Representation stability for homotopy automorphisms

In this paper we describe a construction for finite abstract chiral 4-polytopes with Schlafli type $$\{3,3,k\}$${3,3,k} (with tetrahedral facets) and with an alternating or symmetric group as automorphism group. We use it to prove that for all but finitely many $$n$$n, both $$A_n$$An and $$S_n$$Sn are the automorphism groups of such a polytope. We also show that the vertex-figures of the polytopes obtained from our construction are chiral.

Polytopes of high rank for the symmetric groups

We investigate the properties of graphs whose automorphism group is the symmetric group. In particular, we characterize graphs on less than 2n points with group Sn, and construct all graphs on n + 3 points with group Sn. Graphs with 2n or more points and group Sn are discussed briefly.

This paper surveys some results in the area of maximal subgroups of the finite simple groups and their automorphism groups. The first two sections are concerned with maximal subgroups of the alternating and symmetric groups. We outline the Reduction Theorem and discuss the maximality of primitive groups in the corresponding symmetric groups. In the third section we consider the corresponding questions for the classical groups. The next section contains a brief survey of knowledge of the maximal subgroups of the other almost simple groups. Finally, the last section outlines some recent work on primitive permutation groups of special degrees.

The uniqueness of a certain graph

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A$, $B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{\mathbf {R} }(A\cup B)\le \dim _{\mathbf {R} }A+\dim _{\mathbf {R} }B+1 \end{equation*} for any ring $\mathbf {R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then [a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. [b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. [c.] There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z’$ with $K_{i}\in AE(Z’)$ for all $i\ge 1$ embeds in $Z$ as a closed subset.

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

For a p-toral group G, we answer the question which compact (respectively, open) smooth manifolds M can be diffeomorphic to the fixed point sets of smooth actions of G on compact (respectively, open) smooth manifolds E of the homotopy type of a finite ℤ-acyclic CW complex admitting a cellular map of period p, with exactly one fixed point. In the case where the CW complex is contractible, E can be chosen to be a disk (respectively, Euclidean space).

Galatius’ most striking result is easy enough to state. Let Σn be the symmetric group on n letters and Fn be the free (non-abelian) group with n generators. The symmetric group Σn acts naturally by permutation on the n generators of Fn, and every permutation gives thus rise to an automorphism of the free group. Galatius proves that the map Σn → AutFn in homology induces an isomorphism in degrees less than (n− 1)/2. The homology of the symmetric groups in these ranges is well understood. In particular, in common with all finite groups, it has no non-trivial rational homology. By Galatius’ theorem, in low degrees this then is also true for AutFn: H∗(AutFn)⊗Q = 0 for 0 < ∗ < 2n/3,

Let $${\mathcal{A} = (A; F)}$$ be an algebra with T the set of all its term operations. For any permutation τ of A, the induced mapping $${f \to \tau\circ f\circ\tau^{-1}}$$ defines a permutation $${\tau^{\star}}$$ of the set of all finitary operations on the set A. We say that τ is a weak automorphism of $${\mathcal{A}}$$ if and only if τ*(T) = T. Of course any automorphism α of $${\mathcal{A}}$$ is a weak automorphism, because α*(t) = t for all $${t \in T}$$ . The set of all weak automorphisms of $${\mathcal{A}}$$ forms a subgroup of the symmetric group on A. In this paper, we describe weak automorphisms of the dihedral groups $${\mathcal{D}_n}$$ for n ≥ 3. We show that the weak automorphism group of $${\mathcal{D}_n}$$ is a semidirect product of the group of automorphisms of $${\mathcal{D}_n}$$ and some group related to the group of invertible elements of the ring $${\mathbb{Z}_n}$$ .

According to the results of Fedida and Molino [9], the structure of a G-Lie foliation F on a compact manifold M can be described by means of four locally trivial fibre bundles. In this paper we study the relations that those fibrations imply among the (rational) homotopy groups of: the manifold M, the generic leaf L, its closure N = L, the basic manifold W, the Lie group G, and the structural Lie group H. Also, we prove that those relations are a particular case of an algebraic result concerning generalized homology theories.

group of odd order has only inner automorphisms. The group of automorphisms of the holomorph of an arbitrary group was studied by Gol'fand [3] who found some cases in which the outer automorphism group has order one or two. In the present paper I determine explicitly the outer automorphism group 0 of the holomorph H of an arbitrary finite abelian group G. If G is the direct product of a group of odd order, a group of order two, and a cyclic group of order 2n where n> 2, then 0 is the direct product of a finite number of groups of order two and a non-abelian group 0* of order six or eight. If n =2 then 0 * is isomorphic to the symmetric group of order six, and if n >3 then 0* is the octic group. In all other cases 0 is either trivial or the direct product of a finite number of groups of order two. Let A be the group of all automorphisms of the finite abelian group G, let GB be the group of all automorphisms of H that map G onto itself, and let g be the group of all inner automorphisms of H. Then (B3/J can be identified with the first cohomology group H'(A, G). Thus H'(A, G) can be regarded as a subgroup of 0. Now G is an ilnvariant subgroup of H, and it is known [6] that H has at most four invariant subgroups isomorphic to G. It follows that H'(A, G) has index at most four in 0.

Pseudo-isotopy and invariant theory—I

The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free group Fn and inspect their BNSR-invariants. We prove that for n≥2, all the “positive” and “negative” character classes of PΣAutn lie in Σn−2(PΣAutn)∖Σn−1(PΣAutn). We use this to prove that for n≥2, Σn−2(ΣAutn) equals the full character sphere S0 of ΣAutn but Σn−1(ΣAutn) is empty, so in particular the commutator subgroup ΣAutn' is of type Fn−2 but not Fn−1. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

The structure of the centralizers of finite subgroups in homogenous symmetric groups are given up to isomorphism by Güven-Kegel-Kuzucuoğlu in 2015 and Kuzucuoğlu-Oliynyk-Sushchansky in 2018. In this article, we answer the natural question; what is the structure of the normalizers of finite semiregular subgroups in homogenous symmetric groups? We prove, for some sequence We answer negatively, the following open question: does every automorphism of preserve a level? We show that, there exist uncountably many automorphisms of which does not preserve any level. Finally, we show that the level preserving automorphisms of the homogenous finitary symmetric group is isomorphic to