[Meta]physics

Deck transformations are important because we want to study the fundamental group of the base space (X) by looking at subgroups induced by the image of the covering map (q: E —> X).

(I don’t like the way I phrased the first sentence so I will rewrite this lecture with pictures and come back to it later.

Later: Literally why did I write any of this? This doesn’t explain anything to me. It describes the mechanics but it doesn’t explain or give a purpose behind the construction. I bring this up now because I want to know exactly why we need covering spaces and what they are good for. Does anyone have a concrete example? Particularly from topological neuroscience.)

There are really nice circumstances where the calculation of these subgroups are easier:

1a. When the covering of X is the Universal Cover (C), it’s Automorphism Group (also called group of deck transformations or covering isomorphisms) is the identity group. [Identity is a subgroup of every group but this does not guarantee a universal cover for this space. To overcome that, we created “nice spaces” where everything works such that universal cover for our space exists.

1b. The group of deck transformations of a universal cover (C) for the space (X) is isomophic to the fundamental group of X. So to calculate the fundamental group for a space that is kind of difficult, you go to its universal cover to to see if you can calulate the fundamental group that way. A great example of this is why we take the universal cover of S1 which is R and study the deck transformations from R to R to see if we can calculate the fundamental group. Another example is taking the universal cover of the torus (T) which is S2. You find Aut_q(S2) and will come to the answer ZxZ. Something else that’s useful here is to study the product topology and products of covering spaces.

2. Given a “nice space” X, the subgroups of the fundamental group of the base space (X) will corespond to coverings of X. This is good for a couple of reasons: Calculating subgroups of a group that you know may give you more information on finding a universal cover. Also, if you want to know more about the fundamental group of X, you can look at coverings of X and use a bit more group theory to make deductions about the group structure of pi_1(x).

3. When the covering space (E) is simply connected, the automorphism group is isomorphic to the fundamental group of X.

4a. A covering map defines a conjugacy class of subgroups of the fundamental group of X. Equivalent covers of X define the same conjugacy class of subgroups. By this we mean isomorphic covers will have subgroups that only differ by conjugation. Conjugation in group theory is a “change of perspective.” It’s basically looking at what happens within the group when you “stand from a different view” (sending x to y would be like moving from one side of the room to the other.) If you’ve studied linear algebra, this is what we mean by change of basis. (This is the part where the group action stuff we do in class is important. p. 287-end of chapter 11 is important for chapter 12.)

4b. For any point in X, at any point in the fiber of x, the set of induced subgroups are in exactly one conjugacy class of the fundamental group of X (this is because we are changing the base point throughout different points in the fiber). Conjugacy is again like “shining the light” to a different point on the stage. This time our lights point towards different elements in the fiber and the stage is the covering space.

4c. Normal subgroups are groups that are conjugate to themselves. In other words, if we have two induced subgroups at different respective points in the fiber and they are the same subgroup, the covering map is normal.

5. Normal covering maps make computing fundamental groups easier for this reason [equivalent statements]:

5a. The subgroup induced under the image of q is normal at some point e /in E.

5b. For some points x in X, the subgroups are the same for every point in the fiber over x.

5c: For all points x in X, the subgroups are the same for every point in the fiber over x.

5d. The subgroup induced under the image of q is normal for every point e /in E.

In particular, if the subgroup is normal at every point in the fiber then no matter which point we select as a basepoint, our information about this particular induced subgroup will be the same. This means the fundamental group at one point in the fiber is enough to give us what the induced subgroup will be (this will be important later).

6a. Given two basepoints in two covering spaces of X such that their covering maps both agree on some point in X, then there exists a (necessarily unique) covering isomorphism between the coverings if and only if their induced fundamental groups are the same at those basepoints. [[The covering isomorphism criterion is so important because we can use this when taking covering automorphisms.]]

6b. Two coverings are isomorphic if and only if for some x in X, the conjugacy classes of the induced subgroups of the fundamental group of X based at x are the same (ie both of the subgroups are sitting in the same conjugacy class). Recall: conjugacy classes form a partition of a set. Not to mention, conjugacy corresponds to orbits. Normality corresponds to stabilizers. If two subgroups are in the same conjugacy class, that means they’re in the same orbit under the action of conjugation.

The idea is something like this (assume all of this is happening at specific point):

covering map —> conjugacy class of subgroups of pi_1(X,x)

normal covering map —> normal subgroup of pi_1(X,x)

covering map (over points in fiber) —> conjugate subgroups of pi_1(X,x)

normal covering map (over points in fiber) —> normal subgroup of pi_1(X,x)

Two coverings are the same if their subgroups are in the same conjugacy class. A normal covering gives a normal subgroup (invariant under conjugation). Covering maps over points in the fiber give conjugate subgroups and if you have a normal covering you have exactly one subgroup corresponding to every point in the fiber.

Yes, there are two levels of conjugacy here. For a covering map, conjugate classes are the set of induced subgroups that vary over changing the basepoint. For a normal covering map, no matter where the basepoint was chosen, the subgroup is still the same. For a covering map over the fiber, the conjugate classes are the sets of induced subgroups that vary over taking different points in the fiber (conjugation is an inner-automorphism). For a normal covering map over the fiber, no matter which point we pick (in fiber), the induced subgroup will be the same, ie, the induced subgroup is completely determined by what happens at one point in the fiber.

7. If we take two points in the same fiber, then there exists a covering automorphism if and only if the induced subgroups are the same. This makes sense: it is a special case of 6a.

8. Normal coverings have transitive automorphism groups. This means that for every pair of points in the fiber, there exists a covering automorphism sending one point to another point. The subgroups are the same for every point in fiber over x.

9a. If q is a normal covering, then for any point in X and any point in the fiber over x in X, the group of desk transformations are isomorphic to pi_1(X,x) [mod] (normal subgroup induced by q).

9b. If E is simply connected, Aut(E) is isomorphic to pi_1(X,x).

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Required reading: https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter25/contents.html

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