Topological spin textures in magnetic systems are intriguing objects that exhibit exotic physics and have potential applications in information storage and processing. The most fundamental and exemplary topological spin texture is called the skyrmion, which is a nanoscale circular domain wall carrying a nonzero integer topological charge. The skyrmion texture in magnetic materials was theoretically predicted in the late 1980s, and it was experimentally observed in chiral magnets a decade ago. Since the first observation of magnetic skyrmions, the skyrmion community has focused on a series of topological spin textures evolved from the skyrmion, such as the skyrmionium and bimeron.
In a recent theoretical work carried out by an international team from China, Japan, Australia, Russia, and France. The authors introduced a new type of topological spin textures, which is called the bimeronium. The bimeronium exists in magnets with in-plane magnetization. It is a topological counterpart of skyrmionium in perpendicularly magnetized magnets and can be seen as a combination of two bimerons with opposite topological charges. Therefore, the bimeronium carries a topological charge of zero, like the skyrmionium.
Light not only plays a key role as an information carrier for optical computer chips, particularly for the next generation of quantum computers. Its lossless guidance around sharp corners on tiny chips and the precise control of its interaction with other light are the focus of research worldwide. Scientists at Paderborn University have now demonstrated the spatial confinement of a light wave to a point smaller than the wavelength in a topological photonic crystal. These are artificial electromagnetic materials that facilitate robust manipulation of light. The state is protected by special properties and is important for quantum chips, for example. The findings have now been published in Science Advances.
Topological crystals function on the basis of specific structures, the properties of which remain largely unaffected by disturbances and deviations. While in normal photonic crystals the effects needed for light manipulation are fragile and can be affected by defects in the material structure, for example, in topological photonic crystals, the conditions are protected from this. The topological structures allow properties such as unidirectional light propagation and increased robustness for guiding photons, features that are crucial for future light-based technologies.
Israeli and US researchers have developed a new, highly efficient coherent and robust semiconductor laser system: the topological insulator laser.
The findings are presented in two new joint research papers, one describing theory and the other experiments, published online today by the prestigious journal Science.
Topological insulators are one of the most innovative and promising areas of physics in recent years, providing new insight into the basic understanding of protected transport. These are special materials that are insulators in their interior but conduct a “super-current” on their surface: the current on their surface is not affected by defects, sharp corners or disorder; it continues unidirectionally without being scattered.
The studies were conducted by Professor Mordechai Segev, of The Technion–Israel Institute of Technology, and his team: Dr. Miguel A. Bandres and Gal Harari, in collaboration with Professors Demetrios N. Christodoulides and Mercedeh Khajavikhan and their students Steffen Wittek, Midya Parto and Jinhan Ren at CREOL, College of Optics and Photonics, University of Central Florida, together with scientists from the US and Singapore.
Topological effects, such as those found in crystals whose surfaces conduct electricity while their bulk does not, have been an exciting topic of physics research in recent years and were the subject of the 2016 Nobel Prize in physics. Now, a team of researchers at MIT and elsewhere has found novel topological phenomena in a different class of systems — open systems, where energy or material can enter or be emitted, as opposed to closed systems with no such exchange with the outside.
This could open up some new realms of basic physics research, the team says, and might ultimately lead to new kinds of lasers and other technologies.
The results are being reported this week in the journal Science, in a paper by recent MIT graduate Hengyun “Harry” Zhou, MIT visiting scholar Chao Peng (a professor at Peking University), MIT graduate student Yoseob Yoon, recent MIT graduates Bo Zhen and Chia Wei Hsu, MIT Professor Marin Soljačić, the Francis Wright Davis Professor of Physics John Joannopoulos, the Haslam and Dewey Professor of Chemistry Keith Nelson, and the Lawrence C. and Sarah W. Biedenharn Career Development Assistant Professor Liang Fu.
