Related papers: The spine which was no spine

The well-rounded retract for $\mathrm{SL}_n(\mathbb{Z})$ is defined as the set of flat tori of unit volume and dimension $n$ whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal…

We classify spin structures on the noncommutative torus, and find that the noncommutative n-torus has 2^n spin structures, corresponding to isospectral deformations of spin structures on the commutative n-torus. For n>3 the classification…

It is shown that a $d$-dimensional classical SU(N) Yang-Mills theory can be formulated in a $d+2$-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. In this paper we present an explicit proof…

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichm\"uller space of a non-orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the…

For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichm\"uller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli…

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we…

In previous works, the universal mapping class group was taken to be the group PPSL(2,Z) of all piecewise PSL(2,Z) homeomorphisms of the unit circle S^1 with finitely many breakpoints among the rational points, and in fact, the Thompson…

This paper shows that there is a mapping class group-equivariant deformation retraction of the Teichm\"uller space of a closed, orientable surface onto a cell complex of dimension equal to the virtual cohomological dimension of the mapping…

Earlier work took as universal mapping class group the collection PPSL(2,Z) of all piecewise PSL(2,Z) homeomorphisms of the unit circle S^1 with finitely many breakpoints among the rational points. The spin mapping class group P(SL(2,Z))…

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb…

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space…

The scalar curvature for the noncommutative four torus $\mathbb{T}_\Theta^4$, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the…

This paper contains some results about the topology of $\M_{0,n+1}/\Sigma_n$, where $\M_{0,n+1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that $\M_{0,n+1}/\Sigma_n$ is not a topological manifold for…

Let $F(M,k)$ be the configuration space of ordered $k-$tuples of distinct points in the manifold $M$. Using the Fadell-Neuwirth fibration, we prove that the configuration spaces $F(M,k)$ are never contractible, for $k\geq 2$. As…

The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of…

We prove that the set of symplectic lattices in the Siegel space $\mathfrak{h}_g$ whose systoles generate a subspace of dimension at least 3 in $\mathbb{R}^{2g}$ does not contain any $\mathrm{Sp}(2g,\mathbb{Z})$-equivariant deformation…

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that…

We reformulate chiral higher-spin Yang-Mills and gravity on $\mathbb{R}^4$ as 'CR-holomorphic' theories of Chern-Simons type; in the most general case, these are Moyal deformed to become non-commutative. They are defined on the space of…

We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially…

Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$-triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the…