The h Bar

10:01 AM

and not to mention, atomic precision control of a huge ensemble of subsystems to coerce them to give the desired outcome. It's pretty much as if reality is a computer and you are the programmer that has backdoor access

...this discussion also touched upon a related topic of digital physics in various manga and games:

anonymous

Is there any way to see my last few deleted questions on Chem SE ?

Secret

We now seemed to suspect that the physical reality is consists of information. But information theory does not really have a notion of a programmer that can freely manipulate this information. So far the physics laws only allow us to manipulate information in limited ways using other information, and information is conserved

ACuriousMind

@anonymous If you deleted them recently, there's a link to them at the bottom left of your question tab on your profile

Moses

Does this make sense "A practical blackbody can be constructed by taking a hollow cavity whose internal walls perfectly reflect electromagnetic radiation (e.g., metallic walls) and which has a very small hole on its surface. Radiation that enters through the hole will be trapped inside the cavity and gets completely absorbed after successive reflections on the inner surfaces of the cavity." It is a perfect reflector yet there is absorption?

Secret

I am not very sure if it is even theoretically allowed to manipulate information arbitrarily so that you can e.g. Make a car to materialise in midair

anonymous

10:07 AM

@ACuriousMind Uh no! (chemistry.stackexchange.com/users/39812/anonymous) I can't see it. I had asked it about 1 week back. It contained some useful information about E2 reaction mechanism.

Can mods on Chem SE see it?

Secret

But anyway, given my haphazard knowledge, despite I tried, it might turned out my analysis above is completely flawed

ACuriousMind

@anonymous I repeat, there should be a link at the bottom left saying "deleted recent questions"

But yes, if you can't find it, a mod definitely can

anonymous

@ACuriousMind Okay! Yay! I found it :) (chemistry.stackexchange.com/questions/68372/…)

It was on the questions tab

Not on the main page

ACuriousMind

5 mins ago, by ACuriousMind

@anonymous If you deleted them recently, there's a link to them at the bottom left of your question tab on your profile

told you so :P

@anonymous Also, that link of course 404s foreveryone with less than 10k on chem.SE

anonymous

@ACuriousMind LOL, I didn't notice the "questions tab" thing earlier :P Thanks a lot :)

Secret

10:12 AM

Browsing PSE does not give any question regarding on whether information can be directly manipulated.

Loong

@anonymous yes

ACuriousMind

@Secret What does "manipulating information directly" even mean? Information in the information-theoretic sense is a property of things, not a substance

It's like asking to "directly manipulate" temperature - doesn't make sense to me

anonymous

@Loong I see. I was going to ping you just before ACM replied :). BTW you seem to be active in so many chat rooms at once :P!

skullpetrol

He is a jack of all sciences :P

Secret

@ACuriousMind according to a PSE link, the top voted answer describes information as the number of yes no question one has to ask to define a system. Since Shannon entropy is the -Ln of information, and entropy in general count the number of microstates that realise a given macrostate, it does seemed to suggest that information is the number of parameters in the system need to be specified to define the system. Since the electron coooling experiment last year have demonstrated information can cool a

system, it should not be too far fetched to consider whether it is possible to change the yesno questions asked so that a given system will be transformed into a completely different system?

Moses

10:23 AM

@ACuriousMind Would it be correct to say that a black body would absorb electromagnetic radiation of say a single frequency and emit a spectrum of black body radiation of equal energy as the incident wave, thus maintaining thermal equilibrium?

ACuriousMind

@Moses What does "radiation of equal energy" mean. Radiation has power - only radiation measured over a specific time interval has energy.

anonymous

@Loong I undeleted my previous question. I thought earlier that I know the answer but it seems my intuition was wrong. But now the problem is that the question isn't appearing on the "active" or "newest" page. Unless people see the question on active on newest page I don't think it will receive any attention. Can something be done ? Here is the link (chemistry.stackexchange.com/questions/68372/…)

Loong

@anonymous I'll have a look.

