Topic
Poincaré complex
About: Poincaré complex is a research topic. Over the lifetime, 20 publications have been published within this topic receiving 232 citations. The topic is also known as: Poincare complex.
Papers
113 citations
16 citations
TL;DR: In this article, it was shown that odd-dimensional manifolds have even Euler characteristic zero unless the dimension of the manifold is a multiple of 2πk+1 + 1.
Abstract: It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We prove here a generalisation of these statements: a $k$-orientable manifold (or more generally Poincare complex) has even Euler characteristic unless the dimension is a multiple of $2^{k+1}$, where we call a manifold $k$-orientable if the $i^{th}$ Stiefel-Whitney class vanishes for all $0
9 citations
TL;DR: The unstable tangent fibration of a Poincare complex is defined in this paper, which is consistent with the manifold case, and it exists uniquely for each PoINCare complex X up to fibrewise homotopy equivalence.
Abstract: The unstable tangent fibration of a Poincare complex is defined so that it is consistent with the manifold case. It exists uniquely for each Poincare complex X up to fibrewise homotopy equivalence and, furthermore, if a Poincare embedding structure exists on the diagonal X→X×X, its normal fibration is the tangent fibration.
8 citations
1 Jan 1993
TL;DR: The n-dimensional quadratic L-group Ln(A) was introduced by Mishchenko as mentioned in this paper to describe the symmetric part of the surgery obstruction, and are not 4-periodic in general.
Abstract: for any oriented covering X with group of covering translations π. If X is the universal cover, σ∗(f, b) is the surgery obstruction, and σ∗(f, b) = 0 if (and for n ≥ 5 only if) (f, b) is normally bordant to a homotopy equivalence. The n-dimensional quadratic L-group Ln(A) was expressed in Ranicki [4,6] as the cobordism group of n-dimensional quadratic Poincare complexes over A, which are chain complexes C of finitely generated free A-modules with an n-dimensional quadratic structure ψ inducing Poincare duality isomorphisms (1+T )ψ0 : Hn−∗(C) ∼= H∗(C). The symmetric L-groups L(A) (n ≥ 0) were introduced by Mishchenko [1] to describe the symmetric part of the surgery obstruction, and are not 4-periodic in general. The n-dimensional symmetric L-group L(A) is the cobordism group of n-dimensional symmetric Poincare complexes over A, which are chain complexes C of finitely generated free A-modules with an n-dimensional symmetric structure φ inducing Poincare duality isomorphisms φ0 : Hn−∗(C) ∼= H∗(C). A geometric Poincare complex X determines its symmetric signature (or “higher signature”)
8 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 2 |
| 2018 | 2 |
| 2017 | 1 |
| 2011 | 1 |
| 2003 | 1 |
| 2000 | 1 |