Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$ -groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$ , the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$ -homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$ -acyclic $\mathbb Z /3$ -homology manifold of dimension less than $2n$ , and if $n\ge 3$ , then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$ , then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$ , which is $\mathbb Z /2$ .     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

Functions whose set of critical points is an arc

Let $M$ be a $C^{\infty }$ connected closed manifold with $\mathrm{dim }(M)\ge 2$ . Using tools developed by Körner in (J Lond Math Soc (2) 38(3):442–452, 1988) we prove that the subset of functions $f$ in $C^1(M,\mathbb R )$ such that the set of critical points of $f$ is an arc is dense in $C^{0}(M,\mathbb R )$ . We then present applications in dynamics.     

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let $A$ be a (possibly unbounded) self-adjoint operator on a separable Hilbert space $\mathfrak H .$ Assume that $\sigma $ is an isolated component of the spectrum of $A$ , that is, $\mathrm{dist}(\sigma ,\Sigma )=d>0$ where $\Sigma =\mathrm spec (A)\setminus \sigma .$ Suppose that $V$ is a bounded self-adjoint operator on $\mathfrak H $ such that $\Vert V\Vert <d/2$ and let $L=A+V$ , $\mathrm{Dom }(L)=\mathrm{Dom }(A).$ Denote by $P$ the spectral projection of $A$ associated with the spectral set $\sigma $ and let $Q$ be the spectral projection of $L$ corresponding to the closed $\Vert V\Vert $ -neighborhood of $\sigma .$ Introducing the sequence $$\begin{aligned} \varkappa _n=\frac{1}{2}\left(1-\frac{(\pi ^2-4)^n}{(\pi ^2+4)^n}\right), \quad n\in \{0\}\cup {\mathbb N }, \end{aligned}$$ we prove that the following bound holds: $$\begin{aligned} \arcsin (\Vert P-Q\Vert )\le M_\star \left(\frac{\Vert V\Vert }{d}\right), \end{aligned}$$ where the estimating function $M_\star (x)$ , $x\in \bigl [0,\frac{1}{2}\bigr )$ , is given by $$\begin{aligned} M_\star (x)=\frac{1}{2}\,\,n_{_\#}(x)\,\arcsin \left(\frac{4\pi }{\pi ^2+4}\right) +\frac{1}{2}\,\arcsin \left(\frac{\pi ( x-\varkappa _{n_{_\#}(x)})}{1-2\varkappa _{n_{_\#}(x)})}\right), \end{aligned}$$ with $n_{_\#}(x)=\max \left\{ n\,\bigr |\,\,n\in \{0\}\cup {\mathbb N }\,, \varkappa _n\le x\right\} $ . The bound obtained is essentially stronger than the previously known estimates for $\Vert P-Q\Vert .$ Furthermore, this bound ensures that $\Vert P-Q\Vert <1$ and, thus, that the spectral subspaces $\mathrm{Ran }(P)$ and $\mathrm{Ran }(Q)$ are in the acute-angle case whenever $\Vert V\Vert <c_\star \,d$ , where $$\begin{aligned} c_\star =16\,\,\frac{\pi ^6-2\pi ^4+32\pi ^2-32}{(\pi ^2+4)^4}=0.454169\ldots . \end{aligned}$$ Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic $\sin 2\theta $ estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Global injectivity conditions for planar maps

We prove that a planar $C^1$ -smooth map $f:D\longrightarrow \mathbb{R }^{2n}$ , where $D\subseteq \mathbb{R }^{2n}$ is a convex open set, is injective if $\mathbb{R }\cap \mathrm{Spec}(df)_z=\emptyset $ for all $z\in D$ . We continue by showing that the triangulability of the differentials $(df)_z$ , $z\in D$ , ensure the global injectivity as well.     

Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle ξ over a finite connected CW-complex X is by definition the largest integer ${k, 0 \leq k \leq \mathrm{dim}(X)}$ , such that every cohomology class ${x \in H^{j}(X;\mathbb{Z}_2), 0 \leq j \leq k}$ , is a polynomial in the Stiefel–Whitney classes w _i (ξ). In this note we compute the characteristic rank of vector bundles over the Stiefel manifold ${V_k(\mathbb{F}^n), \mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}}$ .     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

On the largest conjugacy class length of a finite group

Let $G$ be a finite group and $\mathrm{bcl}(G)$ the largest conjugacy class length of $G$ . In this note we slightly improve He and Shi’s lower bound for $\mathrm{bcl}(G)$ , showing that $|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$ .     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .