Related papers: A short proof for tc(K) = 4

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological…

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma(X)$ can be computed from data about $Y$, $B$ and the local degrees of…

It is well-known (cf. K.-Pudl\'ak 1989) that a polynomial time algorithm finding tautologies hard for a propositional proof system $P$ exists iff $P$ is not optimal. Such an algorithm takes $1^{(k)}$ and outputs a tautology $\tau_k$ of size…

We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one $4\times 4$ Hermitian matrix is uniquely determined by its values as a Hermitian form on this collection of…

Cook's theorem is commonly expressed such as any polynomial time-verifiable problem can be reduced to the SAT problem. The proof of Cook's theorem consists in constructing a propositional formula A(w) to simulate a computation of TM, and…

Circular and non-wellfounded proofs have become an increasingly popular tool for metalogical treatments of systems with forms of induction and/or recursion. In this work we investigate the expressivity of a variant CT of G\"odel's system T…

In this article we study the higher topological complexity ${\sf TC}_r(X)$ in the case when $X$ is an aspherical space, $X=K(\pi, 1)$ and $r\ge 2$. We give a characterisation of ${\sf TC}_r(K(\pi, 1))$ in terms of classifying spaces for…

We note that, for any natural $k$ and every natural $l$ between $k$ and $2k$, there exists a group $\pi$ with $\cat K(\pi,1)=k$ and $\TC(K(\pi,1))=l$. Because of this, we can set up a problem of searching of purely group-theoretical…

We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact…

We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifcations. We provide a simple proof of the fact that a space can have arbitrarily many different complexities in different compactifications.…

A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X.…

In the lattice CP(N) models we studied the problems related to the measure of the topological susceptibility and the string tension . We perfomed numerical simulations at N=4 and N=10. In order to test the universality, we adopted two…

Suppose that there's no transitive model of ZFC + there's a strong cardinal, and let K denote the core model. It is shown that if \delta has the tree property then \delta^{+K} = \delta^+ and \delta is weakly compact in K.

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a…

Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat K\"ahler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact…

We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant $\widetilde{TC}_n(X)$ is used in the proof. We interpret…

We study smooth projective varieties with small dual variety using methods from symplectic topology. We prove the affine parts of such varieties are subcritical, and that the hyperplane class is invertible in their quantum cohomology. We…

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one…

In this paper, we introduce the n-th discrete topological complexity and study its properties such as its relation with simplicial Lusternik-Schnirelmann category and how the higher dimensions of discrete topological complexity relate with…