Fundamental Group

September 10, 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Alexander Givental and Fomenko and Fuchs

We have some space $X$ with a basepoint $x_0$, and define $\pi_1(X, x_0) := \pi((S^1,0),(X,x_0)) = \text{cone of }\Omega X$. path conected.

You might think a group has one operation but a group actually has three operations, one binary which is the composition, one unary which is the inverse, and one nullary which is the identity. Constant loop will play the roll of the identity element, have reverses of loops be called invereses ($\gamma^{-1}(t) = \gamma(1-t)$), and multiplication is defined by composition of loops.

If we have a loop about $x_1$ and a path from $x_0$ to $x_1$, then we can get a loop about $x_0$ by conjugation in this group of loops. By the way, the loop of groups here is what is called the fundamental group. What we can say is that:

$\pi_1(X^{CW},x_0) = \pi_1(\text{sk}_2X^{CW},\text{sk}_0)$ and by definition, if $X$ is simply connected, then $\pi_1(X,x_0) = \{e\}$. In particular, for $n \ge 2$, $\pi_1(S^{n\ge 2}) = \{e\}$.

Coverings (and Fundamental Groups)

Definition. A covering is just a map $p$ from some other space $T$ to $X$ (usually written vertically rather than horizontally). Gneerally we think $X$ is path connected.

Beware that covering is not the same of just a homeomorphism or something I might have to clarify later what the warning was.

Then we overed the Homotopy Lifting Lmema.

So if you break the rectangle into grids, you can find a erfined enough partition, apparently some similar proof to Heine-Borel Theorem, way to use compactness of the cube.

What we can do now is that ?

To prove this, we first take a Lemma where $Z = I^0$, $I^1$, or $I^n$ in general. This Lemma is proved via the diagram I drew in Krita.

A polyhedron is something like $\{x : Ax \le b\}$. $A$ is a $m \times n$ matrix.

We can get a map by just tracing any point in the cylinder back to the base of the cylinder, and then mapping that back via the lemma. So now the question is more verifying that it is continuous.

Something like “if we define the cylinders small enoguh, they each land in a defining neighborhood in $X$“.

Something analogous in multivariable calculus where given $a$ and $b$ we want to find some $f$ such that

$df = a \;dx + b\;dy$ and to get this we can draw a square, integrate first over $y$ and then integrate over $x$.

I think he’s discussing the theorem in 6.6

Moreover: $p^{-1}(x_0) \simeq p_*\pi_1(T_1?) \forwardslash \pi_1(X,x_0)$ right cosets.

Now we do the map-lifting theorem. There is a map defined in the only possible way and we need to consider continuity.

Finally getting to criterion for equivalence of coverings.

If we have two coverings one from $T$ and one from $T'$, then we can take map-lifting theorem to get two homeomorphisms between $T$ and $T'$.

If $X$ is semilocally simply connected, then there exists a unique universal covering from a simply connected space. We can then do a construction of the universal covering.

Next Lecture…

I think $E(X,x_0)$ denotes the universal covering quotiented by something.

This discussion is supposedly related to the third problem on the homework.

What is the inverse image of this thing in the loop space? It consists of all paths which end in $V$ and whichif I connected the endpoint with that point by a path inside $U$, the composite path is homotopic to $\gamma$.

Compact open condition…

From this we get that the universal cover $E(X,x_0)/(\gamma \sim \tilde{\gamma})$ is the universal cover of this space.

Now we define Deck Transformations.

$\pi_1(X,x_0)$ acts freely and discretely on $\tilde{X}$, and $X = \tilde{X}/\pi_1(X,x_0)$. This is supposedly obvious now, but I don’t get it.

Consider $\pi \subseteq \pi_1(X,x_0)$. We have this universal cover $\tilde{X} \rightarrow X$

Moreover, $\pi$ acts ferely and discretely on a simply connectd $Y$.

He drew some tree and said tree contractible, maybe you need some technology and stuff. What is clear is that any finite tree is contractible. Also related to some words. Every loop is compact so it lands in a finite part of it.

We then talked about abstract deck transformations, and mentioned something like $p_*(\gamma_{t_0}^{t_1}) \in N(\pi)/\pi$. There was a theorem, map lifting theorem, existence of map, image of fundamental group here lies in image of that group.

If $\pi \subseteq \pi_1(X,x_0)$ is normal, then the corresponding covering $T,t_0 \xrightarrow{p} X,x_0$ is called regular.

ITS JOVER BRUH HES BRINGING GALOIS THEORY INTO THIS… my worst nightmare comes back 😭

Apparently the relationship was between regular coverings of $x_1,...,x_n - \cup (x_i = x_j)$ over $S_n$ are somehow analogous to normal extensions of $\mathbb{C}(x_1,...,x_n)$ over $\mathbb{C}(a_1,...,a_n)$.