Is there such thing as a p-adic manifold?

There is.

What's fun about that is that a $p$-adic manifold isn't a manifold. This is an example of an "adjective noun" not being a special kind of "noun".

@Thorgott how does a correspondence between complex line bundle and U(1) principal bundle be given?

does the cellular approximation theorem hold in the smooth category? I mean smoothly homotopic to a cellular map.

just follows from smooth approximation?

@onepotatotwopotato ah it seems it follows by using the orthonormal frame bundle

In Rudin's rank theorem, there is a $C^1$ function $F$ from $E\subset R^n$ to $R^m$ that has constant rank $r$ for every $x\in E$. Let $A=F'(a)$ for some $a\in E$ and $P$ be a projection in $R^m$ whose range is that of $A$. The conclusion of the theorem is that there are open sets $U,V$ in $R^n$ with $a\in U$ and 1-1 $C^1$ mapping $H$ of $V$ onto $U$ such that $$F(H(x))=Ax+\varphi(Ax),\quad x\in V.\tag{66}$$I have a couple of questions where Rudin interprets the above decomposition.

(a) Why is $P$ 1-1 on $F(U)$?

(b) What does he mean that $F(U)$ has one point "over" $A(V)$?

(c) Why can we regard $\varphi$ as the graph of $F(U)$?

By the way, $\varphi$ is another $C^1$ function mapping from $A(V)$ into the null space of $P$.

If anyone sees this and its immediately clear to them, I'd be very grateful for a reply. I've been working with rank theorem for some days now. :(

@onepotatotwopotato yes, and the inverse construction is given by taking the product with $\mathbb{C}$ and quotienting by the diagonal $U(1)$-action

I hope a holomorphic map $S\to\Bbb CP^1$ corresponds to a holomorphic line bundle over $S$ under that classifying correspondence but there seems to be no clue.

well, pulling back the universal bundle (which is holomorphic) along a holomorphic map yields a holomorphic bundle

but this won't be a correspondence

Is the universal bundle holomorphic?

@Thorgott If your statement is true then why not?

@onepotatotwopotato Sure

@onepotatotwopotato first of all, it's not clear what equivalence relation on maps would force the pullback bundles to be isomorphic (there is no such thing as holomorphic homotopy)

second of all, not every bundle arises as pullback of the universal bundle over projective space

Is there a way of making the intersection symbol bigger here?

@Joe \Huge{\cap} ?

$\Huge{\cap}$

would not do that

I don't think there's anything out of the box, but nothing's stopping you from using the usual ways of resizing stuff

there's scalerel, resizebox, adjustbox, ...

$\displaystyle\bigcup_{i = 1}^\infty \alpha_i \quad\text{vs}\quad \mathop{\scaleobj{1.25}{\bigcup}}_{i = 1}^\infty \alpha_i$

ah, I guess chatjax does not have scalerel

well, copy that code into your document and adjust to your liking

but honestly I'm not sold on the idea at all

for a start you'll note that the $V$ also looks much too small

that will only be worse if you scale up the operator

the way you avoid problems like this is generally by not having $\left(\frac{\text{big}}{\text{fractions}} + \frac{\text{inside}}{\text{function arguments}}\right)$ like that

$\displaystyle T_p X = \bigcap_{f \in I(X)} V(\partial f / \partial z_1 |_p (z_1 - a_1) + \cdots + \partial f / \partial z_n |_p (z_n - a_n))$ consider this instead

you may want to sprinkle some additional parentheses and perhaps adjust your notation for the differentials

math.stackexchange.com/q/5030749/681666 the community bot can take out bounties?!

oh I guess the bounty was started by the OP and then they deleted their account?

the activity log reads very strangely

Interestingly 4 out of 6 answers are from new contributors.

is there a systematic way to compute the limit $\lim_{N,M \to \infty} \exp(c_1N + c_2M)$ where $c_1$ and $c_2$ are real numbers?

Naively taking the limit one at a time seems to not be a good method as the result then depends on the order you take the limits.

@skullpatrol well it's a (sub?) high school level word problem

True dat.

Taking the limit $\lim_{\delta \to \infty} \exp(c_1(N+\delta) + c_2(M+\delta))$ also seems like a bad idea as this puts constraints on what $c_1$ and $c_2$ can be, which is not supposed to be the case in my problem.

but seemingly the limit is not really well defined then

"is there a systematic way to compute $X$" when $X$ is not a well-defined expression in most cases is a meaningless question

you have to figure out what you want it to mean first

@BenSteffan That is typically what happens. And what then happens is that, after the bounty period, the community bot will award half of the bounty amount to the highest upvoted answer left after the bounty was started.

-_-

In this case, however, there is a good reason to cancel the bounty.

Quality?

the answers are... well

it's a bit of a hot mess :)

Trash bin fire🔥

In the top-right corner of Stack Exchange sites there are several icons. For example, one of them with the tool-tip "A list of all 183 Stack Exchange sites" is used to get to this chat room. The icons have two different functions on left-click; sometimes they open a drop-down (such as to get to this chat room) and sometimes they link to a different web page. I can't figure out how to control which function is performed. It doesn't seem to be holding the left-click for longer or clicking

in a different spot on the icon. I just keep trying until I randomly get the desired outcome. Does anyone know how to control this?

they should just open dropdowns (...?)

the only time I ever get taken to a different website when clicking those is when the site hasn't fully loaded in

in any case, you should be able to access the sites you get redirected to via the dropdowns

You can always get to the other webpage by center-clicking instead of left-clicking, the problem is getting to the drop-down sometimes. Maybe it's just a matter of the page loading slowly sometimes then. That seems consistent with my experiences just a bit strange so I didn't attribute it to that.