The reason theres no precise equation for the perimeter of an ellipse is because the practical, real-world manner in which to produce an ellipse is to cut a cylinder at an angle. This would require third-dimensional rotational geometry, or mathematics that cannot be produced only on a two-dimensional plane. Meaning the originally problem is missing key information. (which can be expressed by numbers on a two dimensional plane, but only after the unique ellipse is produced and measured.)
Sometimes you will be provided a mathematically and fundamentally incomplete data set. You will either have to say that the problem is impossible or guess; depending on the circumstances and moralistic understanding of the problem.
P.S.
Imagine being a 2D creature finding an elliptical object. You can only interact with 2D object, and this thing is immeasurable. Alternatively, how deep is your personal gravitational well?
The reason theres no precise equation for the perimeter of an ellipse is because the practical, real-world manner in which to produce an ellipse is to cut a cylinder at an angle. This would require third-dimensional rotational geometry, or mathematics that cannot be produced only on a two-dimensional plane. Meaning the originally problem is missing key information. (which can be expressed by numbers on a two dimensional plane, but only after the unique ellipse is produced and measured.)
Sometimes you will be provided a mathematically and fundamentally incomplete data set. You will either have to say that the problem is impossible or guess; depending on the circumstances and moralistic understanding of the problem.
P.S.
Imagine being a 2D creature finding an elliptical object. You can only interact with 2D object, and this thing is immeasurable. Alternatively, how deep is your personal gravitational well?
There is a 100% accurate equation for the perimeter of an ellipse, but it requires calculus.
Sorry, I’m an Ancient Greek; I don’t know or care about that kinda nonsense. /s
Actually this is very interesting, and I made a lighthearted joke about it. Another example of limited input affecting a system, though. I have little to no experience with calculus :(
AL HASIRA is inspired by the rise in the need to enjoy the comfort and ambiance of Dubai’s sensational outdoor scenes. During the time of the year when the weather conducive for outdoor activity, it is almost impossible to resist the atmosphere at a park or in the Deserts. On the other hand, when it gets hot in the summer, most people would rather spend their time out on the beach. The Idea for Al Hasira stems from the intrinsic need to seat lower to the ground when at a park, the beach, plaza or square. It has become a common practice to use a combination of mats and pillows / cushions or anything that can be used to get comfortable in certain the outdoor spaces like the beach, a patch of grass, the park or the desert. With Al Hasira, we combine these possibilities into one single panel, which encourages a variety of postures ( i.e. seating, laying and so on), that can be combined together to facilitate interaction in parks, plazas, the desert, the beaches, on the sidewalk and several other public spaces such as the lobby of a building.
AL Hasira is a multi-functional piece of furniture that can be customized to fit a variety of seatingconfigurations and a multitude of colors, materials and texture. This sleek piece consists of two majorcomponents, a base material for support and a cushion material to make seating comfortable.Depending on the client’s preference and the function of a space, the piece could be customized with ajuxtaposition of Fiberglass or Corian (for base material) and leather, rubber or wool (for cushion material) which would be suitable for interior spaces. On the other hand, for outdoor environments, in order to have a more timeless design or material option, clients may prefer a combination of a fiberglass or concrete base with timber (red meranti or teak) seating. Every Urban space is different, be it an interior space, sidewalk or a landscape, they all have their individual character. Al Hasira provides the client with the opportunity to personalize their benches, to make it unique to them and to their spaces.
DSC_7417 OUTPOST Studio/Cyanotype Process Painting by Russell Moreton Via Flickr: Pattern and Chaos/Liminality/Tectonics Architectural surface for a Library, raw materials, light, silence and solitude.
given any loop (of any shape), are there four points that, when connected, form a square? (unsolved) or a rectangle? (solved and proved in this video)
topology is really cool!! and so is this video, it’s got really nice explanations and does so in a way that people with no topology knowledge can understand PS topology was used by this year’s winners of the nobel prize in physics!!
