Description: This book explains the following topics: Some Underlying Geometric Notions, The Fundamental Group, Homology, Cohomology and Homotopy Theory.
Description: This book explains the following topics: The fundamental group and some of its applications, Categorical language and the van Kampen theorem, Covering spaces, Graphs, Compactly generated spaces, Cofibrations, Fibrations, Based cofiber and fiber sequences, Higher homotopy groups, CW complexes, The homotopy excision and suspension theorems, Axiomatic and cellular homology theorems, Hurewicz and uniqueness theorems, Singular homology theory, An introduction to K theory.
Description: The first half of the book deals with degree theory, the Pontryagin construction, intersection theory, and Lefschetz numbers. The second half of the book is devoted to differential forms and deRham cohomology.
Description: First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces.Further it covers metric spaces, Continuity and open sets for metric spaces, Closed sets for metric spaces, Topological spaces, Interior and closure, More on topological structures, Hausdorff spaces and Compactness.
Description: This is one day going to be a textbook on K-theory, with a particular emphasis on connections with geometric phenomena like intersection multiplicities.
Description: This note covers the following topics: Geometric reformulation, The Adams-Novikov spectral sequence, Elliptic cohomology, What is TMF, Geometric and Physical Aspect.
Description: This note provides an overview of various aspects of algebraic K-theory, with the intention of making these lectures accessible to participants with little or no prior knowledge of the subject.
Description: This book covers the following topics: The Mayer-Vietoris Sequence in Homology, CW Complexes, Cellular Homology,Cohomology ring, Homology with Coefficient, Lefschetz Fixed Point theorem, Cohomology, Axioms for Unreduced Cohomology, Eilenberg-Steenrod axioms, Construction of a Cohomology theory, Proof of the UCT in Cohomology, Properties of Ext(A;G).
Description: This book covers the following topics: Cell complexes and simplical complexes, fundamental group, covering spaces and fundamental group, categories and functors, homological algebra, singular homology, simplical and cellular homology, applications of homology.``
Description: This note covers the following topics: Basic Algebra of Polynomials, Induction and the Well ordering Principle, Sets, Some counting principles, The Integers, Unique factorization into primes, Prime Numbers, Sun Ze's Theorem, Good algorithm for exponentiation, Fermat's Little Theorem, Euler's Theorem, Primitive Roots, Exponents, Roots, Vectors and matrices, Motions in two and three dimensions, Permutations and Symmetric Groups, Groups: Lagrange's Theorem, Eul...
Description: This note explains the following topics: Cohomology, The Mayer Vietoris Sequence, Compactly Supported Cohomology and Poincare Duality, The Kunneth Formula for deRham Cohomology, Leray-Hirsch Theorem, Morse Theory, The complex projective space.
Description: This note covers the following topics: Group theory, The fundamental group, Simplicial complexes and homology, Cohomology, Circle bundles.
Description: Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring A instead of over a eld.
Description: This note covers the following topics: The Fundamental Group, Covering Projections, Running Around in Circles, The Homology Axioms, Immediate Consequences of the Homology Axioms, Reduced Homology Groups, Degrees of Spherical Maps again, Constructing Singular Homology Theory.
Description: We will follow a direct approach to differential topology and to many of its applications without requiring and exploiting the abstract machinery of algebraic topology. The course will cover immersion, submersions and embeddings of manifolds in Euclidean space (including the basic results by Sard and Whitney), a discussion of the Euler number and winding numbers, fixed point theorems, the Borsuk-Ulam theorem and respective applications. At the end of the cou...
Description: This book covers the following topics: Cell complexes and simplical complexes, fundamental group, covering spaces and fundamental group, categories and functors, homological algebra, singular homology, simplical and cellular homology, applications of homology.
Description: This note covers the following topics: Background Linear Algebra, Lie Algebras: Definition and Basic Properties, Solvable Lie Algebras and Lie s Theorem, Nilpotent Lie Algebras and Engel s Theorem, Cartan s Criteria for Solvability and Semisimplicity, Semisimple Lie Algebras, root Space Decompositions, Classical Simple Complex Lie Algebras.