We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if and only if it does not contain a rank 3 elementary abelian subgroup. We show that if G is a finite subgroup of U(n), acting freely on U(n)/U(k) for some k>0 and if (|G|,(n-1)!)=1, then the action propagates to a free and smooth action on a product of n-k spheres. A number of explicit examples are discussed.
We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.
A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non simply connected spaces.
We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.
It is proved that for a product action of $(\mathbb Z_p)^k$ on a product of (mod p) homology spheres $N^{n_1}\times...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times...\times N^{n_k}\to \mathbb R^m$ the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}| f(x)=f(gx) \forall g\in(\mathbb Z_p)^k\}$ has dimension at least $n_1+...+n_k-m(p^k-1)$, provided $n_i\ge mp^{i-1}(p-1)$ for all $i (1\le i\le k)$. Moreover, if $n_i\ge mp^{k-1}(p-1)$ for all $i(1\le i\le k)$ then the free action $μ$ can be assumed arbitrary.
In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.
In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.
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.
This paper investigates the moment-angle manifolds whose cohomology ring is isomorphic to that of a connected sum of sphere products. We first give a example of moment-angle manifolds corresponding to a 4 dimentional simplicial polytope. It has the property that its cohomology ring is isomorphic to that of a connected sum of sphere products with one produt of thress spheres. Finally, we give some general properties of this kind of moment-angle manifolds.
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.
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 th…
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 $\mathbb{S}^n \times \mathbb{S}^m,1\leq n\leq m.$
The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial then so is the action on X x S^n. Unlu and Yalcin recently showed that for every finite group G, there is a finite CW complex K with fundamental group G which acts homologicially trivially on the universal cover of K. Thus every finite group acts smoothly, freely, and homologicall…
We compute the monoid of essential self-maps of of the product of two n-spheres fixing the diagonal. More generally, we consider products S x S, where S is a suspension. Essential self-maps of S x S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental 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.
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.
A classical conjecture in transformation group theory states that if $G=(\bbZ/p)^r$ acts freely on a product of $k$ spheres $S^{n_1} \times ... \times S^{n_k}$, then $r\leq k$. We prove this conjecture in the case where the average of the dimensions ${n_i}$ is large compared to the differences $|n_i-n_j|$ between the dimensions.
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…
We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexisten…
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 show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on $M \times \bbS ^{n_1} \times \dots \times \bbS ^{n_k}$ when the set of isotropy subgroups of the $G$-action on $M$ can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if $G$ is an (almost) extra-special $p$-group of rank $r$, then it acts freely and smoothly on a product of $r$ spheres.
The stable systolic category of a closed manifold M indicates the complexity in the sense of volume. This is a homotopy invariant, even though it is defined by some relations between homological volumes on M. We show an equality of the stable systolic category and the real cup-length for the product of arbitrary finite dimensional real homology spheres. Also we prove the invariance of the stable systolic category under the rational equivalences for orientable 0-universal manifolds.
We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.
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 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.
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）
We say that a simply connected space $X$ is pre-c-symplectic if it is the fibre of a rational fibration $X\to Y\to \C P^{\infty}$ where $Y$ is cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. It is a rational homotopical property but not a cohomological one. By using Sullivan's minimal models, we give the necessary and sufficient condition that the product of odd-spheres $X=S^{k_1}\times ... \times S^{k_n}$ is pre-c-symplectic and see some related topics. Also we give a charactarization of the Hasse diagram of rational toral ranks for…
The (non)triviality of Samelson products of the inclusions of the spheres into p-regular exceptional Lie groups is completely determined, where a connected Lie group is called p-regular if it has the p-local homotopy type of a product of spheres.
In this paper we show that most rank two groups act freely on a finite homotopy product of two spheres. This makes new progress on a conjecture by Benson and Carlson which states that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recalling that Qd(p) is the semidirect product of a rank two elementary abelian p-group with SL(2,p), we show that a rank two finite group acts freely on a finite CW-complex homotopic to the product of two spheres if, it does not contain any subquotient isomorphic to Qd(p) for any odd …
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.
We consider the space $ΛM:=H^1(S^1,M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $ΛM/G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $tr:H_*(ΛM/G)\rightarrow H_*(ΛM)$ to define a homology product on $ΛM/G$ via the Chas-Sullivan loop product. We call this product $P_G$ the transfer product. The involution $\vartheta:ΛM\rightarrow ΛM$ which reverses orientation, $ \vartheta\big(γ(t)\big):=γ(1-t)$, is of particular interest to us. We comput…
In this paper we present unpublished work by David Stone on polyhedral smash products. He proved that the polyhedral smash product of the CW-pair $(D^2, S^1)$ over a simplicial complex $K$ is homeomorphic to an iterated suspension of the geometric realization of $K$. Here we generalize his technique to the CW-pair $(D^{k+1}, S^{k})$, for an arbitrary $k$. We generalize the result further to a set of disks and spheres of different dimensions.
In this paper we study the topological structure of moment-angle complexes $\mathcal{Z_K}$. We consider two classes of simplicial complexes. The first class $B_Δ$ consists of simplicial complexes $\mathcal{K}$ for which $\mathcal{Z_K}$ is homotopy equivalent to a wedge spheres. The second class $W_Δ$ consists of $\mathcal{K}\in B_Δ$ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that $B_Δ = W_Δ$. In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding…
We show rational homological stability for the homotopy automorphisms and block diffeomorphims of iterated connected sums of products of spheres. The spheres can have different dimension, but need to satisfy a certain connectivity assumption. The main theorems of this paper extend homological stability results for automorphism spaces of connected sums of products of spheres of the same dimension by Berglund and Madsen.
Chas and Sullivan recently defined an intersection product on the homology $H_*(LM)$ of the space of smooth loops in a closed, oriented manifold $M$. In this paper we will use the homotopy theoretic realization of this product described by the first two authors to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when $M$ is simply connected. The $E_2$ term of this spectral sequence is $H^*(M;H_*(ΩM))$ where the product is given by the cup product on the cohomology of the manifold $H^* (M)$ with coefficients in the Pontryagin ring struc…
Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises n…
The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric produ…
We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley--Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes $K$ with moment-angle complexes $\mathcal Z_K$ having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then…
A Lie group is called $p$-regular if it has the $p$-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into $p$-regular $\mathrm{SO}(2n)_{(p)}$ is determined, which completes the list of (non)triviality of such Samelson products in $p$-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion $\mathrm{SO}(2n-1)\to\mathrm{SO}(2n)$ in the sense of James at any prime $p$.
Fan, Chen, Ma and Wang constructed a moment-angle manifold $Z_K$ whose cohomology ring is isomorphic to that of a connected sum of sphere products with one product of three spheres. In this paper, we show that they are actually diffeomorphic.
Realizing orders in rational sphere product algebras with three generators
Institute of Mathematical Science, Pusan National University（釜山国立大学数学科学研究所）；Department of Mathematics and Statistics, Univeristy of Regina（里贾纳大学数学与统计系）；School of Mathematical Sciences, University of Southampton（南安普顿大学数学科学学院）；Department of Mathematics, The Univeristy of British Columbia（不列颠哥伦比亚大学数学系）
We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Joellenbeck on Golod rings and homotopy…