ACuriousMind

@Secret That sounds completely far-fatched to me. "chang[ing] the yes/no questions" is trivial: Of course you can do some things to a systems to change the number of allowed microstates (heat it, add more particles, etc.).

Moses

@ACuriousMind Yes I meant power (energy per unit time). So it would take in radiation of a single frequency and emit a spectrum of black body radiation of equal power hence maintaining thermal equilibrium?

ACuriousMind

10:28 AM

But then you're doing something to the system. You're either asking for something that can be done trivially to every system or for magic.

@Moses A blackbody always radiates the blackbody spectrum corresponding to its temperature. If the power of incident radiation is higher than the power of emitted radiation, then the blackbody heats up until the two are equal.

Secret

Well if that implication about information being the fundamental constituent of reality as hinted in one of this months new scientist is true, then it will mean everything from electrons to mountains are made of information. Then it is quite possible to have some process that will,say, edit all the yesno questions of an ensemble of electrons so that you end up have a bunch of yesno questions that describes a car instead, thus effectively changing a bunch of electrons to a car via such process.

But

This sounds like magic and I don't think there are physics process that will allow that magnitude of rearranging information by doing something to the system to turn a system a into a completely different system b so long information is conserved

John Rennie

Cor, there's a whopping storm raging through Chester at the moment ...

The sound of the wind howling is straight out of a cod horror film.

ACuriousMind

@Secret What does it mean to be "made of information"? That's a meaningless salad of words to me. Information is a property of a system. A system cannot be "made of" a property. The sky is not made of blue and food is not made of taste

anonymous

@JohnRennie (chesterchronicle.co.uk/news/chester-cheshire-news/…) Looks pretty bad! Stay safe :). Personally I find stormy days fun and a good time to stay at home and enjoy tasty food. :P

ACuriousMind

@JohnRennie Tell us when the first zombie knocks on your door

2

Moses

10:42 AM

@ACuriousMind Yes so the black body will regulate it's temperature until the power of the incident radiation is negated.

John Rennie

@anonymous Doris somehow doesn't sound a very threatening name for a storm :-)

It's a name associated with little old ladies.

ACuriousMind

If you think little old ladies aren't threatening you must not know many little old ladies :P

John Rennie

@ACuriousMind ooh, there's scope there for ban inducing comments ... :-)

Why is a storm like a woman?

Moses

@ACuriousMind So after the black body heats up and emits the black body radiation it goes back to it's original temperature?

ACuriousMind

Didn't they change it a while back so that storms now alternate between male and female names?

@Moses Only if you turn the incident radiation off

John Rennie

10:45 AM

Well come on, someone say I don't know, why is a storm like a woman

anonymous

@JohnRennie Reminds me of Doris Yeh :P

BTW she isn't really old :'D

@JohnRennie Hell hath no fury like a woman scorned :'D

I miss stormy days. Its been a long time. In Meghalaya we used to have rain and storms almost every day of the year!

AccidentalFourierTransform

I don't know, why is a storm like a woman?

John Rennie

@AccidentalFourierTransform When she comes it's wet and wild, and when she leaves she takes your house with her!

Boom, boom!

anonymous

Poor joke ^

AccidentalFourierTransform

[TRIGGERED]

Secret

10:51 AM

@ACuriousMind right, but in that case, I no longer have any idea what they talk about when they say information is the most basic thing

John Rennie

@anonymous I don't know any good jokes ...

anonymous

@JohnRennie That's a good joke :D

ACuriousMind

@Secret Neither do I. I'm with Aaronson's quote in that article: "Does this mean that our universe is made of information, as some headlines claim? “It strikes me as a contentless question,” says Aaronson."

skullpetrol

Storms are all female names in alphabetical order iirc.

ACuriousMind

The article explains pretty well why the concept of information is important, and may be taken as fundamental rather than derived. I don't know why you insist on taking the specific phrasing of stuff being "made of information" and fantasize about it instead of engaging with the information theory itself.

