Make a product amalgamated groups by splitting a closed variety along a codimension 1 subvariety

In the article "A Division Theorem for Varieties" by S. E. Cappell,

https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf

the following "inverse" of Seifert-van Kampen's theorem for closed varieties is indicated in the introduction (see page 71).

Let $ Y $ to be a connected connected dimension variety (say, differentiable) $> 4 $. Suppose that the fundamental group of $ Y $ is written as a merged product $ pi_ {1} (Y) = G_ {1} ast_ {H} G_ {2} $ two groups $ G_ {1} $ and $ G_ {2} $ along a common subgroup $ H leq G_ {1} cap G_ {2} $. Then there is a closed connected codimension $ 1 $ submanifold $ X subset Y $ such as $ Y setminus X $ has two components, let's say with closures $ Y_ {1} $ and $ Y_ {2} $ in $ Y $, such as $ pi_ {1} (X) = H $ and $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.

My question: How can this result be proven?

Cappell states that the evidence is easily derived from the methods developed in section I ยง3 of the document. However, I do not see how these methods can be adapted. In fact, the methods are based on the existence of a homotopy equivalence $ f colon Y rightarrow Y $ to another variety already divided by a submultiple of codimension $ X $, and we consider $ X = f ^ {- 1} (X) $ and wants to make the restriction of $ f $ at $ X rightarrow X $ an equivalence of homotopy by deformation $ f $ (and changing $ X $). In my question, however, it is not clear how to choose a first $ X $ work with. In addition, the sleeve exchange technique only seems useful to make the card induced $ pi_ {1} (X) rightarrow pi_ {1} (Y) $ injective (by killing elements in the kernel), but not to produce the desired groups $ pi_ {1} (X) = H $ and $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.

Ag.algebraic geometry – A class derived from coherent sheaves with a $ geq $ 1 codimension support

Let $ X $ to be a smooth algebraic variety. I would like to understand the relationship between the following two categories:

  1. $ D ^ b_ {cd, 1} text {Coh} (X) subset D ^ b text {Coh} (X) $: the complete subcategory of the category derived from coherent sheaves with cohomologies supported in codimension $ geq 1 $.

  2. $ D ^ b text {Coh} _ {cd, 1} (X) $: The category derived from the category Abelian coherent sheaves supported in codimension $ geq 1 $.

I was trying to show that for a bounded complex $ F ^ bullet $ coherent sheaves with cohomologies supported in codimension $ geq 1 $ there is a quasi-isomorphic complex $ E ^ bullet $ with $ E ^ i $ being supported in codimension $ geq 1 $, which would show that they are equivalent, but I have encountered problems like $ i _ * circ i ^ * $ for $ i $ a closed immersion not being accurate (the idea was to take the union of support cohomologies and restrict the complex to this union). Is it still true that these categories are equivalent? If no, are they at least pleasantly linked?

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 $.