Making matrices better: Geometry and topology of polar and singular value decomposition
Dennis DeTurck
Amora Elsaify
Herman Gluck
Benjamin Grossmann
Joseph Hoisington
Anusha M.Krishnan
Jianru Zhang
2017
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You don’t want to view matrices as 9 entries in an array. So you learn the inner product, rank, and the determinant. Then how do matrices look?
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The two components of O(3) appear as real projective 3-spaces in the 8-sphere, each the core of a open neighborhood of nonsingular matrices, whose cross-sectional fibres are triangular 5-dimensional cells lying on great5-spheres. The common boundary of these two neighborhoods is the 7-dimensional algebraic variety V⁷ of singular matrices.
This variety fails to be a submanifold precisely along the 4-manifold M⁴ of matrices of rank 1. The complement V⁷−M⁴, consisting of matrices of rank 2, is a large tubular neighborhood of a core 5-manifold M⁵ consisting of the “best matrices of rank 2”, namely those which are orthogonal on a 2-plane through the origin and zero on its orthogonal complement.
V⁷ is filled by geodesics, each an eighth of a great circle on the 8-sphere, which run between points of M⁵ and M⁴ with no overlap along their interiors. A circle’s worth of these geodesics originate from each point of M⁵, leaving it orthogonally, and a 2-torus’s worth of these geodesics arrive at each point of M⁴, also orthogonally.
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [30−45 cm] in height. Its height gradually diminished, and after a chase of one or two miles [2–3 km] I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.
—John Scott Russell, 1845
meeting of the British Association for the Advancement of Science, York, September 1844 (London 1845), pp 311-390, Plates XLVII-LVII).
For almost two years, I’ve been skating by, saying “I don’t want to explain homology right now, homology is complicated, ask me later.”.
Later has come, folks. This is the final post of a seven-part sequence dedicated to shining light on this beautiful, mysterious beast. (1234567)
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The aim of this post is to say some words about cohomology.
It is harder to give an intuitive understanding about cohomology from the cellular perspective. Probably the least algebraic way to introduce cohomology is through de Rham cohomology, but even this relies on an extremely solid grasp of multivariable calculus. It’s unreasonable for me to assume you have such a strong intuition, and frankly mine is not very good either.
However, I can say words to convince you that although we no longer have a geometric intuition, things seem more or less the same:
The two basic notions in cohomology are cocyclesand coboundaries
Coboundaries count for nothing.
Every coboundary is a cocycle, but some cocycles are not coboundaries.
Cocycles and coboundaries are special examples of cochains.
Cocycles which differ by a coboundary are called cohomologous, and are said to be of the same cohomology class.
The collection of cohomology classes is called the cohomology of the space
Breaking the cohomology up by dimension, we obtain the cohomology groups of dimension $\bm{d}$, which we notate $H^d(X)$.
However, this is a bit too brisk to explain why we have a whole separate concept for these two things.
The point of contact between homology and cohomology is at the level of chains: every cochain can be thought of as a chain of the same dimension by a fairly standard operation known as vector space duality. This process is kind of uncomfortable for the (co)chains; there is a meaningful sense in which chains and cochains really “don’t want” to be identified with each other. But we can force it to happen anyway, if we so choose.
However, once you start moving away from chains, the two theories diverge. The most notable difference is that while the boundary of a $d$-dimensional chain is $(d-1)$-dimensional, the coboundary of a $d$-dimensional cochain is $(d+1)$-dimensional! By the time we get to the level of homology groups, it seems like the two theories are hopelessly different.
But then, a miracle occurs: for the most reasonable class of $n$-dimensional spaces, $H_d(X)$ and $H^{n-d}(X)$ are essentially the same— for instance, they contain the same number of (co)homology classes. This miracle was first discovered by Poincaré, and it uses a much more natural method of assigning chains to cochains, thus, today this phenomenon is called Poincaré duality.
