Notes on Covering Spaces as Extensions

This post assumes knowledge of fiber bundles, the group action functor, groupoids, and basic vector calculus. I am in the process of learning the topics discussed below, and I deeply appreciate constructive feedback. 

How does a big space cover a little one?

Given a covering space $E \to B$ we can uniquely lift any path in the base space (once you choose a starting point) to a path in $E$. Conversely, we can create a covering space of $B$ by letting the fiber over $b$ be $F(b)$.

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Formally: If $B$ satisfies a few topological constraints*, the functor $\text{Fiber}$ is an equivalence of categories

$\text{Fiber}: \text{Cov}/B \to \text{Set}^{\Pi_1(B)}$

between the category of covering spaces $\text{Cov}/B$ and the functor category $\text{Set}^{\Pi_1(B)}$.

*$B$ is locally path-connected and semi-locally simply-connected

Covering spaces $E \to B$ are classified by functors, $F: \Pi_1(B) \to \text{Set}$. The fundamental groupoid is the group of automorphisms of the appropriate fiber functor. A fiber functor is a forgetful functor from the category of finite sets with $G$-action to the category of finite sets.

Sidenote: $\infty$-groupoids = spaces up to homotopy. Thus, we can think of $\Pi_1$ as capturing the information of a space up to 1-homotopy, and truncating the information about higher homotopies.

What is the analogue of a covering space for groupoids?

It’s called a discrete fibration. This is a functor $p: E \to B$ that satisfies the unique path lifting lemma (for any morphism $f: x \to y$ in $B$ and $\tilde{x} \in E$ lifting $x$, there’s a unique morphism $\tilde{f}: \tilde{x} \to \tilde{y}$).

Sheaves and fibrations are generalizations of the notion of fiber bundles. Discrete fibrations $E \to B$ are also classified by functors $B \to \text{Set}$.

As curious persons, we like to take things apart to see how they work.

We can convert a decomposition of a space in terms of simple pieces $\xrightarrow{into}$ a collection of vector spaces (or modules) and linear transformations (or homomorphisms).

Let’s take something simple apart to get an intuition we can abstract off of: our favourite trivial fiber bundle, the cylinder.

unnamed (5)unnamed (6) The sequence $0 \to F \to E \to B \to 0$ is an example of a chain complex.

The Extension Problem: $0 \to F \to ? \to B \to 0$

The extension problem is essentially the question: Given the end terms $F$ and $B$ of a short exact sequence, what possibilities exist for the middle term $E$?

Screenshot from 2014-10-29 22:19:42

Extensions of the group $B$ by the group $F$, that is, short exact sequences $1 \to F \to E \to B \to 1$, are classified by 2-functor from $B \to 2Aut(F)$.

Examples of this are the more familiar classifications of central group extensions using $H^2$ or $\text{Ext}$.

What does this category $\text{2Aut}(F)$ look like?

  • objects: $F$
  • morphisms: automorphisms of $F$
    Screenshot from 2014-10-29 23:00:00
  • 2-morphisms: morphisms between the automorphisms of $F$
    Screenshot from 2014-10-29 22:59:18

For more on this, I recommend Dr. Baez’s Lectures on n-Categories and Cohomology, which will explains why “…generalizing the fundamental principle of Galois theory to fibrations where everything is a group gives a beautiful classification of group extensions in terms of nonabelian cohomology.”

Excited yet? Let’s look at chain complexes a bit more carefully.

If you have vector calculus running through your veins and at your fingertips, then you’re already familiar with a differential complex:

$\mathbb{H_1} \xrightarrow{grad} \mathbb{H}_{curl} \xrightarrow{curl} \mathbb{H}_{div} \xrightarrow{div} \mathbb{L}_2$

(where $\mathbb{H}_{curl}, \mathbb{H}_{div}$ are the domains for the curl and div operators respectively.)

You’ll note that the composition of any two consecutive maps is zero.  This is a key point of the definition of a chain complex (a sequence of modules connected by homomorphisms such that the composition of any two consecutive maps it zero).

In the category of groups, this is equivalent to the question: What groups $B$ have $A$ as a normal subgroup and $C$ as the corresponding factor group?

Screenshot from 2014-10-26 21:08:36
$ker(f)=0$, $im(f) = ker(g)$, $im(g) = C$, Source

In general, for a short exact sequence: $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$

$A$ is a subobject of $B$, and the corresponding quotient is isomorphic to $C$:

$C \cong B/f(A)$ , (where $f(A) = im(f)$).

