Theorems in algebraic topology | Homological algebra

Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: Hi(X; Z) completely determine its homology groups with coefficients in A, for any abelian group A: Hi(X; A) Here Hi might be the simplicial homology, or more generally the singular homology: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor. For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers bi of X and the Betti numbers bi,F with coefficients in a field F. These can differ, but only when the characteristic of F is a prime number p for which there is some p-torsion in the homology. (Wikipedia).

Video thumbnail

Tutorial - Detrmining the Leading coefficient and degree of a polynomial with a fraction ex 14

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is t

From playlist Find the leading coefficient and degree of a polynomial | equation

Video thumbnail

What is the definition of standard form, degree and leading coefficient of a polynomial

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is t

From playlist Find the leading coefficient and degree of a polynomial | equation

Video thumbnail

How to tell the difference between the leading coefficient and the degree of a polynomial

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is the s

From playlist Find the leading coefficient and degree of a polynomial | expression

Video thumbnail

What is the leading coefficient of a polynomial & degree

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is t

From playlist Find the leading coefficient and degree of a polynomial | equation

Video thumbnail

Calculus 5.3 The Fundamental Theorem of Calculus

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

Video thumbnail

Amaury Pouly

A (truly) universal polynomial differential equation Lee A. Rubel proved in 1981 that there exists a universal fourth-order algebraic differential equation P(y,y',y'',y''')=0 (1) and provided an explicit example. It is universal in the sense that for any continuous function f from R to

From playlist DART X

Video thumbnail

Determining if a equation is a polynomial or not

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Classify a polynomial then determining if it is a polynomial or not

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

How to identify Degree and Leading Coefficient of a polynomial

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is t

From playlist Find the leading coefficient and degree of a polynomial | equation

Video thumbnail

Iwasawa Main Conjecture for Universal Families by Xin Wan

PROGRAM ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (HYBRID) ORGANIZERS: Ashay Burungale (CalTech/UT Austin, USA), Haruzo Hida (UCLA), Somnath Jha (IIT Kanpur) and Ye Tian (MCM, CAS) DATE: 08 August 2022 to 19 August 2022 VENUE: Ramanujan Lecture Hall and online The program pla

From playlist ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (2022)

Video thumbnail

Macroscopically minimal hypersurfaces - Hannah Alpert

Variational Methods in Geometry Seminar Topic: Macroscopically minimal hypersurfaces Speaker: Hannah Alpert Affiliation: Ohio State University Date: March 12, 2019 For more video please visit http://video.ias.edu

From playlist Variational Methods in Geometry

Video thumbnail

Stable Homotopy Seminar, 16: The Whitehead, Hurewicz, Universal Coefficient, and KĂĽnneth Theorems

These are some generalizations of facts from unstable algebraic topology that are useful for calculating in the category of spectra. The Whitehead and Hurewicz theorems say that a map of connective spectra that's a homology isomorphism is a weak equivalence, and that the lowest nonzero hom

From playlist Stable Homotopy Seminar

Video thumbnail

Common Linear Patterns Are Rare - Nina KamÄŤev

Computer Science/Discrete Mathematics Seminar I Topic: Common Linear Patterns Are Rare Speaker: Nina Kamčev Affiliation: University of Zagreb Date: April 03, 2023  Several classical results in Ramsey theory (including famous theorems of Schur, van der Waerden, Rado) deal with finding mon

From playlist Mathematics

Video thumbnail

Stochastic Homogenization (Lecture 1) by Andrey Piatnitski

DISCUSSION MEETING Multi-Scale Analysis: Thematic Lectures and Meeting (MATHLEC-2021, ONLINE) ORGANIZERS: Patrizia Donato (University of Rouen Normandie, France), Antonio Gaudiello (Università degli Studi di Napoli Federico II, Italy), Editha Jose (University of the Philippines Los Baño

From playlist Multi-scale Analysis: Thematic Lectures And Meeting (MATHLEC-2021) (ONLINE)

Video thumbnail

The Ramanujan Conjecture and some diophantine equations - Peter Sarnak

Speaker : Peter Sarnak Date and Time : Faculty Hall, IISc, Bangalore Venue : 25 May 12, 16:00 One of Ramanujan's most influential conjectures concerns the magnitude of the Fourier Coefficients of a modular form. These were made on the basis of his calculations as well as a far-reaching in

From playlist Public Lectures

Video thumbnail

Local (\ell = p) Galois Deformation Rings - Ashwin Iyengar

Joint IAS/Princeton University Number Theory Seminar Topic: Local (\ell = p) Galois Deformation Rings Speaker: Ashwin Iyengar Affiliation: Johns Hopkins University Date: February 10, 2022 I will present joint work with V. Paškūnas and G. Böckle concerning deformation rings for mod p Galo

From playlist Mathematics

Video thumbnail

Stochastic Homogenization (Lecture 3) by Andrey Piatnitski

DISCUSSION MEETING Multi-Scale Analysis: Thematic Lectures and Meeting (MATHLEC-2021, ONLINE) ORGANIZERS: Patrizia Donato (University of Rouen Normandie, France), Antonio Gaudiello (Università degli Studi di Napoli Federico II, Italy), Editha Jose (University of the Philippines Los Baño

From playlist Multi-scale Analysis: Thematic Lectures And Meeting (MATHLEC-2021) (ONLINE)

Video thumbnail

Geometry of the moduli space of curves – Rahul Pandharipande – ICM2018

Plenary Lecture 3 Geometry of the moduli space of curves Rahul Pandharipande Abstract: The moduli space of curves, first appearing in the work of Riemann in the 19th century, plays an important role in geometry. After an introduction to the moduli space, I will discuss recent directions

From playlist Plenary Lectures

Video thumbnail

Determining the leading coefficient and degree of a polynomial

👉 Learn how to find the degree and the leading coefficient of a polynomial expression. The degree of a polynomial expression is the the highest power (exponent) of the individual terms that make up the polynomial. For terms with more that one variable, the power (exponent) of the term is t

From playlist Find the leading coefficient and degree of a polynomial | equation

Video thumbnail

CTNT 2022 - The unbounded denominators conjecture (by Yunqing Tang)

This video is one of the special guess talks or conference talks that took place during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. Note: not every special guest lecture or conference lecture was recorded. More about CTNT: https://ctnt-summer.math.uconn.edu/

From playlist CTNT 2022 - Conference lectures and special guest lectures

Related pages

Cohomology ring | Topological space | Simplicial homology | Torsion (algebra) | Bockstein spectral sequence | Coefficient | Homological algebra | Adjoint | Betti number | Algebraic topology | Cohomology | Chain complex | Tensor product of modules | Eilenberg–MacLane space | Closed manifold | Poincaré duality | Real projective space | Singular homology | Characteristic (algebra) | Connected space | Free abelian group | Field (mathematics) | Orientability | Ext functor | Functor | Manifold | Prime number | Fundamental theorem of finitely generated abelian groups | Tor functor | Abelian group | Splitting lemma