Topics in Algebraic Topology The Sullivan Conjecture
Topics in Algebraic Topology The Sullivan Conjecture
Topics in Algebraic Topology The Sullivan Conjecture
The
goal of this note is to describe some of the tools which enter into the proof of
Sullivan's conjecture. Topics covered includes: Steenrod operations, The Adem
relations, Admissible monomials, Free unstable modules, A theorem of
Gabriel-Kuhn-Popesco, Injectivity of the cohomology of BV, Generating analytic
functors, Tensor products and algebras, Free unstable algebras, The dual
Steenrod algebra, The Frobenius, Finiteness conditions, Injectivity of tensor
products, Lannes T-functor, The T-functor and unstable algebras, Free E-infinity
algebras, A pushout square, The Eilenberg-Moore spectral sequence, Operations on
E-infinity algebras, The Sullivan conjecture.
This PDF Lectures
covers the following topics related to Algebraic Topology : Singular homology,
Introduction: singular simplices and chains, Homology, Categories, functors, and
natural transformations, Basic homotopy theory, The homotopy theory of CW
complexes, Vector bundles and principal bundles, Spectral sequences and Serre
classes, Characteristic classes, Steenrod operations, and cobordism.
This book explains the following topics: Introduction,
Fundamental group, Classification of compact surfaces, Covering spaces,
Homology, Basics of Cohomology, Cup Product in Cohomology, Poincaré Duality,
Basics of Homotopy Theory, Spectral Sequences. Applications, Fiber bundles,
Classifying spaces, Applications, Vector Bundles, Characteristic classes,
Cobordism, Applications.
Author(s): Laurentiu Maxim, University of
Wisconsin-Madison
The contents of this
book include: Topological spaces, General topology: some delicate bits,
Topological manifolds and manifolds, Categories, functors and natural
transformations, Covering spaces and manifolds, Homotopy equivalent topological
spaces, Differential topology, Basics of group theory, The basic Seifert-van
Kampen Theorem , Presentations of groups and amalgamated products, The general
Seifert-van Kampen Theorem , Cones, suspensions, cylinders, Limits, etc .
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 Brouwer’s fixed point theorem, Borsuk Ulam
theorem, fundamental theorem of algebra.
This note covers the
following topics: Important examples of topological spaces, Constructions,
Homotopy and homotopy equivalence, CW -complexes and homotopy, Fundamental
group, Covering spaces, Higher homotopy groups, Fiber bundles, Suspension
Theorem and Whitehead product, Homotopy groups of CW -complexes, Homology
groups, Homology groups of CW -complexes, Homology with coefficients and
cohomology groups, Cap product and the Poincare duality, Elementary obstruction
theory.
This note explains the
following topics: Building blocks and homeomorphy, Homotopy, Simplicial
Complexes,CW-Spaces, Fundamental Group , Coverings, Simplicial Homology and
Singular Homology.
The
goal of this note is to describe some of the tools which enter into the proof of
Sullivan's conjecture. Topics covered includes: Steenrod operations, The Adem
relations, Admissible monomials, Free unstable modules, A theorem of
Gabriel-Kuhn-Popesco, Injectivity of the cohomology of BV, Generating analytic
functors, Tensor products and algebras, Free unstable algebras, The dual
Steenrod algebra, The Frobenius, Finiteness conditions, Injectivity of tensor
products, Lannes T-functor, The T-functor and unstable algebras, Free E-infinity
algebras, A pushout square, The Eilenberg-Moore spectral sequence, Operations on
E-infinity algebras, The Sullivan conjecture.
Much of topology is
aimed at exploring abstract versions of geometrical objects in our world. The
concept of geometrical abstraction dates back at least to the time of Euclid.
All of the objects that we will study in this note will be subsets of the
Euclidean spaces. Topics covered includes: 2-manifolds, Fundamental group and
covering spaces, Homology, Point-Set Topology, Group Theory, Graph Theory and
The Jordan Curve Theorem.
This note covers the
following topics: moduli space of flat symplectic surface bundles, Cohomology of
the Classifying Spaces of Projective Unitary Groups, covering type of a space, A
May-type spectral sequence for higher topological Hochschild homology,
topological Hochschild homology of the K(1)-local sphere, Quasi-Elliptic
Cohomology and its Power Operations, Local and global coincidence homology
classes, Tangent categories of algebras over operads, Automorphisms of the
little disks operad with p-torsion coefficients.
This book explains the following
topics: the fundamental group, covering spaces, ordinary homology and cohomology
in its singular, cellular, axiomatic, and represented versions, higher homotopy
groups and the Hurewicz theorem, basic homotopy theory including fibrations and
cofibrations, Poincare duality for manifolds and manifolds with boundary.
These notes provides a brief overview of basic
topics in a usual introductory course of algebraic topology. Topics covered
includes: Basic notions and constructions, CW-complexes, Simplicial and
singular homology, Homology of CW-complexes and applications, Singular
cohomology, homological algebra, Products in cohomology, Vector bundles and Thom
isomorphism, Poincar´e duality, Homotopy groups, Fundamental group, Homotopy and
CW-complexes, Homotopy excision and Hurewitz theorem.
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).
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.
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.