## 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 $$n-1$$.