Given a semisimple stable autonomous tensor category over a field $K$, to any group presentation with finite number of generators we associate an element $Q(P)\in K$ invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm of Frank Quinn. The new definition allows us to present a relatively simple proof of the invariance and to evaluate $Q(P)$ for some presentations. On the basis of some numerical calculations over different Gelfand-Kazhdan categories, we make a conjecture which relates the value of $Q(P)$ for two different classes of presentations.
We consider a 2-complex in a particular form, called the Quinn model of a 2-complex. It can be sliced in graphs, where a change from one graph to another can be organized by a sequence of local transitions, which are described in a list of F. Quinn [Q1]. The decomposition of that 2-complex into graphs has to be translated into an algebraic context (for example Topological Quantum field theory (TQFT)) to construct suitable potential invariants under 3-deformations. These invariants are accessible for computation by using a supercomputer and the results may yield a counterexample to the Andrews-…
We prove a concordance version of the 4-dimensional light bulb theorem for $π_1$-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if $F_0$ and $F_1$ are such surfaces in a 4-manifold $X$ that are homotopic and there exists an immersed framed 2-sphere $G$ in $X$ intersecting $F_0$ geometrically once, then $F_0$ and $F_1$ are concordant if and only if their Freedman-Quinn invariant $\mathop{fq}$ vanishes. The proof of the main result involves computing $\mathop{fq}$ in terms of intersections in the universal covering …
We show that the Reshetikhin-Turaev-Walker invariant of 3-manifolds can be normalized to obtain an invariant of 4-dimensional thickenings of 2-complexes. Moreover when the underlying semisimple tortile category comes from the representations of a quantum group at a primitive prime root of unity, the 0-term in the Ohtsuki expansion of this invariant depends only on the spine and is the mod p invariant of 2-complexes defined previously from the second author. As a consequence it is shown that when the Euler characteristic is greater or equal to 1, the 2-complex invariant depends only on homology…
The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T^4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T^4 two subsets are identified : T^3, which produces invariants of 4-thickenings normalizable to in…
Two simple homotopy equivalent 2-complexes K2 and L2 are related by an algebraic criterion of their corresponding presentations as stated in [HoMeSier]. Frank Quinn set it into a topological context (see [Qu1]) and call these 2-complexes related by an s-move. Using elementary 3-expansions, K2 extends to 3-cells in K3 respectively L2 to 3-cells in L3. By associating both explorations we obtain a decomposition of the s-move 3-cells into a sequence of 2-cells. We present ideas, sketch of proofs and problems to use it for constructing an Andrews-Curtis invariant on s-move 3-cells.
Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants $τ$, $\varepsilon$, and $Υ$. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type invariants of knots and homology 3-spheres fit perfectly into this conception. In particular, we get a natural explanation why they behave like polynomials.
Let $X$ be a finite aspherical CW-complex whose fundamental group $π_1(X)$ possesses a subnormal series $π_1(X) \rhd G_m \rhd ... \rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invariants $α_n({\tilde X})$ are positive. We further prove that the $L^2$-torsion $ρ^{(2)}({\tilde X})$ vanishes if $π_1(X)$ has semi-integral determinant.
We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexes with unfree fundamental group that improves the previously known bounds in this dimension.
An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.
Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.
This paper introduces an inner product on chain complexes of finite simplicial complexes that is well-adapted to the harmonic study of subdivisions. Its definition utilizes a decomposition of the chain spaces that suggests a sequence of subdivision invariants which we show do not all vanish for non-trivial subdivisions. We exhibit a combinatorial lower bound for these invariants and provide an effective algorithm for their computation. Unfortunately, these invariants cannot distinguish every subdivision nor do they necessarily increase over successive subdivisions.
We show that Tim Cochran's invariants $β^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call {\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $β^i$ invariants for all $i\leq k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whitney tow…
We are concerned with bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to these toral manifolds by toral CW complexes. We observe from primary to tertiary flow-invariant toral CW complex bifurcations for one-parametric systems associated with two most generic cases. In a particular case, a tertiary toral CW complex bifurcates from and resides outside a secondary toral CW complex. When the parameter varies, the secondary internal tora…
For a closed, oriented, odd dimensional manifold $X$, we define the rho invariant $ρ(X,E,H)$ for the twisted odd signature operator valued in a flat hermitian vector bundle $E$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $X$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We show that the twisted rho invariant $ρ(X,E,H)$ is independent of the choice of metrics on $X$ and $E$ and of the representative $H$ in the cohomology class $[H]$. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta …
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into $R^{2n}$ for $n\neq 2$, and it was recently shown to be incomplete for $n=2$. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2-complexes in $R^4$.
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associated with the universal complexes of singular fibers give complete invariants for all these cobordism groups. We also discuss invariants derived from hypercohomologies of the universal homology complexes of singular fibers. Finally, as an application of the theory of universal comp…
We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\bA$ from an element of $\widetilde{K}^{iso}_1(\bA)$ to an element of the absolute group $K_1^{iso}(\bA)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invariants. In the companion paper [1] (joint with Ian Hambleton and Andrew Ranicki) this new invariant is applied to obtain the multiplicativity of the signature of fibre bundles mod 4.
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