What are these 0’s for?

0 represents the zero object, such as the trivial group or a zero-dimensional vector space. The placement of the 0’s forces $f$ to be a monomorphism and $g$ to be an epimorphism.

These chain complexes generalize past where our geometric intuition can follow, but it’s nice to be grounded in examples.

“Two objects $X$ and $Y$ are connected by a map $f$ between them. Homological algebra studies the relation, induced by the map $f$, between chain complexes associated with $X$ and $Y$ and their homology. This is generalized to the case of several objects and maps connecting them.”- Source

Curious as to why we need the the composition of consequent maps to be 0 and how this fits into homology? I recommend Dr. Ghrist’s notes on Homology from Elementary Applied Topology.

How can a big commutative algebra define a little one?

Classifying how a big space will cover a little one amount to classifying how a little commutative algebra can “sit inside” a big one.

We can study the ways a little thing $k$ can site inside a bigger thing $K$ ($k \hookrightarrow K$) by keeping track of the symmetries of $K$ that fix $k$. These “fixing symmetries” form a subgroup of the symmetries of K. $Gal(K/k) \subseteq Aut(K)$.

Let $k$ and $K$ fields such that $k \hookrightarrow K$.

In this case, $K$ is called an extension field of $k$ (denoted $K/k$).

I’m going to interject here and give a bit of context: Field extensions are fiber bundles with zero-dimensional fibers (covering spaces). 

A fiber bundle with zero-dimensional fibers has a total space that “gives multiplicity” to the points of the base space.

(This means that a multiple-valued function on the base space has a good chance of being interpretable as a single-valued function on the total space.)

Screenshot from 2014-10-26 21:28:26

But now I’m getting ahead of myself, without grounding you in classical commutative algebra!

Now that you have a glimpse of the importance of studying extensions, let’s get back to those: 

Let $Aut(K/k)$ be the set of all $k$-automorphisms of $k$

$Aut(K/k) = \{\sigma \in Aut(K) : \sigma\vert_k = Id_k\}$

But why is the restriction of $\sigma$ to $k$ an automorphism of $k$?

If $k$ is a splitting field of a family $\mathcal{P}$ of polynomials, then any action $\sigma$ preserves the set of roots of $\mathcal{P}$, and since $k$ is generated by these roots, the action $\sigma$ preserves $k$ (i.e. $\sigma k = k$).

Then $Aut(K/k)$ is a group, the automorphism group of $K/k$, or the Galois group of $K/k$, denoted as $Gal(K/k)$.

the category $\text{Fields}$:

  • objects: Fields, i.e. $F_1, F_2$
  • morphisms: Field extensions, $F_1 \xrightarrow{\phi} F_2 := F_2/F_1$ (note that morphims are all injective)

The Fundamental Theorem of Galois:  Given a polynomial, the intermediate fields of the splitting field are in one-to-one correspondence with the subgroups of the Galois group.

Suppose $G$ acts transitively on $X$. Pick any figure $x$ of type $X$ and let H be its “stabilizer”: the subgroup consisting of all guys in $G$ that map $x$ to itself. Then we get a one-to-one and onto map

$f: X \to G/H$

sending each figure $gx$ in $X$ to the equivalence class $[g]$ in $G/H$.

We can use this to set up a correspondence between sets on which $G$ acts transitively and subgroups of $G$. This is one of the principles lurking behind Galois theory

— John Baez, The Tale of Groupoidification

 

If we’re looking for amazing applications of Galois theory within, say, arithmetic, then we might as well read Kronecker or Hecke.

If we interpret Galois theory in a very expansive way, then the Erlangen Program, and Cartanian geometry, are legitimate consequences.

Thanks to David Yang and Semon Rezchikov for helping me correct my visualization of a category of representations, and thanks to Aaron Slipper for teaching me basic Galois Theory!

Sources

Classifying Spaces Made Easy
Lectures on n-Categories and Cohom.
Motivic Spectral Sequence
Topology for Physicists
Covering Spaces
Homology: Elementary Applied Topology
Exact Sequence

Postscript: We have the beginnings of a map of things that can be extended to define other things:

Fields $\leftrightarrow$ Groups
Base Spaces $\leftrightarrow$ Covering Spaces
Groupoids $\leftrightarrow$ Discrete Fibrations
Principle Fiber Bundles $\leftrightarrow$ Structure Groups

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