Emre Yüksel - Homology representation of deck groups

Abstract

Talk

Our vector space will be a vector space over the complex field. (Which is a Algebraically closed field) We start with some definitions:

Definition

A $G$ representation is a map $G\to GL(\mathbb{V})$ A subrepresentation is a space $W\subseteq V$ that is $G$-invariant. An Irreducible representation is a representation that does not admit any proper subrepresentation.

Theres also a notion of non-decomposable, which is different from irreducible but the same for finite groups.

Lemma

Any representation is adirect sum of irreducible representations.

Definition

Let $\rho :G\to V$ be a $G$-represetation. The character ${\chi }_V$ is a map which maps a group element of $G$ to the complex trace of the matrix.

Why are we using the trace here? Because the trace is invariant under conjugation. This makes the character a conjugation invariant.

Characters determine the representation.

Lemma

If $V,W$ are $G$-representations, then if the characters are the same, then the two representations and their vector spaces are isomorphic. (This depends on the fact that the base field is complex)

Reviewing covering spaces

The Galois-correspondence tells us that the subgroups of the Fundamental group of a space corresponds to covers of that space This is probably an explanation which only helps you if you already know the subject. Luckily I already know the subject but I’m kinda feeling bad for the others. What should I do in my talk?? Probably take my time and explain everything again… But in a subtle way. I got an idea! We make a deal: I will explain them topology, and they’ll teach me repesentation theory. Hensel explained it the other way around. Imagining the Deck group as something that acts as a quotient of the space.

Lemma

Regular Cover are exactly the covers where the fundamental group is a Normal subgroup.

We describe the fact that the lower fundamental group acts by conjugation on the upper fundamental group and furthermore on the Abelianisation (i.e. the first Homology) of the fundamental group.

Theorem (Chevalley-Weil)

Let $\Gamma$ be a closed oriented genus $g$-surface and let $\tilde{\Gamma }$ be a finite regular cover with Deck group $G$. Then $H_1(\tilde{\Sigma },\mathbb{C})\cong \mathbb{C}[G{]}^{2(g-1)}\oplus {\mathbb{C}}^2$ The dimensionality is correct. The left side of the equation is the homology of the cover, meaning the dimension might be bigger than $2g$ The representation studies here is the Deck group, I think.

Proof: If two representations have the same characters, then they are isomorphic. The character of the regular representation is ${\chi }_{reg}=\{\begin{array}{ll}|G|& g=1\\ 0& \end{array}$ (since the group acts non-trivially on itself and therefore permutes all axis non-trivially). There is some kind of Lefschetzzahl $\tau (f)$, which counts fixpoints of $f$ on surfaces or smth. We take $f$ to the the action of $g$ on the homology. The Lefschetz-number can be written out as a sum over linear maps between the zeroth, first and second homology. (We know that all actions of $g$ preserve the zeroth and second homology of connected surfaces, meaning their characters are both $1$) Doing such, we can recover the value of the character ${\chi }_{H_1}(g)$ if $g\ne 1$. We also use the formula for the Euler characteristic for a covering space $\chi (\tilde{\Sigma })=|G|2(1-g)$ where $g$ is the genus of the base surface.

Definition

The Regular representation is the Group ring $\mathbb{C}(G)$. The regular representation decomposes into all irreducible representation.

Theorem (Gaschütz)

Let $X$ be a wedge of $n\ge 2$ circle with a finite regular cover $\tilde{X}$, then we once again have $H_1(\widetilde{,}\mathbb{C})\cong \mathbb{C}[G{]}^{n-1}\oplus \mathbb{C}$

There is some k