AccidentalFourierTransform

10:58 AM

i.imgur.com/7YKo3w3.gifv

2

neat.

skullpetrol

Oops you were right @ACuriousMind they did change it.

anonymous

@Secret Any idea about deuterium's hyperconjugation effect relative to hydrogen ? Is it more or less?

Anyone here knows?

Wikipedia just says that C-D bond is stronger than C-H without giving any reason :/

John Rennie

11:13 AM

@anonymous the zero point energy of the C-D vibration is lower because of the higher reduced mass.

So the dissociation energy is higher

anonymous

@JohnRennie What is zero point energy ?

And what is C-D vibration?

Zero-point energy

Zero-point energy (ZPE) or ground state energy is the lowest possible energy that a quantum mechanical system may have, i.e. it is the energy of the system's ground state. Zero-point energy can have several different types of context, e.g. it may be the energy associated with the ground state of an atom, a subatomic particle or even the quantum vacuum itself. In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy (known...

Looks too complicated for me^

@skullpetrol What ?

This isn't in JEE anyway -_-

skullpetrol

Yet.

anonymous

@skullpetrol I don' really understand what you are trying to say. Take a chill pill :'D You've gone crazy.

John Rennie

@anonymous: if you look at the energy as a function of the C-H or C-D bond length it looks like this:

OK so far?

anonymous

@JohnRennie Ah, yes!

What does ZPE represent?

John Rennie

11:26 AM

If the system sat at the bottom of the potential then the dissociation energy would be the left of the two vertical dashed lines. Yes?

anonymous

@JohnRennie "dissociation energy would be the left of the two vertical dashed lines" I don't understand this. Dissociation energy is energy needed to break the bond, right?

Why should be the left of the two vertical dashed lines?

John Rennie

@anonymous yes, the dissociation energy is the energy needed to increase the bond length to infinity. If you start at the lowest point on the red curve, i.e. at the equilibrium bond distance, then the energy needed to increase the bond length to infinity is the left dashed line.

anonymous

@JohnRennie Oh okay. I get upto that. Then ?

John Rennie

OK. But the C-H or C-D bond behaves like a simple harmonic oscillator. The atoms don't sit there stationary at the equilibrium distance. They are always vibrating with an energy called the zero point energy.

I've drawn the horizontal dashed line to indicate the zero point energy i.e. it is an energy above the lowest possible energy.

anonymous

@JohnRennie Oh interesting! This "the zero point energy of the C-D vibration is lower because of the higher reduced mass." makes sense now :D. Yay!!!

John Rennie

11:33 AM

Bing!! :-)

anonymous

Reduced mass concept is one of the most fascinating concepts in mechanics :).

Thanks a lot JR!

John Rennie

My neighbour's fence has just blown down in the storm!

Moses

$u(\mu, T) = A \mu^3 e^{-\frac{\beta \mu}{T}}$ given in units energy per unit volume per unit frequency. How do yo interpret the per unit frequency if the function is in terms of frequency as well? Does this not seem strange?

skullpetrol

The storm must be leaving soon.

It's taking the fence with it too.

Moses

If the energy density depends on the frequency then what does energy density per unit frequency mean?

John Rennie

11:36 AM

@Moses It just means that to get the energy in a frequency range you integrate $$E = \int_{\nu_1}^{\nu_2} u(\nu, T) d\nu $$

@skullpetrol It's certainly wet and wild at the moment!

Moses

@JohnRennie That is correct.

John Rennie

@Moses So ... I don't see the problem

Moses

@JohnRennie Was over-thinking something simple. Apologies.

Secret

12:04 PM

@ACuriousMind Yeah, your explanation clear things up. I guess this is the first time I learn about that the phrase "fundamental “stuff” of the universe" can mean said "stuff" is a fundamental and important concept and not necessary a building block of reality. I am guessing I was misleaded by that headline the media used as quoted in Aaronson's and then end up mistakenly trying to rationalise how on earth can information be stuff because as in the context in information theory,

it is a property in systems (that may be dissimilar) rather than some kind of common building block

Yashas Samaga

@anonymous a bottle of frozen honey took more time than a bottel of honey at room temperature to roll down an inclined plane

anonymous

@YashasSamaga That is expected. What is your question?