[ Now, nothing comes for free in this world, including the “naturality” of Poincaré duality. The cost is that this duality does not take $d$-chains to $d$-cochains, but instead takes $d$-chains to $(n-d)$-cochains. ]
——
Poincaré duality is really cool! But, stealing a line from Ghrist’s Elementary Applied Topology: “The beginner may be deflated at learning that cohomology seems to reveal no new information… Students may wonder why they should bother with this… as it is alike to homology in every respect except intuition.”
Indeed, Poincaré duality is almost too much of a good thing; were it not for a second miracle, cohomology may have been relegated to a footnote in the algebraic topology historybooks. And the second miracle is this: while homology classes naturally “want” to be aggregated by dimension, cohomology classes do not. The collection which I called the “cohomology” in the list above is more commonly known as the cohomology ring.
[ Briefly, this is because cochains are actually functions. Such functions can be multiplied, and this product structure provides the “glue” that binds the (direct sum of the) cohomology groups together as a ring. ]
You may be surprised to see the word “ring” showing up here. In this context it roughly means “number system”: two elements of a ring can be added together or multiplied together. Rings are a central object of study in the field of algebra proper, and so a great deal of general theory can now be imported into this situation, making cohomology a secret trove of structure hidden deep, deep beneath the geometry of the space.
For almost two years, I’ve been skating by, saying “I don’t want to explain homology right now, homology is complicated, ask me later.”.
Later has come, folks. This is the sixth post of a seven-part sequence dedicated to shining light on this beautiful, mysterious beast. (1234567)
This post will be a little bit long, because I didn’t see any point in belaboring this calculation into multiple posts. You won’t need to have seen anything from this calculation to read the final post, so if you get impatient you can feel free to move on.
——
Meet the Torus
We have built so much so far out of one-dimensional examples, that I feel obligated to show you at least one interesting two-dimensional hole. A natural example is provided by the torus, which you can think of as the crust (so: hollow) of a donut:
A few remarks before proceeding:
I will do my best to illustrate what is happening here, but since we are now looking at a curved two-dimensional surface, it becomes harder to see what’s happening in a drawing. It would probably help if you actually got a donut (and if not, at least you now have a donut! Yum!)
I will draw some grid lines on the surface of the donut, and I will try to draw them lightly so we do not mistake them for one-dimensional chains.
Finally, it is common parlance in mathematics to remove the word “dimension” whenever possible. For instance, $1$-dimensional chains are called $1$-chains, $2$-dimensional cycles are called $2$-cycles, and so on. \item One that is a bit confusing: a $2$-boundary, for instance, is a $2$-chain, and also the boundary \emph{of} a $3$-chain. So the dimension refers to the dimension of the shape itself, not the dimension of the thing which the shape is the boundary of.
Let’s begin by looking at $2$-cycles. This is not something we had ever considered before, but we will do so now. Actually, there are not many options available to us. One $2$-cycle is the entire surface itself. This has the right dimension and it is a cycle (make sure to convince yourself that it is, indeed, boundaryless). It is, in fact, the only one!
Here is an argument to see why: suppose we wanted to make a $2$-cycle. It would need to contain at least one point, and so in order to have the right dimension, we would need to have some small disk.
However, this disk has a boundary. If we make the disk smaller, we’ll never totally get rid of the boundary, so let’s try to make it larger instead. In order to remove the “upper part” of the boundary, we have to wrap around the torus one way…
… and then to get rid of the rest of it, we have to wrap around the other way:
(Note: this picture obviously still has a boundary; in order to make it disappear we have to keep wrapping the chain around the torus. In the end, the two boundaries will meet up, at which point they cease to be boundaries at all, and the chain is the entire torus, which is what we were expecting.)
Now, is this $2$-cycle a boundary? Remember that this means it would be a $2$-boundary, or in other words the boundary of a $3$-chain. But a moment’s consideration makes us realize that it could not possibly be the boundary of a $3$-chain: the space is itself only two-dimensional! There’s not enough room to fit a $3$-dimensional shape into it.
