8 – How to connect different node bundles in one page

I am building a web site about football teams in Drupal 8 and I have the following content types Team, Player, Manager, Game etc. I would like to connect nodes from Player, Manager content types in the page of a Team.

So when viewing the Team page, below the Team nody/body I would like to have tabs with Manager node/body and Player node/body.

So, how is possible to connect nodes from different content types?

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differential geometry – Inverse of the tangent map for tangent bundles

(Physicist here trying to understand some more involved differential geometry).
In the context of differential geometry we define the tangent bundle as

$ TM = bigsqcup_{p in M} T_p(M) $ where a general element is given by the cartesian product $(p,v_P)$ for $p in M$ and $v_p in T_p(M)$.

We have the natural map $pi: TM rightarrow M$, explicitly $pi(p,v_p)=p$, called the tangent map.

However, we then define the inverse tangent map as $pi^{-1}: M rightarrow TM$ but I dont understand how we can do this. The tangent map is surjective and so its inverse does not uniquely map onto an element in the tangent bundle.

Vector bundles admitting resolution by ample line bundles

Let’s assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $mathcal{O}(n)$ for $nin mathbb{Z}$. I have several related questions:

  1. Which vector bundles on $X$ admit a finite resolution by vector bundles in $C$? There might be obstructions in chern classes. You can assume the vector bundle on $X$ has whatever the chern classes you want it to have. For example you can assume all of its chern classes vanish.

  2. Are there examples of varieties that this happens for all vector bundles?

  3. Is it true that for ACM (arithmetically cohen-macaulay) varieties (2) is true?

at.algebraic topology – Do $mathbb{R}^n$ bundles have unit sphere bundles?

Recall there are multiple ways to define the unit sphere bundle of a vector bundle. One is by constructing a fiberwise vector space metric and declaring the sphere bundle to have fibers the unit spheres in each of the vector space fibers. The other way is to use the equivalence of vector bundles and principal $O(n)$ bundles and then since $O(n)$ acts faithfully on $S^{n-1}$ we may replace the fiber to obtain a sphere bundle.

There is, however, a distinction between vector bundles and $mathbb{R}^n$ bundles, that is fiber bundles with fiber $mathbb{R}^n$ and structure group $Homeo(mathbb{R}^n)$. It is not always possible to assign a coherent vector space structure to the fibers to make it a vector bundle. Is there a notion of an associated sphere bundle in this context?

Now I tend to just believe fiberwise constructions are always possible, so I would believe someone if they said we could pick a metric on the fibers of any $mathbb{R}^n$ bundle and define its unit sphere bundle in the same way as for vector bundles. But on the principal bundle side of things, this seems to me to be asserting that $Homeo(mathbb{R}^n)$ has a subgroup $H$ so that the inclusion $H rightarrow Homeo(mathbb{R}^n)$ is a weak equivalence, and $H$ preserves the unit sphere $S^{n-1}$ while acting faithfully on it. This seems very difficult to believe.

PageSpeed Insights warns me about my AMP pages: “Remove duplicate modules in JavaScript bundles”

I’ve checked the score of my AMP pages with Google PageSpeed Insight, and I received this message in red:
“Remove duplicate modules in JavaScript bundles”

This is the initial piece of code of my AMP pages. The fact is that I started with AMP pages several years ago, and now I’m not sure if I have to remove/update/add something.

<!doctype html>
<html amp lang="en"><head><meta charset="utf-8" />     
<link rel="preload" as="script" href="https://cdn.ampproject.org/v0.js">
<link rel="preload" as="script" href="https://cdn.ampproject.org/v0/amp-experiment-0.1.js">
<link rel="preload" as="script" href="https://cdn.ampproject.org/v0/amp-ad-0.1.js">
<script async custom-element="amp-analytics" src="https://cdn.ampproject.org/v0/amp-analytics-0.1.js"></script>
<script async src="https://cdn.ampproject.org/v0.js"></script>
<script async custom-element="amp-user-notification" src="https://cdn.ampproject.org/v0/amp-user-notification-0.1.js"></script>
<script async custom-element="amp-ad" src="https://cdn.ampproject.org/v0/amp-ad-0.1.js"></script>
<script async custom-element="amp-auto-ads" src="https://cdn.ampproject.org/v0/amp-auto-ads-0.1.js"></script>
<script async custom-element="amp-social-share" src="https://cdn.ampproject.org/v0/amp-social-share-0.1.js"></script>
<script async custom-element="amp-sidebar" src="https://cdn.ampproject.org/v0/amp-sidebar-0.1.js"></script>
<script async custom-element="amp-iframe" src="https://cdn.ampproject.org/v0/amp-iframe-0.1.js"></script>

ag.algebraic geometry – Cancellation property of vector bundles on non-proper varieties

