In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping tori. Here is the construction: take the Cartesian product of a surface with the unit interval. Glue the two copies of the surface, on the boundary, by some homeomorphism. This homeomorphism is called the monodromy of the surface bundle. It is possible to show that the homeomorphism type of the bundle obtained depends only on the conjugacy class, in the mapping class group, of the gluing homeomorphism chosen. This construction is an important source of examples both in the field of low-dimensional topology as well as in geometric group theory. In the former we find that the geometry of the three-manifold is determined by the dynamics of the homeomorphism. This is the fibered part of William Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups. In geometric group theory the fundamental groups of such bundles give an important class of HNN-extensions: that is, extensions of the fundamental group of the fiber (a surface) by the integers. A simple special case of this construction (considered in Henri Poincaré's foundational paper) is that of a torus bundle. (Wikipedia).
From playlist Surface integrals
Flow through a single piece of area
From playlist Surface integrals
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From playlist Surface integrals
Multivariable Calculus | Surface integrals over vector fields.
We introduce the notion of a surface integral over a vector field, derive a computational formula, and give an example. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcol
From playlist Multivariable Calculus | Surface Integrals
Physics Ch 67.1 Advanced E&M: Review Vectors (54 of 113) What is a Surface Integral?
Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn that a surface integral is the integral over a surface multiplied (via the dot product) with a vector field passing thr
From playlist PHYSICS 67.1 ADVANCED E&M VECTORS & FIELDS
Given the circumference how do you find the surface area of a hemisphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Benson Farb, Part 2: Surface bundles, mapping class groups, moduli spaces, and cohomology
29th Workshop in Geometric Topology, Oregon State University, June 29, 2012
From playlist Benson Farb: 29th Workshop in Geometric Topology
How to find the perimeter of a triangle given a lot of tangent lines
Learn how to solve problems with tangent line. A tangent line to a circle is a line that touches the circle at exactly one point. The tangent line to a circle makes a right angle with the radius of the circle at the point of its tangency. Thus, to solve for any missing value involving the
From playlist Circles
How do you find the surface area of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
An introduction to spectral data for Higgs bundles.. by Laura Schaposnik
Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio
From playlist Higgs Bundles
Jordan Sahattchieve: A Fibering Theorem for 3-Manifolds
Jordan Sahattchieve Title: A Fibering Theorem for 3-Manifolds In this talk, I will endeavor to communicate a new fibering theorem for 3-manifolds in the style of Stalling's Fibration Theorem.
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Homology Smale-Barden manifolds with K-contact and Sasakian structures by Aleksy Tralle
Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio
From playlist Higgs Bundles
Pedram Hekmati: What is a cohomological field theory?
Abstract: Many interesting invariants in geometry satisfy certain glueing or factorisation conditions, that are often useful when doing calculations. Topological quantum field theories (TQFTs) emerged in the 1980s as an organising structure for invariants that are governed by bordisms. I
From playlist What is...? Seminars
Andrei Teleman - Instantons and holomorphic curves on surfaces of class VII (Part 4)
This series of lectures is dedicated to recent results concerning the existence of holomorphic curves on the surfaces of class VII. The first lecture will be an introduction to the Donaldson theory. We will present the fundamental notions and some important results in the theory, explainin
From playlist École d’été 2012 - Feuilletages, Courbes pseudoholomorphes, Applications
Alex Wright - Minicourse - Lecture 3
Alex Wright Dynamics, geometry, and the moduli space of Riemann surfaces We will discuss the GL(2,R) action on the Hodge bundle over the moduli space of Riemann surfaces. This is a very friendly action, because it can be explained using the usual action of GL(2,R) on polygons in the plane
From playlist Maryland Analysis and Geometry Atelier
On embeddings of manifolds - Dishant Mayurbhai Pancholi
Seminar in Analysis and Geometry Topic: On embeddings of manifolds Speaker: Dishant Mayurbhai Pancholi Affiliation: Chennai Mathematical Institute; von Neumann Fellow, School of Mathematics Date: October 26, 2021 We will discuss embeddings of manifolds with a view towards applications i
From playlist Mathematics
Area and Perimeter of Geometric Figures
Worked out examples involving area and perimeter.
From playlist Geometry
Floer theory in spaces of stable pairs over Riemann surfaces - Timothy Perutz
Floer theory in spaces of stable pairs over Riemann surfaces Timothy Perutz University of Texas, Austin; von Neumann Fellow, School of Mathematics May 4, 2017
From playlist Mathematics
Spinors and the Clutching Construction
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From playlist Summer of Math Exposition Youtube Videos
Calculus 16.7 Surface Integrals
My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart
From playlist Calculus