define) packages of invariants for 3- and 4-manifolds (including manifolds with boundary and manifolds with certain types of corners). We describe what are known about their relationship, what are expected, and raise some questions along the way.

Monopole classes and Perelman's invariant of four-manifolds

We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.

Hyperbolic Manifolds, Harmonic Forms, and Seiberg-Witten Invariants

New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.

Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds

We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

Computing $ν$-invariants of Joyce's compact $G_2$-manifolds

Crowley and Nordström introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordström, we compute it for many of the closed torsion-free $G_2$-manifolds defined by Joyce's generalized Kummer construction.

The Theta Divisor and Three-Manifold Invariants

In this paper we study an invariant for oriented three-manifolds with $b_1>0$, which is defined using Heegaard splittings and the theta divisor of a Riemann surface. The paper is divided into two parts, the first of which gives the definition of the invariant, and the second of which identifies it with more classical (torsion) invariants of three-manifolds. Its close relationship with Seiberg-Witten theory is also addressed.

Quasitoric Manifolds with Invariant Almost Complex Structure

We prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and hence are trivial.

Flat manifolds, harmonic spinors, and eta invariants

The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors of an infinite family of closed flat manifolds. It consits of some flat manifolds M with cyclic holonomy groups.

Holomorphic triangle invariants and the topology of symplectic four-manifolds

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This …

Eta invariants for flat manifolds

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178

An Ozsvath-Szabo Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, …

The Universal Perturbative Quantum 3-manifold Invariant, Rozansky-Witten Invariants, and the Generalized Casson Invariant

Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a physical level of rigour. We show that Z_{X}^{RW}[M] satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.

Lagrangian matching invariants for fibred four-manifolds: I

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The `Lagrangian matching invariants' are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold. They fit into a field theory which assigns Floer homology groups to fibred 3-manifolds. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres,…