Krull-Schmidt theorem for proper varieties over a field implies that given an isomorphism of vector bundles between $Eoplus F$ and $Goplus F$ we can deduce that $E$ and $G$ are isomorphic. My question is, does this cancellation property hold if I replace the proper variety $X$ with $Xtimes mathbb{A}^1$?

What if I know for the two vector bundles $E$ and $G$ on $Xtimes mathbb{A}^1$ ($X$ is smooth proper or projective) there is a reflexive sheaf $F$ such that $Eoplus Fcong Goplus F$. Then can we deduce that there is a vector bundle $F’$ such that $Eoplus F’cong Goplus F’$. (The reflexive sheaf is
a vector bundle outside a closed subvariety of the form $Ztimes mathbb{A}^1$ where $Z$ is a closed subvariety of $X$ of codimension $2$)

algebraic geometry – Equivalent Definitions of Projective Morphisms with Line Bundles

This is part b) of Vakil 17.3 A, self-study. We say a morphism $pi: X to Y$ is projective if $X$ is isomorphic as $Y$-schemes to the relative $operatorname{Proj}$ of some quasicoherent sheaf of algebras $mathcal L$ on $Y$ (such that $mathcal L$ is finitely generated in degree $1$).

In part a) of this question, we are asked to show that $pi$ is projective iff there is a finite type quasicoherent sheaf $mathcal L_1$ on $Y$ such that there exists a closed embedding of $Y$-schemes $X hookrightarrow mathbb P mathcal L_1$. I believe I have done this.

Part b) takes it a step further: given an invertible sheaf $mathcal L$ on $X$, we want to show $pi$ is projective with $mathcal O(1) simeq mathcal L$ iff there is a finite type quasicoherent sheaf $mathcal L_1$ on $Y$ such that there exists a closed embedding of $Y$-schemes $i: X hookrightarrow mathbb P mathcal L_1$ with $i^* mathcal O_{mathbb P mathcal L_1}(1) simeq mathcal L$.

I am only unable to show that given the latter, we get the isomorphism $mathcal O(1) simeq mathcal L$ of the former. The only observation I have been able to make is that if we could show $i^* mathcal O_{mathbb P mathcal L_1}(1) simeq mathcal O(1)$, we would be done. For example, we could show that $mathcal L$ is very ample and that $X$ is a proper $Y$ scheme, which we have not shown, and, in fact, not even make precise outside the context in which $Y$ is an affine scheme, so I don’t believe this is the intended route.

ag.algebraic geometry – Extending vector bundles from a specific square to its neighborhood

Let’s consider the square $x=0,x=1,y=0,y=1$ in $mathbb{A}^2$, we denote it by $S$. Let $text{Spec}(R)$ be a smooth affine scheme over a field. We consider a vector bundle on $text{Spec}(R)times S$, where on each side is just a pullback of a fixed vector bundle on $text{Spec}(R)$. On intersections of the sides they get identified by an automorphism. Let’s assume the monodromy around the square is not trivial, this prevents us from extending the vector bundle from $text{Spec}(R)times S$ to the whole $text{Spec}(R)times mathbb{A}^2$. Is it possible to at least extend the vector bundle to an open in $text{Spec}(R)times mathbb{A}^2$ that contains $text{Spec}(R)times S$?

If the answer is positive how about replacing $text{Spec}(R)$ by a smooth scheme that is union of two affines and the behavior of vector bundle on the square times each affine is similar to the previous paragraph. (Here we need to extend the gluing automorphism too)

(Topologically) different tangent bundles on a same manifold

Is there an example of a topological manifold in which different smooth structures give rise to tangent bundles which are not isomorphic as topological vector bundles?

My (our) own attempts or remarks, mostly obtained from discussing this question with other people:

  • By the Wu formula, the two tangent bundles would have in any case the same Stiefel–Whitney classes. This is in fact what originally motivated my question.

  • Exotic 7-spheres are no good to produce such examples because they all have trivial tangent bundles.

  • "Different smooth structures correspond to different (stable) linear structures on the tangent microbundle."