Our preprint "Free actions of extraspecial p-groups on S^n x S^n" (arXiv:math/0701558) is now divided into two separate papers. This is the final version of the first part (to appear in the "Quarterly Journal of Mathematics")
We construct a non-abelian extension $Γ$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $Γ$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\times S^{11}$. This gives new actions on $S^{11}\times S^{11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in the literature.
Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.
Let SL(n,Z) be the special linear group over integers and $M =S^r_1 \times S^r_2,T^r_1 \times S^r_2$ , or $T^r_0 \times S^r_1 \times S^r_2$, products of spheres and tori. We prove that any group action of SL(n,Z) on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1$. This confirms a conjecture on Zimmer's program for these manifolds.
We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.
Let p be an odd prime. We prove that every rank three p-group acts freely and smoothly on a product of three spheres. To construct this action, we first prove a generalization of a theorem of L\" uck and Oliver on constructions of G-equivariant vector bundles. We also give some other applications of this generalization.
This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…
Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ in the category of $K(n)$-local spectra. To do this, we build a spectrum which we call the Tate sphere $S(1)$. This is a $p$-complete sphere with a natural continuous action of $\mathbb{Z}_p^\times$. The Tate sphere inherits an action of $\mathbb{G}_n$ via the determinant and smashing Morava $E$-theory with $S(1)$ has the effect of twisting the action of $\mathbb{G}_n$. A large part of this paper consists of analyzing continuous $\mathbb{G}_n$-actions and their homotopy fixed points in the setup…
Using the loop coproduct, Menichi proved that the cup product with the orientation class vanishes for a closed connected oriented manifold with non-trivial Euler characteristic. We generalize this to the sphere spaces, i.e. the mapping spaces from spheres, using two generalizations of the loop coproduct to sphere spaces. One is constructed in this paper and the other in a previous paper of the author.
In this paper, we have determined the orbit spaces of free involutions on a finitistic space having mod 2 cohomology of the product of three spheres $\mathbb{S}^n\times \mathbb{S}^m \times \mathbb{S}^l, 1 \leq n \leq m \leq l$. This paper generalizes the results proved by Dotzel et al. [6] for free involutions on the product of two sphere $\mathbb{S}^n \times \mathbb{S}^m,1\leq n\leq m.$
Let $G=\mathbb{Z}_2$ act on a finite CW-complex $X$ having mod 2 cohomology isomorphic to the product of projective space and sphere $\mathbb{F}P^n\times \mathbb{S}^m,$ where $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}.$ In this paper, we have determined the connected fixed point sets and the orbit spaces of free involutions on $X.$ As an application, we derive the Borsuk-Ulam type results.
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…
Any finite group acts freely and homologically trivially on a product of spheres
主要定理是：如果K是一个有限CW复形，其有限基本群G和泛覆盖同伦等价于球面X的乘积，则G光滑自由地作用在X x S^n上，对任意n大于或等于X的维数.如果G-作用在K的泛覆盖上是同调平凡的，那么作用在X x S^n上也是同调平凡的。Unlu和Yalcin最近证明了，对于每个有限群G，存在一个有限CW复形K，其基本群G同调平凡地作用在K的泛覆盖上.因此，每个有限群都光滑、自由、同调地作用在球面的乘积上。
New families of moment-angle manifolds diffeomorphic to connected sums of products of spheres
Moscow State University（莫斯科国立大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；National Research University Higher School of Economics（国立高等经济大学）；Institute for Information Transmission Problems, Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb{Z}/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk--Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.
We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear $\mathbb{Z}/p$-spheres have both invariants either $2$ or $3$ and calculate exact values in all but two cases for linear actions. On the other hand, we exhibit examples which show that these invariants can be arbitrarily high in the class of smooth $\mathbb{Z}/p$-spheres.
We show that if the moment-angle complex $\mathcal{Z}_K$ associated to a simplicial complex $K$ is homotopy equivalent to a connected sum of sphere products with two spheres in each product, then $K$ decomposes as the simplicial join of an $n$-simplex $Δ^n$ and a minimally non-Golod complex. In particular, we prove that $K$ is minimally non-Golod for every moment-angle complex $\mathcal{Z}_K$ homeomorphic to a connected sum of two-fold products of spheres, answering a question of Grbić, Panov, Theriault and Wu.
This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the orbit space X/G is the integral or mod 2 cohomology quaternion projective space HPn. We have proved that X is the integral or mod 2 cohomology (4n+3)-sphere or the product of 3-sphere and quaternion projective space HPn. Similar results for G = S1 actions are also discussed.
Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres
Moscow State University（莫斯科国立大学）；National Research University Higher School of Economics（国立高等经济大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
Let $X$ be a finite CW-complex having mod $p$ cohomology isomorphic to a wedge of three spheres $\mathbb{S}^n\vee \mathbb{S}^m \vee \mathbb{S}^l,~ 1\leq n \leq m \leq l$. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on $X.$ We also classify the orbit spaces of free actions of $G=\mathbb{Z}_p, p$ a prime or $G=\mathbb{S}^d,~d=1,3,$ on $X$ and derive the Borsuk-Ulam type results.
Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of H k (M ; Z). We prove that the image of this homomorphism contains a congruence subgroup of SL_n (Z) whenever n is at least 3.
