arXiv research

New method for robust fixed-point smoothing without state augmentation.

According to the work of Laitinen, Morimoto, Oliver and Pawałowski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_n$ for $n=3,5,7$, we present a strategy of excluding o…

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, ther…

In this thesis we study the geometry of the fixed point set $Σ$ of a smooth mapping $Φ: M\to M$ on a smooth compact Riemannian manifold $M$ without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator $Δ$ on $M$. We assume that the fixed point set $Σ$ is a…

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic…

Study circle actions with exactly three fixed points on specific manifolds.

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n=1 or 3. Our proof relies on the rigid Hirzebruch genera.

Formula for fixed points on noncompact spaces.

The paper proves group actions on spheres with odd fixed points.

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with…

If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some $x\in X$ whose stabilizer $G_x\leq G$ satisfies $[G:G_x]\leq C$. We say that $X…

Let G be a compact Lie group and X be a compact smooth G-manifold with finitely many G-fixed points. We show that if X admits a G-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of X indexed by the G-fixed points so that each open set is G-stable and G-equivari…

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition $λ=(λ_{1},...,λ_{u})$ of weight $m$ such that the Chern number $(c_{λ_{1}}... c_{λ_{…

We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$…

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincaré homology $3$-sphere $Σ(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on $Σ(2,3,5)$ do not extend to smooth actions on $X_0$ with isolated fixed points for any p…

The study restricts circle actions on certain manifolds to dimensions 4, 8, and 16.

We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…

A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.

The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its si…

A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (qua…

Paper disproves a Smith conjecture about sphere actions.

Extends G-signature theorem to Witt G-pseudomanifolds.

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth l…

The paper proves limitations on actions of a specific group on spheres.

Quantile Temporal-Difference learning proved convergent with proof.

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that…

In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on $S^6$ with fixed-point set an embedded $S^3$.

Classifies real-analytic SL(n,R) actions on closed manifolds.

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any act…

Algorithm improves variational inference in Wasserstein distance.

Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(Ω_{x})$-invariant fun…

Smooth low-regular connections lead to smooth immersions with controlled regularity.

An involutive diffeomorphism $σ$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the constru…

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dy…

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré…

Study efficient derivative computation for nondifferentiable maps in machine learning.

Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces.

We construct a map from the suspension $G$-spectrum $Σ_G^\infty M$ of a smooth compact $G$-manifold to the equivariant $A$-theory spectrum $A_G(M)$, and we show that its fiber is, on fixed points, a wedge of stable $h$-cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible …

Let $Φ\colon \sbat \times M \to M$ be a smooth action of the unit circle $ \sbat$ on a manifold $M$. In this work, we compute the minimal model of $M$ in terms of the orbit space $B$ and the fixed point set $F\subset B$, as a dg-module over the Sullivan's minimal model of $B$.

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…