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Dimensional Downstep
Having handled the $2$-dimensional case, we now move to the harder task of classifying $1$-cycles. We will not be so lucky to just be able to count cycles this time: there is lots of room for us to put loops into the space.
It’s hard to explain, but hard to unsee, that a $1$-cycle is going to be a $1$-boundary unless it “wraps around” the torus somehow: this makes sense with our intuitive definition because it seems like there is a hole in the center:
(In this drawing it kind of looks like you could find a $2$-chain that $\alpha$ is the boundary of. But the dashed part of $\alpha$ is on the back of the torus: if you try to “fill in” the shape you end up on the wrong side of the space.)
In that diagram, you can actually already see another $1$-cycle: all of the grid-lines are 1-cycles! It’s easy to see that the parallel ones are all homologous to each other (take the “rectangle” in between them). It’s less obvious is that none of the “long” ones are homologous to the thickly-drawn $1$-cycle, but this is also true.
We can also see this geometrically, although it is even trickier: notice that $\alpha$ has only one point of intersection with any grid line $\beta$. This poses a problem if we try to find a $2$-chain with boundary $\alpha-\beta$ (which, remember, is also $\alpha+\beta$), because it ends up needing to fill the whole torus:
But we already know that the torus is a $2$-cycle, hence boundaryless, so it couldn’t possibly be that the boundary is $\alpha+\beta$. Therefore, $\alpha$ and $\beta$ cannot be homologous cycles.
——
The One that Got Away
It is natural, at this point, to believe we are done (except that we haven’t mentioned the cycles which are boundaries— now we have). But if we do the calculation for the number of holes, we find that there is an issue. We have $\alpha$ is in a different homology class from $\beta$, and so we’ve found that there are $3$ homology classes (including the class of boundaries). This is a big problem, because $3$ is NOT a power of $2$: It isn’t $2$, and it isn’t $4$ either! We must have missed a cycle.
Indeed we have: I will spare you the trouble of finding it yourself.
But this “missing cycle” raises a more serious concern. How do we know we didn’t miss anything else? Are we absolutely sure there aren’t secretly some more classes hiding around somewhere?
It would surely be impossible to go around checking every single loop: What if you wrap around the center hole four times and then cross the gridline? What if you do the third cycle, but don’t quite close, and then wrap around the center hole and then do the third loop again and then…?
I’m sure that it would be possible to come up with a visual argument like the one I gave above, but you would have to be extremely clever to find it (I do not know one, myself). The point is that when we look at any space that’s even a little bit complicated, it’s not so easy to know when you’re done.
It turns out that there is a method for knowing when to stop counting as well. Unfortunately, that method is quite technical, and this technicality seems to be unavoidable. Actually, the main reason that I delayed writing this sequence so long is because I was trying to find an intuitive way to describe this method; I have not been able to do so. Taking the time right now to build up the required machinery would more than double the length of this sequence, so I am not going to do that. (I am strongly considering writing another sequence, although there is not too much time left >.<)
The final post is a more of an epilogue, which will talk about the highlights from the story of homology’s hotshot cousin: cohomology.
So, for reasons [explained below the break*], it occurred to me that the deadline for submitting abstracts to the Joint Meetings is probably pretty soon (it is— September 26— so if you’re planning to submit, better get your ass in gear!). And while I was looking through the sessions, I was once again reminded why I flinging love the Joint Meetings.
Here are some actual titles of sessions— sessions!There will be multiple talks on these things!
AMS Special Session on Set-theoretic Topology (Dedicated to Jack Porter in Honor of 50 Years of Dedicated Research)
MAA Session on Philosophy of Mathematics as Actually Practiced
Short version for the unaware: there has recently been a growing trend in philosophy of mathematics to shift the discipline closer to studying the practices of professional mathematicians.
This is basically everything I like about philosophy plus everything I like about anthropology/sociology (I’m really bad at distinguishing the two) thrown together into a big pot and seasoned liberally with modern math; delicious!
