The "textbook exercise" on Euler characteristic | Euler characteristic #1

Summary

In this video, Mathemaniac explores the intriguing world of Euler characteristics, focusing on the formula V - E + F and how it applies to different surfaces. Starting with graphs on a plane, the video delves into more complex surfaces like the torus. The theorem indicates that for any graph without self-intersections, V - E + F equals two. However, the number can change based on the surface's topology, particularly its genus, or number of holes. With an informal approach, Mathemaniac attempts to prove this theorem. Along the way, he discusses the nuances of edges and faces in different configurations, ultimately tying it back to algebraic topology concepts like Betti numbers and their higher-dimensional implications.

Highlights

• Discover the relationship between vertices, edges, and faces in Euler characteristics! 🎓
• The torus is a great example of how topological features change Euler characteristics. 🌀
• Learn how non-contractible loops can alter the calculation of V - E + F. 🔍
• Mathemaniac's approach provides a fresh, informal look at a classic mathematical exercise. 👀
• See how Euler characteristics connect to wider mathematical concepts like homology and Betti numbers. 🔗

Key Takeaways

• Euler characteristics and the formula V - E + F can be complex but fascinating! 🌟
• Different surfaces can affect the formula's outcome, especially when holes are involved. 🕳️
• Exploring the torus helps illustrate how holes in surfaces impact Euler characteristics. 🌀
• Understanding non-contractible loops is crucial to the concept of Euler characteristics. 🔄
• Algebraic topology ties into this through things like Betti numbers, which generalize to higher dimensions. 📚

Overview

Euler characteristics are a key concept in topology, touching on how graphs relate to surfaces. At its core, the formula V - E + F offers insights into the properties of a surface by looking at the number of vertices (V), edges (E), and faces (F). Originally applied to simple flat surfaces, the concept expands when considering three-dimensional shapes like the torus.

The torus, with its doughnut-like shape, introduces variables into the Euler characteristic formula, largely because of its 'genus' or the number of holes. This affects how we calculate the difference between V, E, and F. By examining different loops and edges on the torus, Mathemaniac illustrates how surfaces' topology can influence mathematical results.

The video brings attention to the broader connections between Euler characteristics and algebraic topology. Through vivid examples, viewers understand how these concepts are used beyond simple geometry—extending into fields that measure topological features using Betti numbers. Mathemaniac's presentation invites learners to engage with these intriguing ideas in a fun, relaxed manner.

Chapters

• 00:00 - 00:30: Introduction to Euler Characteristic Formula The chapter introduces the Euler characteristic formula, which relates the elements of a planar graph. It explains that for any graph drawn on a plane where all vertices are connected without any crossing edges, the number of vertices (V) minus the number of edges (E) plus the number of faces (F) equals two. This relation holds true for any such graph, demonstrating the consistency and reliability of the Euler characteristic formula.
• 00:30 - 01:30: Euler Characteristic on Different Surfaces The Euler Characteristic, denoted as V - E + F (vertices - edges + faces), can vary depending on the surface on which a graph is drawn. Traditionally, this characteristic equals 2 when drawn on a plane. However, for a torus, the characteristic is not always 2. In the case of a torus, there are three possible Euler Characteristics: it could be 0, as demonstrated in a specific example, or it could be 1, alongside other configurations. The variability of the Euler Characteristic across different surfaces highlights its dependence on the underlying topology of the surface.
• 01:30 - 02:30: General Case and Triangulation The chapter discusses the concept of genus in topology, explained in terms of surfaces with holes. It introduces the formula V - E + F = 2 - 2G, where V represents vertices, E represents edges, F represents faces, and G represents the genus or number of holes in the surface. Examples include the double torus with genus 2, having two holes, and the torus with genus 1, having a single hole.
• 02:30 - 04:00: Proving the Formula on Plane and Torus In "Proving the Formula on Plane and Torus," the chapter discusses the theorem involving the relation V - E + F on the torus, indicating that its values lie between 0 and 2. This is contrasted with most literature that focuses on triangulation graphs, where V - E + F results in the minimum possible value of 2 - 2G for graphs without self-intersections.
• 04:00 - 06:00: Non-Contractible Loops and Decrease in V-E+F The chapter discusses the concept of non-contractible loops and how they relate to the mathematical formula V-E+F (Vertices - Edges + Faces). The speaker mentions that in many cases, proving these concepts is left as an exercise for the reader due to its perceived simplicity. However, the chapter focuses on exploring the speaker's thought process rather than rigorous proofs. The speaker invites readers to judge the difficulty and educational value of this approach.
• 06:00 - 08:30: Exploring Lower Bounds on Torus In this chapter, the discussion begins with exploring the concept of lower bounds on a torus, starting with the simpler context of a plane. The approach involves proving the formula V - E + F = 2, a fundamental result in topology known as Euler's characteristic for planar graphs. The explanation initiates with the base case where the plane is reduced to a single vertex, having no edges, and considered as a single connected region, setting the foundational understanding for further exploration of geometric and topological properties.
• 08:30 - 11:30: Understanding the Topological Equivalence The chapter discusses the concept of topological equivalence through the lens of Euler's formula, V - E + F = 2, which is a fundamental formula in the field of topology for planar graphs. It explores two possibilities when modifications are made to a geometric shape by adding edges. The first possibility involves adding an edge and a vertex, which keeps the topological equivalence unchanged since both V (vertices) and E (edges) increase by one. The second possibility is the addition of an edge to enclose a region within the graph, altering its structure. This further underscores the intrinsic relationship between geometry and topology, illustrating how changes in shape affect the overall system while still maintaining the fundamental formula intact within these operations.
• 11:30 - 14:00: General Inequality and Applications The chapter discusses the geometric concept of forming cycles by joining vertices, specifically focusing on the act of connecting bottom blue and purple vertices to form an additional region with a new cycle. It explores how these actions affect the number of regions (F) and edges (E) while keeping the number of vertices (V) constant, thereby maintaining the invariant V - E + F = 2.

