In Stock Overview Based on lectures to advanced undergraduate and first-year graduate students, this is a thorough, sophisticated and modern treatment of elementary algebraic topology, essentially from a homotopy theoretic viewpoint. The presentation of the homotopy theory and the account of duality in homology manifolds make the text ideal for a course on either homotopy or homology theory. The idea of algebraic topology is to translate problems in topology into problems in algebra with the hope that they have a better chance of solution. The translation process is usually carried out by means of the homology or homotopy groups of a topological space.

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Main branches of algebraic topology[ edit ] Below are some of the main areas studied in algebraic topology: Main article: Homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.

The first and simplest homotopy group is the fundamental group , which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

That is, cohomology is defined as the abstract study of cochains, cocycles , and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology.

Cohomology arises from the algebraic dualization of the construction of homology. Main article: Manifold A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane , the sphere , and the torus , which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions.

Main article: Knot theory Knot theory is the study of mathematical knots. Main articles: Simplicial complex and CW complex A simplicial 3-complex. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points , line segments , triangles , and their n-dimensional counterparts see illustration.

Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. Whitehead to meet the needs of homotopy theory.

This class of spaces is broader and has some better categorical properties than simplicial complexes , but still retains a combinatorial nature that allows for computation often with a much smaller complex. Method of algebraic invariants[ edit ] An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones [2] the modern standard tool for such construction is the CW complex.

In the s and s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups , which led to the change of name to algebraic topology. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statement easier to prove. Two major ways in which this can be done are through fundamental groups , or more generally homotopy theory , and through homology and cohomology groups.

The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a finite simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated.

Finitely generated abelian groups are completely classified and are particularly easy to work with. Setting in category theory[ edit ] In general, all constructions of algebraic topology are functorial ; the notions of category , functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence or, much more deeply, existence of mappings.

One of the first mathematicians to work with different types of cohomology was Georges de Rham. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the s, when Samuel Eilenberg and Norman Steenrod generalized this approach.

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms e. Applications of algebraic topology[ edit ] Classic applications of algebraic topology include: The Brouwer fixed point theorem : every continuous map from the unit n-disk to itself has a fixed point.

A manifold is orientable when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0. The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd.

For n.

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## ALGEBRAIC TOPOLOGY MAUNDER PDF

Main branches of algebraic topology[ edit ] Below are some of the main areas studied in algebraic topology: Main article: Homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. That is, cohomology is defined as the abstract study of cochains, cocycles , and coboundaries.

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## Algebraic topology

Jujinn Algebraic K-theory Exact sequence Glossary of algebraic topology Grothendieck topology Higher category theory Higher-dimensional algebra Homological algebra. Maunder Snippet view — Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence or, much more deeply, existence of mappings. The presentation of the homotopy theory and the account of duality in homology manifolds make the text ideal for a course on either homotopy or homology theory. The translation process is usually carried out by means of the homology or homotopy groups of a topological space. Algebraic topology The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. Account Options Sign in.

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## Algebraic Topology

Grolar K-theory Lie algebroid Lie groupoid Important publications in algebraic topology Serre spectral sequence Sheaf Topological quantum field theory. Cohomology arises from the algebraic dualization of the construction of homology. Homotopy and Simplicial Complexes. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statement easier to prove. This class of spaces is broader and has algeebraic better categorical properties than simplicial complexesbut still retains a combinatorial nature that allows for computation often with a much smaller complex. Topoolgy mathematics, homotopy groups are used in algebraic topology to classify topological spaces.