Related papers: The Standard Complex and the 3-dimensional Poincar…
A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calder\'on problem in dimension 2 via the boundary…
For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), \tau^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is…
We propose a way to derive polynomial invariants of closed, orientable $3$-manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible $3$-manifold $M$, we associate a Heegaard graph $G\subset…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincar\'e complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more…
In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also…
We study the counting function of topological Poincar\'e series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with…
In this article, we are interested in the question whether any complete contractible $3$-manifold of positive scalar curvature is homeomorphic to $\mathbb{R}^{3}$. We study the fundamental group at infinity, $\pi_{1}^{\infty}$, and its…
We introduce LCL covers of closed n-dimensional manifolds by n-dimensional disks and study their properties. We show that any LCL cover of an n-dimensional sphere can be converted to the minimal LCL cover, which consists of 2n+2 disks. We…
This is a first in a series of papers, devoted to the relation betwwen three-manifolds and number fields. The present paper studies first homology of finite coverings of a three-manifold with primary interest in the Thurston $b_1$…
Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of unity of order…
We show that a categorical generalization of the the Poincar\'e symmetry which is based on the n-crossed modules becomes natural and simple when n=3 and that the corresponding 3-form and 4-form gauge fields have to be a Dirac spinor and a…
Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a…
In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, $P^2$-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is…
In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the…
The simplices and the complexes arsing form the grading of the fundamental (desymmetrized) domain of arithmetical groups and non-arithmetical groups, as well as their extended (symmetrized) ones are described also for oriented manifolds in…
We present a complete classification of invariant generalised Killing spinors on three-dimensional Lie groups. We show that, in this context, the existence of a non-trivial invariant generalised Killing spinor implies that all invariant…
The following work demonstrates the viability of Poincar\'e symmetry in a discrete universe. We develop the technology of the discrete principal Poincar\'e bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled…
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic…
We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…