Quandle homotopy invariants of knotted surfaces

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to “regular Alexander quandles”. As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.     

Triple point cancelling numbers of surface links and quandle cocycle invariants

The unknotting or triple point cancelling number of a surface link F is the least number of 1-handles for F such that the 2-knot obtained from F by surgery along them is unknotted or pseudo-ribbon, respectively. These numbers have been often studied by knot groups and Alexander invariants. On the other hand, quandle colorings and quandle cocycle invariants of surface links were introduced and applied to other aspects, including non-invertibility and triple point numbers. In this paper, we give lower bounds of the unknotting or triple point cancelling numbers of surface links by using quandle colorings and quandle cocycle invariants.     

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε_?(X) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε_?(X) where X is the rationalization of a 1-connected CW-complex of finite type.     

Inequivalent surface-knots with the same knot quandle

We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

Homology of dihedral quandles

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies for p odd prime [T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002) 377-572, Conjecture 5.12]. We also show that contains Z_p for n≥3. Furthermore, we show that the torsion of is annihilated by 3. We also prove that the quandle homology contains Z_p for p odd prime. We conjecture that for n>1 quandle homology satisfies: , where f_n are “delayed” Fibonacci numbers, that is, f_n=f_n−1+f_n−3 and f(1)=f(2)=0,f(3)=1. Our paper is the first step in approaching this conjecture.     

Trunks and classifying spaces

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack space BX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Simple homotopy types and finite spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X↘Y of finite spaces induces a simplicial collapse K(X)↘K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K↘L induces a collapse X(K)↘X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

Volume distortion in homotopy groups

Given a finite metric CW complex X and an element \({\alpha \in \pi_n(X)}\), what are the properties of a geometrically optimal representative of \({\alpha}\)? We study the optimal volume of \({k\alpha}\) as a function of k. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an invariant with respect to the rational homotopy of X. We provide a number of examples and techniques for studying this invariant, with a special focus on spaces with few rational homotopy groups. Our main theorem characterizes those X in which all non-torsion homotopy classes are undistorted, that is, their distortion functions are linear.     

Parametrized spaces model locally constant homotopy sheaves

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π₁-sets. We then use these two equivalences to study base change functors for parametrized spaces.     

Finitely stable racks and rack representations

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 strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.     

The essential Lagrangian-Grassmannian and the homotopy type of the Fredholm Lagrangian-Grassmannian

Let H be a separable infinite dimensional Hilbert space endowed with a symplectic structure and let L₀⊂H be a Lagrangian subspace. Using the results of [A. Abbondandolo, P. Majer, Infinite dimensional Grassmannians, math.AT/0307192], we show that the Fredholm Lagrangian-Grassmannian FL₀(Λ) has the homotopy type of Gc(L₀), the Grassmannian of all Lagrangian subspaces of H that are compact perturbations of L₀. It is well known that the latter has the homotopy type of the quotient U(∞)/O(∞). As a corollary, we recover a result by B. Booss-Bavnbek and K. Furutani (see [B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, Contemp. Math. 242 (1999) 53-83; K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004) 269-331]) that the L₀-Maslov index is an isomorphism between the fundamental group of FL₀(Λ) and the integers.     

A classifying invariant of knots,the knot quandle

The two operations of conjugation in a group, $\text{x?y=y}{}^{-1}\text{xy}$ and $\text{x?}{}^{-1}\text{y=yxy}{}^{-1}$ satisfy certain identities. A set with two operations satisfying these identities is called a quandle. The Wirtinger presentation of the knot group involves only relations of the form y^-1xy=z and so may be construed as presenting a quandle rather than a group. This quandle, called the knot quandle, is not only an invariant of the knot, but in fact a classifying invariant of the knot.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

Bott’s periodicity theorem and differentials of the Adams spectral sequence of homotopy groups of spheres

Bott’s periodicity theorem is applied to calculate higher-order differentials of the Adams spectral sequence of homotopy groups π _*(SO). The resulting formulas are used to find higher-order differentials of the Adams spectral sequence of homotopy groups of spheres.     

Quandle cohomology and state-sum invariants of knotted curves and surfaces

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).

A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in -space and knotted surfaces in -space.

Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

     

On processes which cannot be distinguished by finite observation

A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s _n such that s _n (x ₁,…, x _n ) → J(X) in probability for every process X=(x _n ) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure. This research was supported by the Israel Science Foundation (grant No. 1333/04)     

Polyhedra dominating finitely many different homotopy types

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination.We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types. Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r-images up to isomorphism.