Yashas Samaga

just told because I tried it

anonymous

Hmm. That is so obvious.

Yashas Samaga

how is that expected?

anonymous

12:10 PM

Same reason as yesterday. Frozen bottle has more I.

Yashas Samaga

omg no

you said the opposite

anonymous

And more fluid energy dissipation.

Yashas Samaga

you said boiled egg reaches faster than an unboiled egg

anonymous

And fluid friction.

No

I didn't say the opposite.

Yashas Samaga

you said an unboiled egg would have more moment of inertia (which I argued against) than a boiled egg. After that discussion, we started discussing which one would reach first. You said the unboiled egg will take more time because it had the liquid inside which would take energy for internal stuff, etc.

anonymous

12:13 PM

As far as I remember I said boiled egg has more moment of inertia.

You can check the transcript

I1w1=I2w2

Yashas Samaga

@anonymous this one

@anonymous this one too

I am going to try it with the eggs

hopefully the results won't be too close

in the honey case, the frozen one lost by a very small margin but it lost always

Max Payne

Hi everyone !

I have a small doubt regarding electric field.

Yashas Samaga

and the bottles are not symmetrical, they tend to rotate a bit sideways too

anonymous

Oh no. I was right yesterday.

Yashas Samaga

@MaxPayne What's the question?

anonymous

12:18 PM

Sorry. I wrote in a hurry.

Yashas Samaga

@anonymous how?

-__________-

Max Payne

How can electric field exist inside a wire connected in a circuit? Does it exist uniformly in the cross section or in the skin of it?

anonymous

I need to go out. Later.

Yashas Samaga

electric field exists in a wire when it is in a closed circuit

anonymous

I stand by my yesterdays argument still.

I am not sure of your setup.

Yashas Samaga

12:19 PM

if there is potential difference across the wire, there is an electric field (there must be? the electric field is responsible for increasing/decreasing the potential)

@MaxPayne it exists everywhere inside the wire

Max Payne

Ok Yashas, if we know that inside a hollow conductor, if we put charge +q, in the inner surface a charge -q appears, and in outer surface +q, applying gauss law, we get electric field 0 inside the meat of conductor.

anonymous

The Law of Inertia Experiment to assort boiled egg and raw egg

►

Yashas Samaga

current in a wire != electrostatics

Secret

@anonymous Following up on Johnrennie's answer, since CH bonds are weaker, hyperconjugation is stronger for hydrogen relative to deuterium.

anonymous

@YashasSamaga You can watch the video.

You don't need to conduct the experiment.

Also I am not sure of the effectiveness of your experimental setup

Max Payne

12:22 PM

So @Yashas If we connect an emf across the hollow conductor, will electric field then exist there?

Yashas Samaga

even I doubt my experimental setup

I had awful lot of problems doing it

but the frozen bottle always won even tho it was by a tiny margin

there could've been slipping too

anonymous

The reason I said that raw egg has more moment of inertia is because the during rotation its contents will be far away from the centre due to centrifugal/centripetal (whatever you call it) force.

Max Payne

I mean in the inner surface we connect negative terminal, and in outer surface i connect the positive terminal. Then will electric field exist in the meat of conductor>

?

Yashas Samaga

@MaxPayne yes

If there is no electric field, what else would drive the current?

anonymous

After boiling it hardens so the contents come closer to the centre

Max Payne

12:23 PM

Thats right

But I thought how can it be different in different cases.

anonymous

Raw egg ---- > 1)Contents far from centre 2) Fluid friction 3) Energy dissipation of fluid. Boiled egg ----> Contents close to centre. $$I_{boiled}w_{boiled}=I_{raw}w_{raw}$$. $$I_{boiled}<I_{raw}$$ So $$w_{boiled}>w_{raw}$$.