The "textbook exercise" on Euler characteristic | Euler characteristic #1 Transcription

• 00:00 - 00:30 if you draw a graph on a plane where all vertices are connected to each other and the edges don't cross then there is a relation between its vertices edges and faces the number of vertices V minus the number of edges e plus the number of faces F has to equal to two no matter what graph it is this is known as the oiler characteristic formula however the
• 00:30 - 01:00 right hand side really depends on the surface you draw the graph on it's to on a plane and if you draw the graph on a Taurus it can still be two but it's not always two for example in this case which you can pause and check V minus E + f is zero it turns out that for the Taurus there are only three cases it could be zero like here or it could be one like this case which again you can
• 01:00 - 01:30 pause and check or it could be two in general V minus E + f is between 2 - 2 G and 2 where G is known as the genus or very roughly the number of holes on the surface for example the genus of this double Taurus is two because there are two holes in it the Taurus has G equal 1 because there is just one hole so for
• 01:30 - 02:00 the Taurus V minus E + f is between 0 and 2 and therefore it can only be 0 1 or two this theorem might be different from what you have seen because while this is true for all graphs without self intersections most literature deals with a specific kind of graph known as triangulation for those kinds of graphs V minus E + f takes the smallest part possible value 2 - 2 G in the rare
• 02:00 - 02:30 occasion that the general case is mentioned the proof is basically left as exercise because it's supposed to be just an exercise I thought this was easy and attempted it this video is about my thought process When approaching this rather than riger I will focus more on the inition of where the result comes from I'll let you be the judge of whether this should just must be an
• 02:30 - 03:00 exercise let's start with the plane because I didn't know how to prove V minus E + fals 2 in this case the usual way is to start with just one vertex the base case in this case there is just one vertex so V is one there are no edges so e is zero the whole plane is a connected region itself so f is one so in the base case
• 03:00 - 03:30 V - e + f = 2 and if we subsequently add more edges to it that is the inductive step there are two possibilities the first possibility is we add an edge and a vertex in which case v and E both increase by one so V minus E + f remains unchanged the second possibility is that we add an edge to close off a region
• 03:30 - 04:00 let's say we join the bottom blue and purple vertices then we form a cycle to close off a region in addition to the original phase we now have two phases so F increases by one but of course e also increases by one because we have added an edge but V remains unchanged so V minus E + f is still two so whether you
• 04:00 - 04:30 connect to a new Vertex or form a cycle every time you add an edge V minus E plus F remains unchanged does this kind of argument extend to the Taurus well the base case is still exactly the same there is only one vertex so V equal 1 there are no edges yet so e is zero and the whole Taurus is not separated by edges yet so there is one continuous region and F is
• 04:30 - 05:00 one so in the base case for the Taurus V minus E + f is still two if we subsequently add more edges to it one possibility is to connect to a new vertex in which case both V and E increase by one and so V minus E + f remains unchanged at two this possibility is similar to the case for the plane
• 05:00 - 05:30 the other possibility is that the new Edge doesn't connect to a new vertex and forms a cycle instead for example in this case when we connect the yellow and pink vertices this way e goes up by one but because we didn't add any new vertices V remains unchanged however this cycle doesn't help us create a phase there is still just one phase because you can go to The
• 05:30 - 06:00 Other Side by going under so with f and V unchanged but e increasing by one the value of V minus e+ F actually decreases by one and is no longer two I've deliberately chosen this cycle because it's a non-c contractable loop that is you can't contract this Loop down to a point continuously on the Taurus no matter what you do
• 06:00 - 06:30 you can't eliminate that hole this is a key feature that separates a Taurus from a plane but that's not the full reason why we decrease V minus E + f when we complete the loop let's say we have this graph already and want to close the lower loop but before that let's check the v e and f there are six vertices Al together so V is six for reges there are three on top that form the upper Loop
• 06:30 - 07:00 two edges on the incomplete lower loop and one Edge that connects the two so altogether we have e equal 6 finally there is still just one continuous region even if you are apparently stuck here you can still go around the Taurus and get out so it is still one big continuous region altogether V - e + f is 1
