There has been renewed interest in $\mathbb{S}^3$-bundles over $\mathbb{S}^4$ since K. Grove and W. Ziller constructed metrics on nonnegative curvature on the total spaces of these bundles. In this paper we write down necessary and sufficient conditions for a CW complex to be homotopy equivalent to such a bundle. We also show that for a manifold homotopy equivalent to such a bundle, in certain cases, there is no obstruction to homeomorphism. We use this to show that the Berger manifold, $\text{Sp}(2)/\text{Sp}(1)$, is PL-homeomorphic to such a bundle. This question was raised in the paper by Grove and Ziller since this manifold, which admits a normal homogeneous metric of positive sectional curvature, has the cohomology ring of such a bundle.
The principal technique used is the study of the Serre spectral sequence of various fibrations.
We show that coloring properties of a simplicial complex K are reflected by splitting properties of a bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra of K.
Finitely dominated chain complexes over a Laurent polynomial ring in one indeterminate are characterised by vanishing of their Novikov homology. We present an algebro-geometric approach to this result, based on extension of chain complexes to sheaves on the projective line. We also discuss the K-theoretical obstruction to extension.
In previous papers, we used the standard simplices $Δ^p$ $(p\ge 0)$ endowed with diffeologies having several good properties to introduce the singular complex $S^\dcal(X)$ of a diffeological space $X$. On the other hand, Hector and Christensen-Wu used the standard simplices $Δ^p_{\rm sub}$ $(p\ge 0)$ endowed with the sub-diffeology of $\rbb^{p+1}$ and the standard affine $p$-spaces $\abb^p$ $(p\ge 0)$ to introduce the singular complexes $S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$, respectively, of a diffeological space $X$. In this paper, we prove that $S^\dcal(X)$ is a fibrant approxima…
For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we obtain an elliptic $C^*$-complex in the sense of Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^*$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles
Given any topological group $G$, the topological classification of principal $G$-bundles over a finite CW-complex $X$ is long-known to be given by the set of free homotopy classes of maps from $X$ to the corresponding classifying space $BG$. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when $X$ has dimension $2$, it seems there is a case in which such explicit classification has not been explicitly considered. This is the case where $G$ is a Lie group, whose group of components acts non-trivially on its funda…
The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology gro…
The theorem of the title is deduced from the equivalence between crossed complexes and cubical $ω$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.
In this paper, we will give the polyhedron property role to satisfy fiber homotopically equivalent relation in fiber bundle theory over suspensions of polyhedron space.
We consider Γ-equivariant principal G-bundles over proper Γ-CW-complexes with prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups with finite covering dimension Γand G such that G is almost connected.
We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.
Complexes equivalent to $S^{2k-1}$-fibrations over $S^{2k}$
Wenzhou University（温州大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院；中国科学院大学）；University of Chinese Academy of Sciences
本文研究了CW复形顶细胞的粘贴映射 $f$ 具有 $S^{2k-1}$-纤维 over $S^{2k}$ 的同伦类型条件，并确定了总空间的粘贴映射阶数，当 $k\ne 2,4$ 时，通过稳定同伦类对总空间进行分类。
The classification problem for principal fibre bundles over two-dimensional CW-complexes is considered. Using the Postnikov factorization for the base space of a universal bundle a Puppe sequence that gives an implicit solution for the classification problem is constructed. In cases, when the structure group $G$ is path-connected or $π_{1}(G) = 0$, the classification can be given in terms of cohomology groups.
For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar question are also disc…
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
We prove that for a simplicial complex $K$ whose Taylor resolution for the Stanley-Reisner ring is minimal, the following four conditions are equivalent: (1) $K$ satisfies the strong gcd-condition; (2) $K$ is Golod; (3) the moment-angle complex $\mathcal{Z}_K$ is homotopy equivalent to a wedge of spheres; (4) the decomposition of the suspension of the polyhedral product $\mathcal{Z}_K(C\underline{X},\underline{X})$ due to Bahri, Bendersky, Cohen, and Gitler desuspends.
Given a locally trivial fibre bundle $E \to B$ (with fibres and base finite complexes), an orthogonal real line bundle $λ$ over $E$ and a real vector bundle $ξ$ over $B$, we consider a fibrewise map $f : S(λ) \to ξ$ over $B$ defined on the unit sphere bundle of $λ$. Following the fundamental work of Jaworowski and Dold on the parametrized Borsuk-Ulam theorem, we investigate lower bounds on the cohomological dimension of the set of points $v$ in $S(λ)$ such that $f(v) = f(-v)$.
In this article, we prove that there is a canonical Verdier self-dual intersection space sheaf complex for the middle perversity on Witt spaces that admit compatible trivializations for their link bundles, for example toric varieties. If the space is an algebraic variety our construction takes place in the category of mixed Hodge modules. We obtain an intersection space cohomology theory, satisfying Poincaré duality, valid for a class of pseudomanifolds with arbitrary depth stratifications. The main new ingredient is the category of Künneth complexes; these are cohomologically constructible co…
Among the classical variants of the Prüfer surface, some are homotopy equivalent to a CW-complex (namely, a point or a wedge of a continuum of circles) and some are not. The obstruction comes from the existence of uncountably many `infinitesimal bridges' linking two metrizable open subsurfaces inside the surface. We show that any non-metrizable surface that possesses such a system of infinitesimal bridges cannot be homotopy equivalent to a CW-complex. More than for the result on its own, we were motivated by trying to blend elementary techniques of algebraic and set-theoretic topology.
Dochtermann introduced the loop space construction of a based graph $(G,v)$ whose basepoint is a looped vertex. He showed that the complex $C(Ω(G,v))$ is homotopy equivalent to the loop space $Ω(C(G),v)$ of $C(G)$. Here we write $C(G)$ to mean the clique complex of the maximal reflexive subgraph of $G$. In this paper, we consider its bigraph version. A bigraph is a graph equipped with its 2-coloring. We introduce the loop space construction $Ω_{/K_2}(X,x)$ of a based bigraph $(X,x)$. This is a graph such that $C(Ω_{/K_2}(X,x))$ is homotopy equivalent to the loop space of the box complex $B_{/K…
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …
We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of $\mathbb{Z}_2$-spaces. A weak equivalence is a graph homomorphism which induces a $\mathbb{Z}_2$-homotopy equivalence between their box complexes. The box complex is a $\mathbb{Z}_2$-space associated to a graph, considered in the context of the graph coloring problem. In the proof, we discuss the universality problem of the Hom complex.
