Mathematics - 2025-02-12

hbghlyj

01:46

The deficiency of a finite presentation $\langle S | R \rangle$ is $|S| - |R|$.
The deficiency of a finitely presented group $G$ is the maximum of the deficiency over all presentations of $G$.
https://math.stackexchange.com/questions/478841/finitely-presented-group-with-fewer-relations-than-generators proved that a finitely presented group with positive deficiency has to be infinite. I wonder if there exists an infinite finitely presented group with negative deficiency.

Thorgott

01:58

surely the answer should be yes

I would assume GL(2,Z) is an example, though I can't prove it

northerner

I have a bit of a strange observation regarding postfix and prefix notation. Aren't spaces a necessary part of the notation? For example given postfix 153- isn't it ambiguous if this would be 15-3 or 1-53?

But 15 3 - would remove the ambiguity.

hbghlyj

08:44

Does there exist a closed 3-manifold with fundamental group $\mathbb Z^2$?

Anacardium

09:36

Are convex topological spaces necessarily path connected? I am wondering whether the line joining any two points will be continuous with respect to the underlying topology.

In topological vector spaces this is true.

Alessandro Codenotti

09:59

@Anacardium How do you define convex in a space which is not a TVS?

Thorgott

11:01

@hbghlyj no

16

A: Closed 3-manifolds with free abelian fundamental groups

(I assume all occuring 3-manifolds to be orientable and closed) A manifold with a free abelian fundamental group cannot be a connected sum of non-trivial 3-manifolds since its fundamental group is not a free product. A prime manifold is either $S^1\times S^2$ or irreducible (Hatcher's notes on 3...

Ryder Rude

12:52

if a civilisation has an oracle to solve the Halting problem, can they know all truths about the natural numbers? or can they know more truths than we can know, but not all?

can this civilisation breach some fundamental limitations that we have, or is the Halting problem irrelevant here

suppose the civilisation can perform countably infinitely many computations in a finite time (which solves the Halting problem)

Ben Steffan

13:33

If I have to draw the Borromean rings in my 3-folds exam I will fail

one potato two potato

13:53

what is a 3-fold exam?

Ben Steffan

an exam that consists of three parts, duh :)

...an exam about 3-manifolds

Soumik Mukherjee

Which 3?

:)

Thorgott

@BenSteffan more importantly, can you calculate their link group

one potato two potato

@BenSteffan Who is the instructor?

Ben Steffan

@SoumikMukherjee probably $S^3$, $L_{7, 1}$ and, uhh, the Poincare homology sphere

@Thorgott I don't know what that is so I'm going to say yes and hope you don't test me on it :)

@onepotatotwopotato Paula Truöl

Thorgott

14:06

it's just the fundamental group of the complement

Ben Steffan

she's a (visiting?) postdoc at the MPI here

@Thorgott ah, well I can tell you its abelianization :)

and mumble something about wirtinger presentations

one potato two potato

as far as I know, Ben is interested in high dimensional manifold, dimension at least 4.

Thorgott

@BenSteffan yes!

Wirtinger presentations are so cool

Ben Steffan

@onepotatotwopotato "high dimensions, such as 4"

as the joke goes, homotopy theorists are interested in smooth manifolds ($=$ spaces $M$ with a preferred map $M \to \mathrm{BO}$)

Thorgott

high starts at 5 and/or 7, depending on whom you ask

Ben Steffan

14:12

on unstable days you may also demand that $M$ is a Poincare complex

@Thorgott we don't talk about dimension 6

I've also seen people refer to things up to dimension 8 as low-dim

maybe we should strike a compromise and call manifolds of dimension 5 through 8 "medium-dimensional"

one potato two potato

someone told me once that people count low up to dim 4 because after that, the theory can be formalized algebraically completely.

Thorgott

"algebraically completely" sounds like a way too strong statement for anything remotely true

@BenSteffan why??

Ben Steffan

don't ask me

one potato two potato

it's about 3 or 4 years ago so my memory could be distorted

Thorgott

actually, I may have been wrong about 7

hbghlyj

14:30

https://topospaces.subwiki.org/wiki/Fundamental_group_determines_homology_groups_for_compact_connected_orientable_3-manifold : For a compact connected orientable manifold of dimension 3, the fundamental group is sufficient to determine the isomorphism classes of all the homology groups.

I wonder if the fundamental group is sufficient to determine orientability

for a compact connected manifold of dimension 3.

one potato two potato

well by looking at the top dimensional homology?

Ben Steffan

14:45

@onepotatotwopotato what do you have in mind?

one potato two potato

Hmmm maybe I should assume closed 3-manifolds

Ben Steffan

you probably should, but I'm still failing to see how this would follow from homology alone

one potato two potato

btw I recently learned that some people study 3-manifolds with infinitely generated fundamental group

Ben Steffan

sure, in one case $H_3$ is $\mathbb{Z}$ and in the other its 0, but for the nonorientable case you only have PD with $\mathbb{F}_2$-coeffs., so...

one potato two potato

It's $\Bbb Z$ or $0$ depending on orientability.

Ben Steffan

14:52

@onepotatotwopotato yes, but how does that help? :^)

one potato two potato

Oh I didn't know the usual PD assume orientability.

well stay in orientable world

Ben Steffan

@onepotatotwopotato sure, if you assumed your manifolds were orientable to begin with then the fundamental group does determine orientability, vacuously :^)

hbghlyj

Yes, the usual PD with $\mathbb{Z}$ coefficients assumes orientability

Thorgott

15:11

do you get something from F_2-Poincaré and UCT

Ben Steffan

I don't think so

Ben Steffan

15:35

not sure whether I actually believe this is true or not, but I know too little about non-orientable 3folds to say

here's a partial result: all non-orientable closed 3-folds have infinite fundamental group

in fact, have infinite $H_1$

Ah, it's false: There is a non-orientable 3-fold $M$ with $\pi_1 M \cong \mathbb{Z}$

see mathoverflow.net/questions/304326/non-orientable-3-manifolds

Thorgott

oh duh

Avv

17:41

Hello Guys

there used to be a professor at Princton with famous youtube course

He does not come here anymore?