• 07:00 - 07:30 if we now complete the lower loop then e will increase by one because we have added an edge but actually F will also increase by one this is because closing the lower loop genuinely creates a new region if you are stuck inside the strip you can't go outside without passing through the edges so with both E and F increasing by One V minus E + f F remains unchanged so even if you have an
• 07:30 - 08:00 extra non-c constructible Loop you can't decrease the value of vus e + f by more than 1 but maybe that's because the two Loops are kind of the same what if the two non-contract Loops are genuinely different so one of these Loops can't be deformed to another a more technical jargon for this is the two Loops are not homo topic let's count v e and f before
• 08:00 - 08:30 closing the second Loop there are altogether five veres so V is five for the edges there are three on top that form a complete Loop and there are two that form an incomplete Loop so altogether e is also five the whole Taurus is still one big continuous region so f is one which means that V minus e + f is equal to 1 now let's
• 08:30 - 09:00 complete the loop by completing the loop e has increased by One V hasn't changed because we haven't added any new vertices but f is actually still one so the value of V minus E + f has decreased by one the reason why f is still one is that if you want to go from the blue point to the orange point you can go under the the Taurus to get to this
• 09:00 - 09:30 purple Point first and then go around the Taurus to get to the Orange Point throughout your journey you haven't crossed any of the edges so this whole thing is still one big continuous region by considering different non-contract Loops we can make V minus E + f decrease by two from 2 to zero can it decrease even further well we we know the answer
• 09:30 - 10:00 should be no because in the case of a Taurus this theorem says it can't be below zero but what's stopping us from closing a non-contractible loop again the best way to think about this is that this phas after separated by edges is topologically equivalent to a plane these two sides separated by the loop can't connect with each other unless we go the long way around the Torus so
• 10:00 - 10:30 topologically it's the same as cutting along the loop and unwrapping the whole thing as a cylinder but then by the exact same logic these two sides are still separated by an edge so again topologically it is the same as cutting along this line and unwrapping it to become just a normal plane because we previously established that on the plane V minus E e+ F remains unchanged when we
• 10:30 - 11:00 add edges when we add any further edges on the Taurus on the left V minus E + f also will not decrease any further this particular configuration has vus e plus F to be zero because this quantity doesn't decrease further when we add more edges it will never dip below zero of course we haven't covered all the possible cases but now at least
• 11:00 - 11:30 we have the very rough intuition that V minus E + f decreases when we complete a different non-c constructible Loop using a similar intuition if we introduce more holes for the Taurus then roughly speaking each hole can contribute two of these non-contractible Loops so in general V - e + f is bounded below by 2 - 2 G
• 11:30 - 12:00 Technically when we have more holes Loops like this in the middle are still non-c contractable but they won't decrease the value of V minus E + f because closing this Loop creates a new phase these two regions are now actually separated by this Loop so 2 - 2G is really the lower bound but is my idea for this supposed textbook exercise even remotely valid
• 12:00 - 12:30 I Googled a bit harder and found this stack exchange answer basically it states that there are three cases when we add an edge we have already Illustrated each of them separately but this answer phrases them a bit more precisely in a different way so my thought process was actually not bad I put the link to the answer in the description while we did prove this more General inequality which will be useful in the next video there are plenty of
• 12:30 - 13:00 reasons why most people only considered the special case this lower bound known as Oiler characteristic usually denoted as Kai is useful in distinguishing surfaces with different gener yes the plural of genus is gener not only is it useful in differentiating between different surfaces oil characteristic appears in places seemingly unrelated to graphs from ve fields to something called moris
• 13:00 - 13:30 functions you can go to the description to see what those are but I want to highlight one other place where Oiler characteristic comes up these B notot B1 and B2 are called Betty numbers defined using homology an important Concept in algebraic topology these Betty numbers generalize to higher Dimensions so you can use this alternate ating sum of Betty numbers to
• 13:30 - 14:00 Define oiler characteristic for higher dimensional spaces not just Services if you want to know more about homology which Betty numbers are defined with check out atus new video on his channel Alf zero as always thanks for the patrons like And subscribe I'll see you soon bye-bye
• 14:00 - 14:30 [Music]