Borsukulam is considered a great theorem because it has several di erent equivalent versions, many di erent proofs, many extensions and generalizations, and many interesting applications. Researchers solve ham sandwich mystery education the. Detailed syllabus for 2003 qualifying exam in topology. In many instances, we can identify a pattern in the solutions. In italy, the theorem is also known as theorem of carabinieri, better known as the 12 theorem the squeeze theorem is used in calculus and mathematical analysis. In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.
The proof of the theorem relies on algebraic topology pertaining to antipode preserving maps and homotopy theory. Some combinatorial and algorithmic applications of the. The theorem is called the ham sandwich theorem because sandwiches are made of three dimensional pieces of bread that are cut in half with a flat sheet, like a slice of ham. Unfortunately, the intermediate value theorem does not suffice to prove these higherdimensional analogs. One application is to show that the two dimensional and three dimensional. The volumes of any ncompact sets in rn can always be simultaneously bisectedbyann. Many combinatorial problems for example, the the ham sandwich theorem, the kneser conjecture and evasiveness of graph properties, can be rephrased and put into a topological setting which is suitable for applying results and tools from algebraic topology. Sometimes these are detailed, and sometimes they give references in the following texts. A ham sandwich theorem is derived for n complex borel measures on cn.
With this, we o er a proof of the two dimensional polygon case of the ham sandwich theorem. A note on the ham sandwich theorem hugo steinhaus and others from mathesis polska xi, 1938, pp. Algebraic topology paul yiu department of mathematics florida atlantic university summer 2006 wednesday, june 7, 2006 monday 515 522 65 612 619. Like the borsukulam theorem the ham sandwich theorem has many different formulations, though not all are equivalent. Ham sandwich theorem and other adventures in topology.
We discuss the borsukulam theorem concerning a continuous map from the sphere to the plane, and the ham sandwich theorem. Then there is a plane a cut of a knife that will cut all three in half at the same time. This is not the same as the squeeze law, which is a result for computing limits. The ham sandwich theorem takes its name from the case when n 3 and the three objects of any shape are a chunk of ham and two chunks of breadnotionally, a sandwich which can then all be simultaneously bisected with a single cut i. In the plane, examples of this class of problems are finding, e. Ham sandwich theorem simple english wikipedia, the free. To distinguish topological spaces we will consider topological invariants such as the fundamental group, which is a powerful way of using an algebraic invariant to detect topological features of spaces. The proof of theorem 1 uses the polynomial method of dvir. Pdf leftovers from the ham sandwich theorem researchgate. I would like to see a constructive proof of this theorem, but i do not know of one. This is a basic note in algebraic topology, it introduce the notion of fundamental groups, covering spaces, methods for computing fundamental groups using seifert van kampen theorem and some applications such as the brouwers fixed point theorem, borsuk ulam theorem, fundamental theorem of algebra. In this section, we develop the tools that we will use.
Lecture notes were posted after most lectures, summarizing the contents of the lecture. In mathematical measure theory, for every positive integer n the ham sandwich theorem states. In particular, we will use a polynomial generalization of the ham sandwich theorem, proven by. This selfcontained introduction to algebraic topology is suitable for a number of topology courses. The early examples include the fundamental theorem of algebra, brouwers xed point theorem and the domain invariance theorem. The ham sandwich theorem takes its name from the case when n 3 and the three objects of any shape are a chunk of ham and two chunks of breadnotionally, a sandwichwhich can then all be simultaneously bisected with a single cut i. The main new idea in the paper isa new approachforadaptingdvirs method to rn. A familiar consequence is the hamsandwich theorem given d nite continuous. The delightful hamsandwich theorem is discussed along with a proof of the lusternikschnirelmanborsuk theorem. Emphasis is put on easy and intuitive proofs using discrete geometry and algebra. Pdf some combinatorial and algorithmic applications of.
We will omit these details and assume that borsukulam holds. The two major applications under consideration, the ham sandwich theorem and the kneser conjecture, come from. In computational geometry, this ham sandwich theorem leads to a. Ham sandwich theorems we will build our polynomial cell decomposition using a tool from topology, the ham sandwich theorem.
The con guration spaces of the partitions we use for the proof of theorems 1. From cutting pancakes to szemereditrotter and ham sandwiches. This paper continues the search, started in 10, for relatives of the ham sandwich theorem. Here we choose to appeal to 2 big machinery in algebraic topology, namely. The theorem guarantees that one can always simultaneously bisect the ham and the two slices of bread by a. It rescues the careless sandwich maker by guaranteeing that it is always possible to slice the sandwich with one cut so that the ham and both slices of bread are each divided into equal halves, no matter how. A screenshot pdf which includes algtop11 to 20 can be found at my. The ham sandwich theorem has been a treat and a spur to mathematicians for more than half a century. For larger values of r, the lower bounds we obtain for mn. Suppose you have three regions in space think of them as the ham, the cheese, and the bread.
