Note about Stiefel-Whitney classes on real Bott manifolds

关于4维流形的Dold-Whitney可平行化性

On Dold-Whitney's parallelizability of 4-manifolds

本研究证明闭可定向4维流形可平行化等价于其二阶Stiefel-Whitney类、一阶Pontryagin类和欧拉示性数均为零，详细梳理了Kirby基于4维球面$SO(4)$-丛分类的论证思路。

不可定向 $4$ 流形的分支覆盖表示

Branched covering representation of non-orientable $4$-manifolds

本文证明每个闭连通不可定向 PL $4$ 流形 $X$ 是 $\RP^4$ 的简单分支覆盖，且 $X$ 是扭曲 $S^3$ 丛 $S^1 \simtimes S^3$ 的简单分支覆盖当且仅当第一 Stiefel-Whitney 类 $w_1(X)$ 有整提升，覆盖度 $d \geq 4$ 在 $\RP^4$ 情形需与 $w_1^4[X]$ 同奇偶，分支集在 $d \geq 5$ 时非奇异、$d=4$ 时仅有节点奇点。

Spin structures of flat manifolds of diagonal type

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

On $5$-manifolds with free fundamental group and simple boundary links in $S^5$

We classify compact oriented $5$-manifolds with free fundamental group and $π_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $π_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.

Knot concordance, Whitney towers and L^2 signatures

We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain {\em universal} groups. Finally, we…

具有指定Thom-Boardman奇点的同维流形之间的映射

Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities

本文将Eliashberg的h-原理推广到同维光滑流形的一般通用映射，给出连续映射同伦于具有指定Thom-Boardman奇点的通用映射的充要条件，并在三维情形用Stiefel-Whitney类和上同调类重新表述。

关于一个经典嵌入定理的新视角

New perspectives on a classical embedding theorem

本文将Kishimoto与Matsushita关于三角化流形的最新结果与判断光滑闭流形嵌入欧氏空间的经典Stiefel-Whitney类判据相联系，并回顾了Atiyah的K理论条件。

A note on the characteristic classes of negatively curved manifolds

We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite covers of) the Gromov-Thurston examples of negatively curved manifolds. We mention some geometric corollaries: the MinVol question, a lower bound for degrees of covers having tangential maps to the non-negatively curved duals, and estimates for the complexity of some representation…

Spin-structures on real Bott manifolds

Let $$M_{n}\stackrel{\mathbb R P^1}\to M_{n-1}\stackrel{\mathbb R P^1}\to\ldots\stackrel{\mathbb R P^1}\to M_{1}\stackrel{\mathbb R P^1}\to M_0 = \{ \bullet\} $$ be a sequence of real projective bundles such that $M_i\to M_{i-1}$, $i=1,2,\ldots,n$, is a projective bundle of a Whitney sum of a real line bundle $L_{i-1}$ and the trivial line bundle over $M_{i-1}$. The above sequence is called the real Bott tower and the top manifold $M_n$ is called the real Bott manifold. There are a few ways to decide whether there exists a Spin-structure on an oriented flat manifold $M^n$. An oriented flat man…

Conjugation spaces and equivariant Chern classes

Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in the Z/2-equivariant cohomology of X with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of h, forgetting the involution on X, and the Stiefel-Whitney classes of the real bundle of fixed points.

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids…