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Register a new userFree cyclic group actions on highly-connected $2n$-manifolds
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In this paper we study smooth orientation-preserving free actions of the cyclic group $mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $sharp g (S^n times S^n)sharp Sigma$, where $Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$.
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Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$, and let $fin text{Mod}(S_g)$ be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of $ text{Mod}(S_g)$. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation $Psi: text{Mod}(S_g) to text{Sp}(2g; mathbb{Z})$.
Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. Given a finite subgroup $H leq text{Mod}(S_g)$, let $text{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichm{u}ller space $text{Teich}(S_g)$. The Nielsen realization problem, which was answered in the affirmative by S. Kerckhoff, asks whether $text{Fix}(H) eq emptyset$, for any given $H$. In this paper, we give an explicit description of $text{Fix}(H)$, when $H$ is cyclic. As consequences of our main result, we provide alternative proofs for two well known results, namely a result of Harvey on $text{dim}(text{Fix}(H))$, and a result of Gilman that characterizes irreducible finite order actions. Finally, we derive a correlation between the orders of irreducible cyclic actions and the filling systems on surfaces.
For $k ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Sigma^{4k-1}$ such that $M sharp Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.
Denote by $DC(M)_0$ the identity component of the group of compactly supported $C^infty$ diffeomorphisms of a connected $C^infty$ manifold $M$, and by $HR$ the group of the homeomorphisms of $R$. We show that if $M$ is a closed manifold which fibers over $S^m$ ($mgeq 2$), then any homomorphism from $DC(M)_0$ to $HR$ is trivial.
As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice manifolds with corners. Based upon this, we investigate what kinds of closed manifolds admit locally standard $(Z_2)^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;Z_2)=0$, it is shown that $M$ admits a locally standard $(Z_2)^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some $Z_2$-homology sphere $N$, and if further assuming $M$ is irreducible, then $M$ must be homeomorphic to $S^3$. In addition, the argument is extended to rational homology 3-sphere $M$ with $H_1(M;Z_2) cong Z_2$ and an additional assumption that the $(Z_2)^3$-action has a fixed point.
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