Generalized Normal Bundles for Locally-Flat Imbeddings

On the Existence of Cross Sections in Locally Flat Bundles

1. A principal bundle 33(X, G) with base space X and group G is called "locally flat" if the structural group G can be reduced to a totally disconnected subgroup GiCG, or in other words: if there exists a totally disconnected subgroup GiQG, a principal bundle 33i(X, Gi) and an injection j: Soi—>33 [l ]. For instance, if a differentiable bundle 33 admits an infinitesimal connection whose curvatu...

Locally definable fiber bundles

Let η = (E, p, Y, F, K) be a locally definable fiber bundle and f, h : X → Y two locally definable maps. If f and h are locally definably homotopic, then f ∗(η) and h∗(η) are locally definably fiber bundle isomorphic. 2000 Mathematics Subject Classification. 14P10, 14P20, 03C64.

Locally trivial W∗-bundles

Flat Bundles on Affine Manifolds

We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine m...

Generalized Product Theorems for Torsion Invariants with Applications to Flat Bundles

This note announces generalizations of the product theorems for Wall invariants and Whitehead torsions due to Gersten [5], Siebenmann [7, Chapter VII], and Kwun and Szczarba [6], and applies these theorems to study torsion invariants of the total space of a flat bundle. The generalized product theorems are described in §§1 and 2. The applications are found in §3. These theorems were discovered ...