Perturbative

Anyway, it's been nice chatting to you all once again, I gotta head out now :)

For the lectures this semester, I'll actually just be happy if I can absorb like 70% of the material, my background is admittedly quite lacking in some areas

Yeah we had an information talk where they went over all that with us

Yeah makes sense, it's pretty cool how much flexibility you get too in general

Ah cool

Yeah for some reason I thought you had to do the thesis over a year

Ohh sorry I know what you mean now

Ahh did you load up on credits on your earlier semesters, in that case, to be able to do that?

@AlessandroCodenotti Did you get affected much by Covid towards the end of your studies over there?

Bodigheimer for Alg Top I and Huybrechts for AG

@AlessandroCodenotti

I'm attending Alg Top I, Adv Geom I, Topological Manfifolds, Intro to Surgery Theory and Algebraic Geom I for the moment but I might not write the exam for all

Thanks! @TedShifrin @JoeShmo

Just following online for the time being, things are bad in Germany at the moment so not sure when I'll actually get around to moving there

I was really excited to be able to register and stuff

Yep! @AlessandroCodenotti

Hey @AlessandroCodenotti :), yeah been a long time

Zoom classes should be the only option at every university where infection rates are high in my opinion

Lol, at least the chances of getting infected are lower than if you were teaching

How are you doing these days by the way? @Ted :)

Hey @Ted, you were right yesterday, that map $F$ had nothing to do with it, I just assumed it did

Hey everyone

Say I have a smooth map between manifolds $F : M \to S^n$ how can I show that there is an embedded open disc $D$ in $M$ small enough such that $M \setminus D$ is homotopy equivalent to $M \setminus \{p\}$ where $\{p\}$ is some point in $D$?

Hey everyone!

@TedShifrin Oh yes you're right, I was making some silly error which made me think it would be undefined for odd $n$

Ah okay I see, I was skimming through some stuff in a couple books and it seemed on the surface like it was defined only for even $n$

Is the Hopf invariant only defined for even $n$ in maps $f : S^{2n-1} \to S^n$?

Thanks for that, I'll take a bit of time to digest that though

Oh well I guess I should've realized that

Ah I didn't realize that the statement was only for vector bundles over the same base

Like they're equal up to isomorphism but not literally equal

For Stiefel-Whitney classes, it's said that one of the immediate consequences of the axioms is that for isomorphic vector bundles $\xi$ and $\eta$, it follows that $w_i(\xi) = w_i(\eta)$, but isn't it the case that there's just some isomorphism $f^* : H^i(B(\xi), \mathbb{Z}/2) \to H^i(B(\eta), \mathbb{Z}/2)$ for which $f^*(w_i(\xi)) = w_i(\eta)$?

lmao

@TedShifrin I think that's all for now :)

Take my star @AlessandroCodenotti

@TedShifrin Milnor and Stasheff but they mention after they define it that the fibers are never empty so it's equivalent to surjectivity

Oh lol I meant surjective

Do some people define the projection map of a vector bundle $\pi : E \to M$ without having the condition that $\pi$ must be subjective?

Ah thanks for clearing that up @AlessandroCodenotti @MikeMiller@LeakyNun

But "a category with a single object" isn't exactly a group. It's the Hom set of a category with a single object that is a group

Isn't there a joke that a group is just a category with a single object?

Ah okay thanks @MikeMiller

When people talk of Smale's proof of the h-Cobordism Theorem are they referencing his paper "On the structure of manifolds"?

@AminIdelhaj @MikeMiller Thanks for your replies! Stolz and Schommer-Pries are closest to my interests I would say, hence my interest at Notre Dame. It's nice to have Behrens there as well for algebraic topology stuff.

@MikeMiller Do you know if Notre Dame is one of the stronger places in Geometry and Topology in the US at the moment?

Though I do need a bit more background before I could tackle some of those things I guess

Thanks for those suggestions!

Thanks for linking that paper btw

Oh I see, I just assumed moduli spaces were from algebraic geometry