@YashasSamaga

Yashas Samaga

2) Fluid friction 3) Energy dissipation of fluid <= same thing

Boiled egg ----> Contents close to centre. <= needn't be, it can be anywhere

depends how it was before it was boiled

anonymous

@YashasSamaga Not really. Fluid has many more degrees of freedom in which it looses energy.

@YashasSamaga I have never seen such an abnormal egg. In most eggs it is close to the centre of mass.

You are talking about exceptional cases.

Yashas Samaga

It depends on the location of the yolk before you start boiling it

anonymous

@YashasSamaga In all of these the COM of yolk is very very close to the centre of mass of egg.

Yashas Samaga

12:29 PM

There is also a good chance that the yolk remains in the center when you rotate it

or it oscillates near the COM

if the yolk started off very close to the COM

anonymous

Also the density of yolk is greater than the white part

So COM is close to egg's COM.

Yashas Samaga

The yolk needn't always go to the wall

in an unboiled egg

even when you rotate

0celo7

why are you people talking about this

Yashas Samaga

if the yolk is a perfect sphere and it is located exactly at the center of the egg, then it won't move from its place

anonymous

@0celo7 Physics Of Eggs :D

Yashas Samaga

12:30 PM

even if you rotate it at 112312313 radians/second

0celo7

that's pretty fast

Yashas Samaga

if it is slightly off, it will mostly oscillate

if it is way too off, it might get stuck at the wall

moment of inertia of an unboiled egg is unpredictable

Secret

http://www.andersoninstitute.com/models-of-time-in-science-fiction.html
Just when I thought you are going to summarise all known models of scifi time travel (which in that case should give a article not so different from wikipedia's), the more I read down the paragraphs, the more it does not sound right.

The way it is written strongly reminds of many cranks I encountered in sciforums before, with unatural capitalisation of words, and near the end, once again pulling in the concept of mind and wrote about stuff that sound like new age esoterics.

anonymous

Centrifuge

A centrifuge is a piece of equipment that puts an object in rotation around a fixed axis (spins it in a circle), applying a potentially strong force perpendicular to the axis of spin (outward). The centrifuge works using the sedimentation principle, where the centripetal acceleration causes denser substances and particles to move outward in the radial direction. At the same time, objects that are less dense are displaced and move to the center. In a laboratory centrifuge that uses sample tubes, the radial acceleration causes denser particles to settle to the bottom of the tube, while low-density...

Yashas Samaga

-.-

0celo7

12:31 PM

@JohnRennie ::chemical bat sigal::

Yashas Samaga

Read my messages again @anonymous

anonymous

Don't forget to consider centrifugal effect.

Yashas Samaga

In a centrifuge, the test tubes are kept at the SIDE.

if you keep the test tube in the center, it'll have no effect

John Rennie

@0celo7 :: John rouses from his slumber ::

anonymous

From where do test tubes come in?

0celo7

12:32 PM

@JohnRennie uh, that was quick

anonymous

In eggs?

0celo7

I have to eat breakfast and walk to work before I can explain

Yashas Samaga

pff

0celo7

you can go back to sleep

Yashas Samaga

I am talking about the centrifuge

John Rennie

12:33 PM

:: John's head slumps onto the desk once more ::

Yashas Samaga

When I said the yolk needn't move to the edge becaz the yolk could be at the center, you quoted the wiki link on centrifuge

in a centrifuge, the test tubes are at the sides

which is why the contents get separated

if you keep the test tube in the center, it won't work

anonymous

@YashasSamaga In raw egg if you spin it then the liquid yolk will stick to the walls while spinning at high rate.

Try it with a bottle.

Yashas Samaga

@anonymous it depends on the location of the yolk

in that image if the smaller sphere is the yolk

and the outer sphere is the egg

if you rotate the egg at 12312313 radians per second

the yolk won't move

anonymous

@YashasSamaga The yolk isn't a solid mass.

In raw eggs

Yashas Samaga

it will get stretched but the yolk as a whole won't move to the edge

@anonymous

the yolk has to be placed unsymmetrically for it to completely move to the edge @anonymous

John Rennie

12:37 PM

@YashasSamaga: I think you're being cute here. The yolk will never be exactly symmetrically placed so if you spin the egg it will always move towards the outside.

