I apologize for the quality of this blog so far, but I am proud that we have renewed for another year!
I suppose there will be a sort of overhaul of the site, but I won’t remove anything I’ve posted so far. For the readers who have been around since the beginning, thank you very much for your support. It’s hard to connect with others and I really hope this blog does the job.
Have a good holiday.
I apologize for the quality of this blog so far, but I am proud that we have renewed for another year!
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).
On Giving Unsolicited Advice
Perceptions and Illusions
Batman is the Real Villain
Carb-Induced Mood Swings
Law of Attraction (Telekinesis)
Raw Egg Yolks and Protein Denaturation
Lipid Metabolism and Skin Hydration
Sulfur and Testosterone
Diet, Ethics and Morality
Energy and Calories
The Urinary System and Kidney Filtration
Menstruation and Fertility
Cholesterol, Cancer and Vitamin D
Lactobacilli and the Colon
The Theory of Relativity
Time and Growth (Wisdom Teeth)
Cellular Respiration (Banach-Tarski Paradox)
Seasonal Autoimmune Flares
Pottenger’s Cats – Raw vs Cooked Food Experiment
On Intuitive Eating
An Interpretation of Pregnancy Cravings
Why You Cheat (Emotional Malnutrition)
The Role of the Liver and Blood Detoxification
Milk, Stem Cells and IGF-IV
What is the function of the appendix?
Should human rights extend to animals?
What makes the Impossible Burger?
Plant Matter and the B-Vitamins
Chickpeas are not chicken!
Goitrogens and the Cabbage Patch Kids
What is Money?
Vitamin E and Antioxidation
Skin as an Endocrine Gland
What is Endometriosis?
Early Signs of Rheumatoid Arthritis
What is Time?
Comparing Human Suffering to Animal Suffering (What is Time Part 2)
Insects are Animals
The Lymphatic System and the Heart Chakra
The Urinary System and the Chakral Chakra
What is Mucous? (the Immune System)
Ketosis vs Starvation
Pregnancy Duration and Prematurity
The Brain, Nervous System and Third Eye Chakra
The Respiratory System and the Throat Chakra
Vitamin A vs Beta-Carotene
NaCl and Nervous System Activity
Food is Information
Pineapple Enzymes and Hyperpigmentation
Miscarriage, Infertility and Red Blood Cells
Hair Loss Post-Pregnancy
What is Momentum?
Bipolar Disorder as Type IV Diabetes
Seasonal depression can be really difficult for me and normally I have friends or family to keep me company or it’s the fall and I’m in school and don’t think that much about it, so this isolation on top of working this job kind of gets to me, but I know it’ll end when the winter is over. I don’t have many clothes to prepare for the winter here, I suppose that’s my fault. I mean, it is my fault. I’ve donated, recycled and thrown away so many clothes for so many seasons. I would have been set.
This depression this year is really sobering. I know it’s because I am sober, and the last time I was sober this time of year was in 2014. Wow, I didn’t even realize it had been that long.
I said I’m not going to force things anymore and I meant it. You can’t force people to respect you. You can’t force people to like you. You don’t force people to leave you alone. You can’t force them to admire you. You can’t force them to acknowledge your intelligence.
My mom would say that I’m too good to work at McDonald’s or a lot of people will say that, but when you’re depressed like this, the fact I’m even getting out of bed to work at all is good enough for me right now. What I’m doing is good enough, and since it’s a very active job and I get to be on my feet all day, it’s good for my health ultimately. I don’t eat the food there, I just drink water during break. Once you learn all there is to know about the store, you just sort of go through the motions. It’s not as active as a teaching job, and maybe in some ways that’s making it worse for me, but I get to just be quiet, do my job and leave. There are men who hit on me or harass me or whatever, and they’re lonely so I understand why. But there’s no ring on my finger. There isn’t a man who wants to be with me or be married to me, so I may as well just let the lonely men talk to me, at least they feel better and understood for a while.
I’ve lost sight of what I want in life. I do still miss my son, who doesn’t technically exist, but I’ve accepted I’ll never be a mom anyway. I had my chance at motherhood and aborted the baby because the father didn’t want it, so that’s my fault. I didn’t get the career I wanted either, so what do I know about anything? Not like I can do anything really.
I do have stability finally and somewhat of a routine, so I’m building myself back up. If I have to start at McDonald’s that’s fine with me. I used to fear working in fast food because I thought I wouldn’t be able to control myself and would end up really sick from eating the food but I haven’t had one bite of it since I’ve been there and I’m proud of myself for having that discipline and even if I’m hungry, I wait until I get home.
I don’t know what I’m doing with myself right now, all I’m trying to do is take care of myself and not commit suicide. I’m doing my best to nurture myself and my soul and really listen to myself.
I do feel like I let a lot of people down but that doesn’t matter ultimately. Everyone is working on their own stuff. I was such a rockstar in academia, so happy and full of energy and spirit. I enjoyed going to my classes daily and loved interacting with my professors and friends. It was still the happiest days of my life and I had a husband who really supported me as a friend and would do anything for me when I needed it.
I don’t miss him romantically, at all, but I do miss the genuine companionship we had and my depression was in remission back then. I mean it wasn’t all roses, obviously, otherwise I wouldn’t have left, but I left to chase another relationship that didn’t work either. Both of them were true friends to me, and it makes me very sad that I gave up another life that I enjoyed for someone who is not at all my friend. Someone who likes to routinely tell me he hates me, who lies to me and who poisons me with his spiritual venom.
It makes me really sad that I haven’t gotten the strength to never speak to him again. And maybe that’s what is contributing to my depression and underachievement now. I’ve hurt myself a lot. I keep telling myself sorry. I keep saying sorry. But if I’m not leaving, then maybe I don’t mean it.
Even thinking about him now makes me want to cry. I pray that he never speaks to me again and I can move on with my life. I pray that I get the strength to never speak to him again and move on with my life.
But he did the final thing he could do to break my trust, so it shouldn’t be much longer before it’s over for good. He doesn’t even want children, he just said that because he thought he did. He tried to project onto me that I didn’t want them. But I could see it was him realizing he didn’t want them with me.
It’s no matter. All the men eventually realize they don’t want children with me and leave. Even the one I aborted my child for, broke up with me because he said he wanted to have a family some day… but it was too soon when I was pregnant.
So now I’m here again. Crying at the computer screen, because there is still some underlying illness that hasn’t been cured. Even though I’m in ketosis and have eliminated almost all of the anxiety, something is still not computing and I’m still not executing what I have in my mind to do.
I’ve given up on relationships. I’ve given up on my dreams. I stopped believing in love. I stopped believing in myself.
I’m sorry Zack that I’ve let you down.
Required reading: https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter25/contents.html
There’s no point in it. I’m done forcing it. I don’t believe in myself and don’t have the courage to ask anyone to write me letters of recommendation.