Geometric topology | Continuous mappings | 3-manifolds | Theorems in topology
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M with boundary ∂M there is a map with not nullhomotopic in , then there is an embedding with the same property. The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco): Let be a 3-manifold and let be a connected surface in . Let be a normal subgroup such that .Let be a continuous map such that and Then there exists an embedding such that and Furthermore if one starts with a map f in general position, then for any neighborhood U of the singularity set of f, we can find such a g with image lying inside the union of image of f and U. Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the sphere theorem (3-manifolds), which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial embedding of a sphere. There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction. A proof not utilizing the tower construction exists of the first version of the loop theorem. This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by , Marc Lackenby, and Iain Aitchison with Hyam Rubinstein. (Wikipedia).
Basic intro to answer the question, "What's a Loop?" If there's a set of data, (we'll use an array for this example, but it doesn't have to be,) and you want to perform the same manipulation to every piece in that set of data, you can use something called a loop. Loops have some complex
From playlist Computer Science and Software Engineering Theory with Briana
Overview of Loops in Graph Theory | Graph Loop, Multigraphs, Pseudographs
What are loops in graph theory? Sometimes called self loops, a loop in a graph is an edge that connects a vertex to itself. These are not allowed in what are often called "simple graphs", which are the graphs we usually study when we begin studying graph theory. In simple graphs, loop ed
From playlist Graph Theory
How to Make a For Loop in Python
This video explains the basics of for loops in Python including looping over lists, numerical ranges, the continue keyword and the break keyword.
From playlist Python Basics
For Loop In Python | Python For Loop Tutorial | Python Tutorial | Python Programming | Simplilearn
This Python tutorial will help you understand what is for loop and how to use for loop in Python. In programming, statements are executed sequentially. The first statement in a code is executed first, followed by the second one, and so on. There may be a situation when you need to execute
Python 3 Programming Tutorial - For loop
The next loop is the For loop. The idea of the for loop is to "iterate" through something. For each thing in that something, it will do a block of code. Most often, you will a for loop's structure very much like for eachThing in thisThing: do this stuff in this block So, again, wh
From playlist Python 3 Basics Tutorial Series
Graph Theory FAQs: 01. More General Graph Definition
In video 02: Definition of a Graph, we defined a (simple) graph as a set of vertices together with a set of edges where the edges are 2-subsets of the vertex set. Notice that this definition does not allow for multiple edges or loops. In general on this channel, we have been discussing o
From playlist Graph Theory FAQs
This video states and investigates the triangle inequality theorem. Complete Video List: http://www.mathispower4u.yolasite.com
From playlist Relationships with Triangles
More videos like this online at http://www.theurbanpenguin.com We take a moment to look at for loops within Python using Python 3 installed on an openSUSE Linux desktop. First we iterate through modules installed on the system using sys.modules; then we move into reading files line by line
From playlist Python
The Definition of a Graph (Graph Theory)
The Definition of a Graph (Graph Theory) mathispower4u.com
From playlist Graph Theory (Discrete Math)
Fixed points in digital topology
A talk given by Chris Staecker at King Mongkut's University of Technology Thonburi, Bangkok, Thailand, on October 11 2019. This is the second in a series of 3 talks given at KMUTT. Includes an introduction to graph-theoretical ("Rosenfeld style") digital topology, and some basic results a
From playlist Research & conference talks
Vertex gluings and Demazure products by Nathan Pflueger
PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t
From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)
Macroscopic loops in the Spin O(N), double dimer and related models - Lorenzo Taggi
Probability Seminar Topic: Macroscopic loops in the Spin O(N), double dimer and related models Speaker: Lorenzo Taggi Affiliation: Sapienza Università di Roma Date: October 10, 2022 We consider a general system of interacting random loops which includes several models of interest, such a
From playlist Mathematics
Lecture 8: Bökstedt Periodicity
In this video, we give a proof of Bökstedts fundamental result showing that THH of F_p is polynomial in a degree 2 class. This will rely on unlocking its relation to the dual Steenrod algebra and the fundamental fact, that the latter is free as an E_2-Algebra. Feel free to post comments a
From playlist Topological Cyclic Homology
Partitions of n-valued maps: a meal in four courses
A research talk presented at the Farifield University Mathematics Research Seminar, February 12, 2021. Should be accessible to a general mathematics audience. The paper: https://arxiv.org/abs/2101.09326
From playlist Research & conference talks
Intro to the Fundamental Group // Algebraic Topology with @TomRocksMaths
In this video I teach the amazing @TomRocksMaths a little bit of algebraic topology, specifically the fundamental group. Tom also taught me some really cool fluid dynamics and you can find our collab over at his channel here: ►►► https://www.youtube.com/watch?v=bpeCfwY4qa0&ab_channel=TomR
From playlist Collaborations
Stable Homotopy Seminar, 1: Introduction and Motivation
We describe some features that the category of spectra is expected to have, and some ideas from topology it's expected to generalize. Along the way, we review the Freudenthal suspension theorem, and the definition of a generalized cohomology theory. ~~~~~~~~~~~~~~~~======================
From playlist Stable Homotopy Seminar
Benson Farb, Part 3: Reconstruction problems in geometry and topology
29th Workshop in Geometric Topology, Oregon State University, June 30, 2012
From playlist Benson Farb: 29th Workshop in Geometric Topology
Yonatan Harpaz - New perspectives in hermitian K-theory II
Warning: around 32:30 in the video, in the slide entitled "Karoubi's conjecture", a small mistake was made - in the third bulleted item the genuine quadratic structure appearing should be the genuine symmetric one (so both the green and red instances of the superscript gq should be gs), an
From playlist New perspectives on K- and L-theory
Seeing that a while loop can do the same thing as a for loop
From playlist Computer Science
Gödel's Second Incompleteness Theorem, Proof Sketch
In order for math to prove its own correctness, it would have to be incorrect. This result is Gödel’s second incompleteness theorem, and in this video, we provide a sketch of the proof. Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Brandon Chen, Zachary Greenb
From playlist Infinity, and Beyond!