Yashas Samaga

@anonymous and in the honey experiment, it was a common liquid inside so I doubt if that is a good replacement for an egg

@JohnRennie The more closer it is to the center, the more time it'll take to reach the edge?

Is is possible to give an answer to the question "Which one reaches first: boiled egg or an unboiled egg"?

Wouldn't it depend on too many factors?

John Rennie

I would try the experiment and see what happens

Yashas Samaga

There is a possibility that the yolk is at the edge in a boiled egg too, so it starts with an unfair disadvantage which might help it reach after the unboiled egg?

Secret

[[[WARNING: The following analyses concerns fringe material. It is to be reminded our goal is to deduce why it is fringe and then move on]]]

http://www.andersoninstitute.com/quantum-tunneling.html
No, quantum tunneling is not time travel. The evenescent wave is not travelling per sec. Also the rebound wave is the reflected wave, and there are no waves from the other side of the barrier that enters it (because there are no sources there)

http://www.andersoninstitute.com/near-lightspeed-travel.html
That's just time dilation. Theoretically solar sails can get us 50% closer to this goal (if I recall correctly, its max speed is 0.5 c)

http://www.andersoninstitute.com/alcubierre-warp-drive.html
https://en.wikipedia.org/wiki/White%E2%80%93Juday_warp-field_interferometer
The warp field experiment is STILL inconclusive.

http://www.andersoninstitute.com/wormholes.html
We still don't have viable ways to construct them

http://www.andersoninstitute.com/cosmic-strings.html
No cosmic strings found yet

http://www.andersoninstitute.com/tipler-cylinder.html
Not practical, either you need infinite length or heaps of exotic matter. [Recent 2016 calculations](http://link.springer.com/article/10.1007%2Fs10714-016-2117-3) even suggest it will be enveloped by an event horizon

heather

1:02 PM

this egg debate intrigues me - you'll have to tell us the results @YashasSamaga

Secret

http://www.andersoninstitute.com/casimir-effect.html
Nope, we need to make a wormhole first, it is unsure how. Also see http://chat.stackexchange.com/transcript/71?m=30357265#30357265

0celo7

1:16 PM

@JohnRennie ok, chemistry signal again

@BalarkaSen listening

Balarka Sen

hm?

ACuriousMind

@Secret Honestly, examining non-mainstream material for where exactly it fails is a pointless exercise. There's so much garbage out there you could spend your whole life debunking it and it still wouldn't be enough. It's easier to produce nonsense than to refute it. Don't waste your time refuting garbage, go learn something true instead.

0celo7

@BalarkaSen What is this?

I must say, I'm not a fan of this.

Balarka Sen

Oh, you mean the album

0celo7

It's noise!

Balarka Sen

1:21 PM

It's a coherent story :P

Physics Meta

-2

Q: Why is there so little answering of elementary physics questions?

I have begun to notice that it's nearly impossible to get answers for queries on classical mechanics concepts on this website (despite them involving advanced ideas). While say a quantum mechanics question which asks a basic concept is buzzing in comparison. Why is this? Surely if people can answ...

1

Q: The number of questions/answers and upvotes/downvotes per day

I am curious to know what are the number of questions/answers and upvotes/downvotes per day on http://physics.stackexchange.com Can someone who who have some spare time can make something like this ? Thanks.

discussion questions answers voting

0celo7

@BalarkaSen what is the story at 15:14?

Balarka Sen

I don't remember. What's the song title?

0celo7

beats me!

sounds like noise!

I can't tell what the title is.

the track listing is messed up

Balarka Sen

Ah, A Small Plot Of Land.

It's about a victim of the murderer/artist

0celo7

1:26 PM

christ

19:13

what is going on???

Balarka Sen

That's Baby Grace talking isn't it?

Ah, no, the next one. Hallo Spaceboy is classic

0celo7

9

A: Why does the density matrix $\rho$ obey a wrong-signed Heisenberg equation of motion?