AMS Special Session on Alternative Proofs in Mathematical Practice
Besides just being generically interesting, the title of this session is based on the title of a book that came out recently (which is really darn expensive, sorry). So I’m interested to see how the session interacts with that.
MAA Session on Good Math from Bad: Crackpots, Cranks, and Progress
AMS Special Session on A Showcase of Number Theory at Liberal Arts Colleges
This one’s a little subtle. If this were by the MAA, this would be whatever. But it’s not: it’s by the AMS. This is very outside the usual AMS fare and I’m hoping that it succeeds because that would indicate that at least one of teaching, undergrad research, and non-R1 research are being actively supported by both major American mathematical communities, which… fucking finally.
And then there are all the the sessions I’m excited about because people! Math friends!
AMS Special Session on Research from the Rocky Mountain-Great Plains Graduate Research Workshop in Combinatorics
:Ditme! (we already presented our stuff at last year’s JMM, so it won’t actually be me. but still)
AMS Special Session on Research in Mathematics by Early Career Graduate Students
People like me except marginally better at not procrastinating? yes please?
(Also it’s an AMS Special Session so there’s definitely some expectation of quality there… at least with the work. With the presentation style… we’ll see.)
AMS Special Session on Special Functions and Combinatorics (in honor of Dennis Stanton’s 65th birthday)
MAA Project NExT Lecture on Teaching and Learning — Changing Mathematical Relationships and Mindsets: How All Students Can Succeed in Mathematics Learning by Jo Boaler
NO FUCKING WAY
Jo Boaler is a relentless advocate of IBL techniques in the K12 setting
And in particular wrote What’s Math Got To Do With It? that book was my bible on math ed for a good three years, dam.
AMS Invited Address — Algebraic Structures on Polytopes by Federico Ardilla
* So an idea I’ve had kicking around for a little while is that maybe I could try to take some of the lessons that I’ve learned from writing this blog and give a little contributed talk at the Joint Meetings this year. I’ve never really thought to do this earlier because it just doesn’t feel right to eat up a spot just to do some cheap advertisement. But I think I’m more comfortable with the idea now, since I’ll be several months out from writing the blog by then, and I don’t have a follow-up project planned. I mean, I do, it’s called “writing my damn thesis already”, but I haven’t figured out how to publicize that yet* :P
Part of the reason I want to do this is definitely because I’d love the blog to “count” for something CV-wise, and a contributed presentation at a conference isn’t much but it isn’t nothing.
But another reason is that over the course of my writing OTAM is that I’ve talked with a lot of people in real life who speak longingly about blogging. It’s something they’ve wanted to do for a while— maybe they even did do it for a month or two— but have never really found the time/drive/whatever. And I’ve never really had anything useful to say to comments like this. I think that the effort of preparing a conference talk might force me to straighten out my perspective on the value of this whole enterprise, which I think would allow me to have more meaningful conversations.
(This sounds rather falsely noble, but it’s also a bit self-serving: I suspect that I’ll find myself being one those people ten years from now, if not for blogging then for something else. It would be nice to have something of value to say to future-me, too.
[ * I thought pretty seriously about streaming thesis work. But in the end I came to the conclusion that I would need a much more flexible camera setup so that I could film my paper, or alternatively to get some kind of comfortable whiteboard setup in my room. The issue is that it’s easy to stream TeX, but I can’t do all my actual work in TeX; I need some flexibility to draw faster than I can TikZ. So I got myself a cheap tablet (and wow, that’s a story: ask me about it IRL sometime) thinking that I could get used to drawing on that. But in the end I really just couldn’t; I haven’t figured out how to think on the tablet the way I can think on paper or whiteboards**. I have no doubt that I could learn to do it, but I also have no particular interest in learning right now. ]
[ ** Unpopular opinion: chalkboards are great for teaching and giving talks, but when I do math that I have to actually think about, whiteboard every time. ]
For almost two years, I’ve been skating by, saying “I don’t want to explain homology right now, homology is complicated, ask me later.”.