This talk aims to give a picture of what algebraic topology is. Equivariant algebraic topology applied to some problems in. A classical application of the borsukulam theorem is the ham sandwich theorem. Lecture notes assignments download course materials. Lecture notes algebraic topology ii mathematics mit. The sandwich theorem is a result of algebraic topology which says the following. If there were three pieces of bread, it would be possible to make one cut along a plane to divide each of bread into two equal pieces. The borsukulam theorem is a fundamental result in algebraic topology, with applications to various areas of mathematics. For any collection of three solids in the threedimensional space there exists a plane which simultaneously bisects all of them, i. A first course in algebraic topology by czes kosniowski. The proof of the theorem involves algebraic topology, which we will omit. Given two polygons in r2 there exists a line that simultaneously.
We now turn to the proof of the polynomial hamsandwich theorem. Gerrymandering, sandwiches,an d topology pablosoberon communicatedbycesare. The ham sandwich theorem, or stonetukey theorem, is a classical result that appears in many introductory books on algebraic topology. The aim of this module is to explore properties of topological spaces. The bread was made by extruding a closed sketch, and the ham slice was made as a sheet metal part for reasons that will become clear later. Another is the ham sandwich theorem, which says that given any collection of n.
This paper will demonstrate this by rst exploring the various formulations of the borsukulam theorem, then exploring two of its applications. Ham sandwich is equivalent to borsukulam drops schloss. Pdf the ham sandwich theorem, or stonetukey theorem, is a classical result that appears in many introductory books on algebraic topology. It consists of about one quarter general topology without its usual pathologies and three quarters algebraic topology centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is. Sini sa vre cica, university of belgrade adam mickiewicz. This book gives a proof of the theorem, and it also discusses interesting applications of the borsuk ulam theorem, for example to knesers conjecture in combinatorics. In two dimensions, the theorem is known as the pancake theorem because of having to cut two infinitesimally thin pancakes on a plate each in half. It rescues the careless sandwich maker by guaranteeing that. Silva in may, i gave a talk about fair division problems in a. We prove among other results, the following implications fx211 wherekn, k is an important instance of the knasters conjecture so thatkn, n. The proof of the ham sandwich theorem for n 2 n2 n 2 is essentially the same but requires a higherdimensional analog of the borsukulam theorem. Introduction the ham sandwich theorem states something like the following ordinary language proposition. Suppose xis a topological space and a x is a subspace.
To lose no time, they wish to make a single cut with a knife so that each of the ingredients is split equally between the two halves. With this in hand, we explore the implications of generalizing the szemer edi and trotter theorem to algebraic curves instead of lines. The second part of the book introduces the beginnings of algebraic topology. Free algebraic topology books download ebooks online. It turns out this is possible, and the result is known as the ham. There was a bit of a kerfuffle about who invented it, but that question did get settled. Ham sandwiches of the many theorems that follow from the borsukulam theorem, the ham sandwich theorem has some of the best applications to combinatorics. The ham sandwich theorem and the continuum algebraic topology. A first course in algebraic topology, with emphasis on visualization, geometric intuition and simplified computations. If g e g then the subgroup generated by g is the subset of g consisting of all integral. No matter where one places the pieces of the sandwich in the kitchen, or house, or.
In a popular form this result is stated as the fact that it is possible to cut fairly an open hamsandwich consisting of two pieces of bread and a piece of ham with a single. Sure, it took a couple thousand out of our manufacturing budget, but. There is a generalization to higher dimensions, using algebraic topology. Starting from a cute little theorem, we end out with some big tools, and so it justi. Its the business of topology to describe more precisely such phenomena. Note that a ham sandwich typically consists of a ham and two slices of bread in r3. On applications of algebraic topology the applicability of the results and methods of algebraic topology throughout the mathematics is its crucial and one of the most signi cant properties. Take a sandwich made of a slice of ham and two slices of bread. The ham sandwich theorem and the continuum algebraic. The title from topology refers to a subdivision of notes that deals with topology. Borsukulam theorem is an interesting theorem on its own, because of its numerous applications and admits many kinds of proof. After reading the adams book, if you want to see some more serious applications of algebraic topology to knot theory, this book is a classic.
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