John R. Klein
Let M be a Poincare duality space of dimension at least four. In this paper we describe a complete obstruction to realizing the diagonal map M -> M x M by a Poincare embedding. The obstruction group depends only on the fundamental group and the parity of the dimension of M.
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Connected power operations and simplicial Poincaré duality
本研究在链复形上引入连通循环对角结构以诱导稳定幂运算，借助标准增广单纯形整链的庞加莱对偶代数结构，为大类增广单纯对象的循环对角构造提供有效方法，可用于表示无上积的谱的幂运算。
Poincaré Complex Diagonals and the Bass trace Conjecture
本文研究了有限支配的 Poincaré 对偶空间中对角映射的 Poincaré 嵌入存在性总障碍与恒等映射 Reidemeister 迹的关系，并证明了在偶数维且基本群为二阶循环群直积时该嵌入存在。
We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional mani…
We make explicit Poincaré duality for the equivariant $K$-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the $K$-theory orientation.
For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal map of a manifold X, one recovers a result of Chas and Sullivan about the homology of the free loop space LX.
Methods are developed to relate the action of a principal fibration to relative Whitehead products in order to determine the homotopy type of certain spaces. The methods are applied to thoroughly analyze the homotopy type of the based loops on certain cell attachments. Key examples are (n-1)-connected Poincare Duality complexes of dimension 2n or 2n+1 with minor cohomological conditions.
Baues and Bleille showed that, up to oriented homotopy equivalence, a Poincare duality complex of dimension $n \ge 3$ with $(n-2)$-connected universal cover, is classified by its fundamental group, orientation class and the image of its fundamental class in the homology of the fundamental group. We generalise Turaev's results for the case $n=3$, by providing necessary and sufficient conditions for a triple $(G, ω, μ)$, comprising a group, $G$, $ω\in H^{1}(G;\mathbb{Z}/2\mathbb{Z})$ and $μ\in H_{3}(G, \mathbb{Z}^ω)$ to be realised by a Poincaré duality complex of dimension $n$ with $(n-2)$-conn…
Loop spaces of $n$-dimensional Poincaré duality complexes whose $(n-1)$-skeleton is a co-$H$-space
本文研究具有余 $H$-空间 $(n-1)$-骨架的单连通 $n$ 维 Poincaré 对偶复形，在特定条件下证明其环路空间分解，统一已有结果并给出新例子，还推出收缩核关系与单关系代数同调描述。
Local Inertness of Poincaré duality complexes
该研究证明特定同调条件下庞加莱对偶复形顶胞附着映射局部化后惰性，改进了已有结果，还给出6维庞加莱对偶复形的环空间分解，并得到非惰性p-局部映射的新例子。
Loop Space Splittings of Sphere Bundles over Highly Connected Poincaré Complexes
研究球面丛的总空间循环空间分解条件，基于高连通性庞加莱复形和实向量丛的结构，确定循环空间分解的核心方法与贡献。
Suspension splittings of 5-dimensional Poincaré duality complexes and their applications
本文研究了5维庞加莱对偶复形的悬垂分裂性质，探讨其同调类型与代数数据的关系，并应用于计算余cohomotopy群及cohomotopy集合。
We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for Poincare duality complexes in dimension three to the relative case of Poincare duality pairs. The results for Poincare duality complexes are recovered by restricting the results to the case of Poincare duality pairs with empty boundary. Up to oriented homotopy equivalence, three-dimensional Poincare duality pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the d…
In this paper we focus on maximal complex De Concini-Procesi models associated to root arrangements of types A, B, C, D and we compute inductive formulas for their Poincare' series.
Top cell attachment for a Poincare Duality complex
本文研究单连通闭庞加莱对偶复形中骨架包含映射的基于环路空间何时具有右同伦逆，给出整系数条件，发展新方法得到 p-局部版本并恢复有理结论，应用于矩角流形和拟环面流形。
We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows …
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 define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's and develop enough algebraic topology in this setting to prove the Poincare duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.
In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{Δ_X},$ the homotopy cofibre of the diagonal map $Δ_X:X\rightarrow X\times X.$ We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.
Permutohedral complex and complements of diagonal subspace arrangements
本文研究了对角子空间排列补集的同调结构，并通过排列体与立方体的投影关系，揭示了余代数环乘法的构造方法。
We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on $H^{n-1}(X;\mathbb{Z}_p)$ for each odd prime $p$. A stronger result is obtained when localized at odd primes.
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré duality with coefficients in the group ring $Λ$ ($Λ$-complexes). Standard homology theory implies that $X$ is a $\mathbb{Z}$-PD complex. Therefore by Browder's theorem, $X$ has a Spivak normal fibration which in turn, determines a Thom class of the pair $(N,\partial N)$ of a mapping cy…
Loop space decompositions of $(2n-2)$-connected $(4n-1)$-dimensional Poincaré Duality complexes
本文利用\cite{BT2}的结果，极大地简化并增强了Beben和Wu关于特定Poincaré对偶复形环路空间同伦分解的工作，并将其沿多个方向进行了推广。
We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
We factor the virtual Poincare polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as $t^{2u} (t^2-1)^r Q_{G/H}(t^2)$ for a polynomial $Q_{G/H}$ with non-negative integer coefficients. Moreover, we show that $Q_{G/H}(t^2)$ divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.
We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when $X$ is a finite CW-complex of dimension $n$ and $π_1(X)$ is virtually a Poincaré duality group of dimension $\geq n-1$, then $π_i(X)$ is not finitely generated for some $i$ unless $X$ is homotopy equivalent to the Eilenberg--MacLane space $K(π_1(X),1)$; when $M$ is an $n$-dimensional closed manifold and $π_1(M)$ is virtually a Poincaré duality group of dimension $\ge n-1$, then for some $i\leq [n/2]$, …
The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.
Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$. Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev's conjecture on classification. We prove Turaev's conjectures on realization and splitting. We sho…
This paper is an introduction to the use of the cobordism of chain complexes with Poincaré duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of Surveys in Surgery Theory, Ann. of Maths. Studies, Princeton, 2001) which is an introduction to algebraic surgery using forms and formations.
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