$\rho_\psi$, the density matrix, is not an observable/operator evolving in the sense of the Heisenberg equation of motion $$ \mathrm{i}\hbar\frac{\mathrm{d}}{\mathrm{d}t} A = - [H,A]$$ since it is defined, as you correctly write, as a projector on states, hence it is time-dependent in the Schrödi...

@ACuriousMind Is the derivative of the density matrix rigorously defined?

I guess it's bounded...

ACuriousMind

@0celo7 Sure, it's trace-class/Hilbert-Schmidt and hence lives in a Banach space ($H^\ast \otimes H$)

0celo7

@ACuriousMind So I shouldn't even ask what the first $dA/dt$ is supposed to mean?

ACuriousMind

@0celo7 It's some bounded operator. :P

It's also irrelevant for the question as such

Balarka Sen

1:31 PM

If you don't like this you probably should not hear the next album! It's total chaos.

0celo7

I like friendly music

Balarka Sen

The wilder the better for me. What a fantastic death abyss!

Did you figure out the Deck transformation action on $\widetilde{\Bbb{RP}^2 \vee \Bbb{RP}^2}$?

0celo7

@BalarkaSen I think so. I'll write it down tonight or tomorrow

Balarka Sen

Gotcha.

0celo7

@BalarkaSen How would one show invariance of boundary using nonrelative (co)homology?

Balarka Sen

1:40 PM

Invariance of boundary?

What's that?

0celo7

@BalarkaSen $M$ a topological manifold, $\partial M$ and $\mathrm{int}\,M$ are disjoint.

1

Q: Boundary of a topological manifold invariant?

Let $M=(X,\tau)$ be a topological manifold with boundary. One can proof that the interior $Int(M)$ and boundary $\partial M$ of the manifold are distinct sets. I was wondering if someone knows a good reference to cite the proof (Lee only presents this proof as an exercise)? Thanks

manifolds manifolds-with-boundary

Balarka Sen

I am not sure why you want to use (co)homology. It's obvious from the definition, because interiors are locally R^n whereas boundary are locally H^n

But you can use local cohomology

0celo7

It is obvious, but that's apparently not a proof.

You need to prove that "locally Rn" and "locally Hn" are mutually exclusive

Balarka Sen

Ah, fair point.

Yeah, so local (co)homology

0celo7

@BalarkaSen Assuming $M$ is smoothable, I'd like to use de Rham.

So take $x\in M$. Does $x$ have a neighborhood homeo to both an open ball and a half ball with $x$ on the boundary?

ACuriousMind

1:45 PM

@0celo7 If you just need to show that $\mathbb{R}^n$ and $H^n$ are not homeomorphic, consider the behaviour under removing a point from both (from the boundary in the latter case).

0celo7

@ACuriousMind No, that's not enough.

At least, I don't think.

I don't think that $x$ has a neighborhood homeo to both $R^n$ and $H^n$...

@ACuriousMind It changes the fundamental group, sure.

ACuriousMind

You need to show that if there is one open set containing $x$ that's homeomorphic to one of the two, there is no open set containing $x$ that's homeomorphic to the other (with $x$ on the boundary of $H^n$), right?

0celo7

Yeah, that sounds right.

@BalarkaSen Local (co)homology?

ACuriousMind

@0celo7 Okay, so do what the MSE answer you linked says: Use relative homology of the point.

0celo7

@ACuriousMind I don't know relative homology, and I have reason to believe it's not necessary.

John Rennie

1:54 PM

I'm being nagged by an AI

If I wanted to be nagged I would have got married!!

ACuriousMind

@0celo7 What reason?

0celo7

@ACuriousMind It's an exercise in two books that do not talk about relative (co)homology.

ACuriousMind

Are they reasonably advanced books about their topic? You should know by now that they often assume you know more than what's contained in their pages :P

0celo7

@ACuriousMind No, they are not.

Yashas Samaga

@JohnRennie you aren't married? aw

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