Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi/p$. For every $2\leq p\leq8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Isom$(\mathbf H^5)$, whose limit set is homeomorphic to the index-$p$ Pontryagin surface $\Pi_p$. Dimension five is optimal, since $\Pi_p$ does not embed in $\mathbb S^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.
We construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo con...
Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $\pi _{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic...
Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $\alpha\ge 1$, we construct a finitely generated group $G_{n,\alpha}$ that embeds into the group of $C^\alpha$-diffeomorphisms of $M$, but does not embed into the group of $C^\beta$-diffeomo...
Maximiliano Escayola, Carl-Fredrik Nyberg-Brodda· 0 citations
This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable an...
We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component i...
We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\operatorname{Area}(M) \geq \frac{14\zeta(3)}{\pi} \approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary...
Le Chen, Xiao-Long Li, Yi-Min Zhong· 0 citations
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