Later has come, folks. This is the fifth post of a seven-part sequence dedicated to shining light on this beautiful, mysterious beast. (1234567). This is a very important post for us, since we have built up all the background and we will finally see how to use homology to determine the number of holes in a space!
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Recall that in the last post we gave an algebro-geometric justification which permitted us to understand these two cycles as “essentially the same”:
We thought this was a good thing because both $A$ and $B$ “surround the same hole”, and we only want to count each hole once.
Motivated by this argument, we make the following definitions. If the difference* $A-B$, between two cycles $A$ and $B$, is a boundary, we say that the cycles are homologous, or in other words that they are of the same homology class. This allows us to define the homology of a space as the collection of all homology classes.
Since cycles of different dimensions look qualitatively different, and so they probably surround different-looking holes, it’s natural to want to split up the homology by dimension. We say that the collection of homology classes having dimension $d$ is the $\mathbf{d}^\textbf{th}$ homology group, and is usually written $H_d(X)$, where $X$ is the space in which the chains live.
[ There’s a technicality here: how do we define the dimension of a homology class? After all, homology classes aren’t curves or surfaces or anything; they’re these giant unwieldy collections of many, many chains. However, this “issue” turns out not to be, because any two homologous chains have the same dimension. (This is not totally obvious; it’s worth the effort to draw a picture or two to convince yourself.) Because of this, we can define the dimension of a homology class by determining the dimension of a cycle of that class: any choice will give the same number. ]
[ * You may wonder why I said “difference” instead of “combination”, since $A+B$ and $A-B$ are the same thing, after all, and the former is more geometrically intuitive. The answer is that the subtraction definition works even outside of characteristic $2$; the addition definition does not. ]
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However, even restricting our attention to a single homology group, we’re still not counting holes quite yet! The number of $d$-dimensional holes in the space is actually not the number of $d$-dimensional homology classes. For instance, when $d=1$, we can use a slight variant on our example space: instead of a disk with one hole, take a disk with two holes. See if you can convince yourself that no two of these four loops are homologous.
[ You may be concerned with the innermost square loop: this is the class containing the boundaries. We include it for the same reason we include zero as a number: it’s true that boundaries count for nothing, but nothingness, itself, counts for something. We make this convention for convenience, because it makes the theory nicer (like, way nicer, you’d actually be amazed); this is exactly the same reason that we consider zero to be a number. ]
You can keep playing this game: if there are three holes, the picture looks like the one below. (Despite my best efforts, this picture kept turning out a mess; this one I think is at least somewhat legible):
So it turns out that, if you include the class containing all boundaries, the number of classes in a homology group is always a power of $2$; if that number is $2^h$, then we might say that the space has $h$ holes of the corresponding dimension.
By the way: If you think the $2$ in $2^h$ has something to do with being in characteristic $2$, you’re right! But there is also another way to explain it, which perhaps lines up more reasonably with the picture.
People with more advanced mathematics backgrounds may know that $2^h$ is the number of elements in the power set of a set of size $h$, or in other words, is the number of subsets. Applying this knowledge to the situation at hand, it stands to reason that a homology class consists of loops which surround a particular collection of holes. If you believe that the chains I’ve chosen for the pictures above represent “typical” members of their homology classes, then this probably makes sense :) For instance, the fourth cycle in the two-holed space represents the collection “both holes”, and the little squares in both spaces represent the “empty collection” of holes.
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We’ve made it!
We have described holes, things which “aren’t there”, with genuinely intrinsic features of the space. This was no small feat, and it is worth celebrating.
If this is all you know about homology, you’re in a pretty good place (at least as far as not-being-freaked-out-when-it-gets-mentioned goes). However, we may wish to see some more interesting examples. In the next post, we will grant that wish by peering into some higher-dimensional holes.
