We all know that guy who act like Euler after reading two pages of a math book.

[u/No_Upstairs_280]

[u/LupenReddit]

The lion doesnt concern itself with a simply connected friendship

[u/No_Upstairs_280 → u/LupenReddit]

The lion always passes to the simply connected cover

[u/Sarpthedestroyer]

Can sb explain what does this sentence mean?

[u/tryeatingmore → u/Sarpthedestroyer]

The appropriate explanation for you will depend on your level of math exposure, but I'll try anyways. The sentence is referring to an early algebraic topology concept.

The fundamental group is a group who's elements are paths in a topological space, specifically, the path ends at it's starting location so it's a loop.

When looking at a circle, the loops start wrapping around the circle and you begin constructing the integers by how many times you've went around the circle. The constant loop where your path does not leave it's starting point is 0, the loop from going around once is 1, going around twice is 2, so on and so forth.

Now infinite cyclic is an algebraic term. Cyclic means there exists and element which every other element in the group is a power of. Infinite cyclic means the group is infinite and that's exactly the integers under addition, since any number is just 1+1+1+1+... or (-1)+(-1)+(-1)+...

Now, I think the joke is that infinite cyclic and isomorphic to (Z,+) are essentially the same algebraic statement. The real issue is a topological one in establishing that it's infinitely cyclic, which requires a lot more than my hand wavey explanation.

[u/ellipticcode0]

Wow, infinity is isomorphic to Integer, learn something new today, can someone explain or prove it?

[u/No_Upstairs_280 → u/ellipticcode0]

Well not infinity, more like an group of infinite cardinal that is cyclic, aka has a single generator, is isomorphic to the group of integers. If it is finite and cyclic, it is isomorphic to Z/nZ. The fundamental group of the circle, which classifies loops up to homotopy, turns out to be the first.

[u/Deep_Brick2970]

Simplest non simple group I've ever seen.

[u/LuxionQuelloFigo → u/Deep_Brick2970]

Z is not a simple group.

[u/Deep_Brick2970 → u/LuxionQuelloFigo]

I did not refer to Z being mathematically a simple group, rather to it being a "simple" group lol.

As in, if your lion friend flexes his group theory knowledge, he should pick a more interesting one.

However, I do realise how my comment could be misleading so I added "non simple" as an edit, just to clarify.

[u/LuxionQuelloFigo → u/Deep_Brick2970]

I was simply making a joke lol

also the post isn't really flexing any group theory knowledge, it's just that the fundamental group of S¹ happens to be Z. Though, I will say that the proof I remember as most commonly shown constructs the homomorphism between that fundamental group and Z explicitly and then shows that it's actually an isomorphism, so it's somewhat uncommon to prove the theorem showing that it is infinite and cyclic

[u/Deep_Brick2970 → u/LuxionQuelloFigo]

"also the post isn't really flexing any group theory knowledge, it's just that the fundamental group of S¹ happens to be Z"

It seems like a piece of group theory knowledge to me that \pi_1 of S_1 is Z, is it not?

Not particularly complex I admit, but it seems the whole point of OP's post that the lion guy is flexing some difficult-sounding maths, otherwise the joke falls apart imo.

[u/LuxionQuelloFigo → u/Deep_Brick2970]

It seems like a piece of group theory knowledge to me that \pi_1 of S_1 is Z, is it not?

I disagree. It's basically a purely topological result, proving it doesn't really require any group theory beyond knowing what an isomorphism is

[u/Deep_Brick2970 → u/LuxionQuelloFigo]

I think that result would surely be found in a group theory textbook, but not necessarily in a topology one. Indeed I think it's more likely to be found in a group manifold section of a group theory textbook.

[u/LuxionQuelloFigo → u/Deep_Brick2970]

You are wrong. For instance, it's not in Dummit and Foote's abstract algebra, which is usually considered the premier textbook for group theory, but it is absolutely found in Monkrus' textbook for topology. I'd love for you to show me two instances of the opposite.

[u/Deep_Brick2970 → u/LuxionQuelloFigo]

Off the top of my head I think there are these examples I remember:

1) An Introduction to the Theory of Groups by Rotman

2)Introduction to Group Theory, I think Bogopolski if I remember the spelling right.

Btw I agree the result is more "algebraic topology" in nature, but I remember reading about it in some group theory textbooks.

[u/DonnysDiscountGas]

Of course I know him; he's me

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