I forgot his name :/

Thorgott

the only professor with a course on youtube that used to frequent this chat was Ted Shifrin

but Ted was at UGA, not Princeton

flowian

19:28

Hello everyone. Is it true that for $n>2$ the shape $D^n\setminus \{0\}$ is not deformation retractable onto its boundary $S^{n-1}$. Any book suggestions where I could read further about it?

Ben Steffan

It's false.

leslie townes

yeah can't you just write down the map

or a map, not to imply uniqueness

Ben Steffan

You very much can, and should.

flowian

It's true for $D^2\setminus \{0\}$ though, right?

Ben Steffan

No.

leslie townes

19:31

i don't see where 2 would come up as a dividing point, for the kinds of maps i would be thinking about writing down

flowian

Sorry, I meant to say: You can deformation retract $D^2\setminus \{0\}$ to $S^1$ !

Ben Steffan

Yes.

flowian

Thank you. And for $n>2$ is the retraction map similar - you map every point non-injectively to the boundary through a straight line which crosses the origin of $D^n$ and $\partial D^n?$

Ben Steffan

Your description is a bit sketchy but I think you mean the right thing

flowian

hmm

Ben Steffan

19:37

$D^n \setminus \{0\} \cong S^{n - 1} \times (0, 1]$, and then the retraction is given by sending $(x, t)$ to $(x, 1)$.

flowian

oh nice!

thank you :)

Ben Steffan

Welcome.

flowian

Hey @BenSteffan, do you have alg. topo. book recomendations besides Hatcher's?

Ben Steffan

Ah, the dreaded question :) Tom Dieck is pretty good; it's certainly very different from Hatcher. May's "A concise course" is very helpful, but more as a reference than for a first read. The lecture notes by Davis & Kirk have the benefit of covering more advanced topics than most other books, but I've only used them for references purposes.

(the question is dreaded because there is no single great book I feel I can endorse wholeheartedly)

I think if I had to choose a book nowadays I would probably choose tom Dieck, but it's definitely a good idea to have a few books available to cross-reference things.

flowian

Ah makes sense, thanks. I'm looking to mainly strengthen my understanding of concepts I've seen during graduate studies of the course

Ben Steffan

19:45

Well, one of the benefits of Hatcher is that the book has a lot of exercises

flowian

True. I feel it's a bit all over the place though and each chapter is very packed

Ben Steffan

Actually Hatcher does not cover a lot of material for the length of the book; or rather, he covers the expected breadth, but he's often rather superficial and doesn't bother too much with the details

But everybody seems to dislike Hatcher in their own way, somehow :)

his book, not the man himself, in case that wasn't clear

flowian

oh is he? I felt I need to dedicate lots of time and dig from different sources to understand him

:)

Ben Steffan

well, yes, that's common

but I think that's indicative of what I said, no? You have to do this because he is often superficial

flowian

yeah I guess you're right..I felt I spend lots of work with that book

Ben Steffan

19:56

anyways, if you dislike Hatcher I recommend checking out tom Dieck, or maybe May

flowian

Thanks a lot

Ben Steffan

You're welcome, again :)

flowian

is alg topo your research area btw?

Ben Steffan

my future research area, I guess

I'm in my masters' right now

flowian

oh awesome. Good luck!

Ben Steffan

19:59

Thanks!

Thorgott

I love tom Dieck, but I always feel awkward recommending it cause most people looking for an alg top textbook don't want to learn about cofibrations before, like, almost anything else

Ben Steffan

yeah

but if you read hatcher you basically won't learn what a cofibration is at all...

Thorgott

you do, but he refuses to name them that lol

Ben Steffan

well yeah, :)

Well he does define them, in section 4.H...

putting your discussion of cofibrations into the Eckmann-Hilton duality bonus section few people ever look at seems a littlle... strange

Hatcher really doesn't prepare you too well for reading actual papers

flowian

20:24

To show that $\Bbb R^n\setminus \{0\}$ deformation retracts to the unit sphere S^{n-1}, is homotopy $H(x,t):= x(1-t) + \frac{x}{x}t$ sufficient?

Ben Steffan

I assume you forgot a norm there

flowian

20:50

thank you

Ben Steffan

...but yes, then that will work

X4J

What’s an intuition to find $f:\mathbb{R^2} \rightarrow \mathbb{R}$ and $x \in \mathbb{R^2}$ s.t for any sequence $(v_n)$ which approaches zero, the limit $\frac{f(x+v_n)-f(x)}{||v_n||}$ exists but $f$ is not differentiable at x?

Thorgott

@BenSteffan he defines the HEP all the way in chapter 0, to be fair, but the actual role and structure of cofibrations never gets quite made clear

Ben Steffan

yes, of course

Thorgott

and something important like the Barratt-Puppe sequence is introduced too little too late

Ben Steffan

21:01

the whole homotopy theory section feels rather superficial to me

Thorgott

the section in tom Dieck on all this material is great, but I think the right audience for that section is one that has already experienced homology theories and the like

@BenSteffan nah I do think there is a bunch of good stuff in there

Ben Steffan

all the big theorems are there

it's just that everything feels trimmed down to just that

the bonus sections are nice though

especially in chapter 4

X4J

@leslietownes i guess you know the answer

Thorgott

the appendices are like half of chapter 4

but yeah, that's where the real good stuff is

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