manifolds – bundles of trivial lines outside codimension 3

Let $ X $ to be a complex CW (possibly a topological / smooth variety) of dimension $ n $, $ L to X $ a complex bundle of lines and $ Y subset X $ a sub-complex (possibly a submultiple) contained in the codimension 3 skeleton of $ X $, such as $ L $ is trivial about $ X-Y $. Is it true in this case that $ L $ is trivial about the whole $ X $?

This obviously applies to the linear holomorphic bundles of smooth manifolds, in which case this corresponds to the complex codimension 2, but I would like to be able to replace holomorphic by smooth or continuous.

Geometric topology – Wild character of the codimension 1 subvarieties of the Euclidean space

This question arose from this pile trading post. I am writing a thesis on the $ s $-Cobordism and Siebenmann's work on the end obstructions. Combined, they give a quick proof of the uniqueness of the smooth structure for $ mathbb {R} ^ n $, $ n geq $ 6. Siebenmann's theorem says pretty much that for $ n geq $ 6 a contractible $ n $-collecteur $ M $ which is simply connected to infinity integrates (smoothly) inside a compact collector. Since this compact variety is contractible, by the $ s $-cobordism, it is diffeomorphic to the standard $ n $-disk $ D ^ n $ (see Minor Conferences on the $ h $-cobordisme for example). It follows that $ M = text {int} D ^ n $ is diffeomorphic to $ mathbb {R} ^ n $.

The problem is that the case $ n = $ 5 is not covered. I am aware of Stallings beautifully written On the piecewise linear structure of the Euclidean space but I'm looking for a way to deal with the $ n = $ 5 case via Siebenmann's end theorem and the good $ s $theorem of -cobordism (see link to the question mse). This brings me to the next question, which is interesting in itself

Since codimension 1 is well integrated and smooth $ S subset mathbb {R} ^ {n + 1} $, is there a diffeomorphism auto $ mathbb {R} ^ {n + 1} rightarrow mathbb {R} ^ {n + 1} $ who wears $ S $ in a region limited to one dimension $ mathbb {R} ^ n times (-1, 1) $ ?

Now if $ M $ is a multiple that is homeomorphic to $ mathbb {R} ^ $ 5, the product $ M times mathbb {R} $ is homeomorphic to $ mathbb {R} ^ $ 6, and therefore also diffeomorphic. Subject to the existence of diffeomorphism in my question, we could find a diffeomorphism $ f: M times mathbb {R} rightarrow mathbb {R} ^ $ 6 that cards $ M times $ 0 in $ mathbb {R} ^ 5 times (-1, 1) $. This would produce a good $ h $-cobordism between $ M $ and $ mathbb {R} ^ $ 5 taking the area between $ f (M times 0) $ and $ mathbb {R} ^ 5 times $ 1 in $ mathbb {R} ^ 5 times mathbb {R} $. Since $ M $ is simply connected, the good $ s $-cobordism theorem applies and shows that $ M $ and $ mathbb {R} ^ $ 5 are really diffeomorphic.

Ag-algeic geometry – Calculation of the codimension of the variety defined by a system of quadratic forms

Suppose I have a $ m times n $ matrix $ L $where each entry is $ L_ {i, j} (x_1, …, x_s) $ which is a linear form on $ mathbb {C} $. Let $ mathbf {y} = (y_1, ldots, y_n) $. Consider the system of quadratic equations
$$
L. mathbf {y} = mathbf {0};
$$

it's a system of $ m $ quadratic equations and leave $ V subseteq mathbb {C} ^ {s + n} $ to be the affine variety defined by these equations. I'm interested in calculating the codimension of $ V $. Since the equations have a special form (it can be expressed as above), I thought there might be an easier way to calculate the codimension … Since I'm not really sure I would appreciate any comment on how to proceed. on the calculation of the codimension of $ V $. Thank you.

Geometry of complements to codimension 2 compacts

Let $ K subset R ^ n $ be a compact (non-empty) cover size $ n-2 $. In particular, K $ do not separate $ R ^ n $ (even locally). I will equip $ M = R ^ n-K $ with the distance function $ d $ associated with the limited flat Riemannian metric of $ R ^ n $. The metric space $ (M, d) $ is incomplete, leave $ ( bar {M}, bar {d}) $ denote its completion of Cauchy. Since the integration of identity $ (M, d) to R ^ n $ decreases the distance, it extends to a continuous map $ f: bar {M} to R ^ n $.

Question. is $ f $ injective?

Less formally: if points in $ M $ are close in the Euclidean metric, as a result they can be connected by a short path in $ M $?

Note. The answer is positive if I assume that the Hausdorff dimension of K $ is inferior to $ n-1 $.