We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluationg the state-sum invariants defined from quandle cohomology.
We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.
We study Coxeter racks over $\mathbb{Z}_n$ and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the {\em strong} representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the {\em Pontryagin} dual of a rack as an Abelian group which, in the finite involutive c…
GL-racks and coloring invariants of Legendrian knots
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all $y\in X$, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all $x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arb…
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots, for which we also find a lifting by using braided crossed modules.
We introduce a modified rack algebra Z[X] for racks X with finite rack rank N. We use representations of Z[X] into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are strictly stronger than the unenhanced counting invariant and are not determined by the Jones or Alexander polynomials.
Given a presentation for a rack $\mathcal R$, we define a process which systematically enumerates the elements of $\mathcal R$. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for $n$-quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt for the Todd-Coxeter process. We prove that the process terminates if and only if $\mathcal R$ is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot t…
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(τ,σ,ρ)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, and central extensions of groups. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots. We construct a class of Reidemeister pairs which constitute a lifting of the Eisermann invariant, and show through an example that this class is strictl…
It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.
The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group $G$ of a hyperbolic three-manifold satisfies a certain stability property. The stability property is the statement that if all the quotients $G^i/G^{i+1}$ of the derived series $G^i$ of $G$ are finite, then the derived series stabilises. The proof involves basic facts regarding finite group actions on homology spheres.
We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its homology. It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. From the very beginning of the rack homology (between 1990 and 1995) the analogous result was suspected. The first general results in this direction were obtained independently about 2001 by R.A.Litherland and S.Nelson, and P.Etingof and M.Grana. In Litherland-Nelson paper it is proven that if (Q;*) is a finite homogeneous rack (this includes quasigroup racks) then the torsion of ho…
We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2…
We prove that every finitely presented group with positive first $\ell^2$-Betti number that virtually surjects onto $\mathbb Z$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first $\ell^2$-Betti number as well as groups of deficiency at least $2$.
In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For each link, these Betti numbers are supported on a finite subset of $\mathbb{Z}^4$. One can easily recover from these Betti numbers the Poincaré polynomial of the middle HOMFLYPT homology. We explain why the Betti numbers can be viewed as a generalization of the reduced HOMFLYPT h…
Suppose X is any finite complex with vanishing L^2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and are improved in the presence of a spectral gap.
Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results are approximation theorems for $L^2$-multiplicities which extend the approximation theorems for $L^2$-Betti numbers of Lück, Farber and Elek-Szabó respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each Ω-invariant is contained in the complement of a union of …
Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except…
We show that if M is a surface bundle over S^1 with fiber of genus 2, then for any integer n, M has a finite cover tilde(M) with b_1(tilde(M)) > n. A corollary is that M can be geometrized using only the `non-fiber' case of Thurston's Geometrization Theorem for Haken manifolds.
We compute the l^2-Betti numbers of the complement of a finite collection of affine hyperplanes in complex space. At most one of the l^2-Betti numbers is non-zero.
We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $β_1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop [KL], by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal syst…
Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n…
In this paper, we study the expectation values of topological invariants of the Vietoris-Rips complex and Čech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.
A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation for…
We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the squa…
This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distinctive shape. Both vortex nerve shapes (bounded planar surfaces with nonempty interior) and holes (bounded planar surfaces with empty interior that live inside and define shapes) have boundaries that are path-connected cycles. In the context of CW complexes, the usual Betti numbers $…
We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers.
This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve is a maximal collection of triangles with a common vertex in the triangulation of a finite, bounded planar region. I…
We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.
We study quotients $Γ\backslash \mathbb H^n$ of the $n$-fold product of the upper half plane $\mathbb H$ by irreducible and torsion-free lattices $Γ< PSL_2(\mathbb R)^n$ with the same Betti numbers as the $n$-fold product $(\mathbb P^1)^n$ of projective lines. Such varieties are called fake products of projective lines or fake $(\mathbb P^1)^n$. These are higher dimensional analogs of fake quadrics. In this paper we show that the number of fake $(\mathbb P^1)^n$ is finite (independently of $n$), we give examples of fake $(\mathbb P^1)^4$ and show that for $n>4$ there are no fake $(\mathbb P^1)…
Two main theorems are proved in this paper. Theorem 1: There is a constant C(n, D) depending only on n and D such that for a closed Riemannian n-manifold satisfying Ric > -(n-1) and Diam < D, the ith bounded Betti number is bounded by C(n, D). Here the ith bounded Betti number is defined as the dimension of the image of the ith bounded cohomology in the ith real cohomology. Theorem 2: There are only finitely many isometric isomorphism types of bounded cohomology groups among closed Riemannian n-manifolds satisfying K > -1 and Diam < D. The dimension of the ith bounded cohomology might be infin…
We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform lattice of $\Isom(X)$. As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form $\H^4/Γ$ where $\H^4$ is real hyperbolic 4-space and $Γ<\Isom(\H^4)$ is discrete and isomorphic to a subgroup of the fundamental group…
Recently the so-called Atiyah conjecture about l^2-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalizations of l^2-Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology g…
