Algebraic topology. Front Cover. C. R. F. Maunder. Van Nostrand Reinhold Co., – Mathematics Bibliographic information. QR code for Algebraic topology . Based on lectures to advanced undergraduate and first-year graduate students, this is a thorough, sophisticated, and modern treatment of elementary algebraic. Title, Algebraic Topology New university mathematics series · The @new mathematics series. Author, C. R. F. Maunder. Edition, reprint. Publisher, Van Nostrand.
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K-theory Lie algebroid Lie groupoid Important publications in algebraic topology Serre spectral sequence Sheaf Topological quantum field 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.
In the s and s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groupswhich led to the change of name to algebraic topology. Knot theory is the study of mathematical knots. Account Options Sign in. For the topology of pointwise convergence, see Algebraic topology object.
algebraif Retrieved from ” https: A CW complex is a type of topological space introduced by J. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
Simplicial complexes should not be confused with the more abstract notion of a simplicial topolovy appearing in modern simplicial homotopy theory. Cohomology arises from the algebraic dualization of the construction of homology.
Much of the book is therefore concerned with the construction of these algebraic invariants, and with applications to topological problems, such as the classification of surfaces and duality theorems for manifolds. Homotopy Groups and CWComplexes.
Algebraic topology – C. R. F. Maunder – Google Books
The author has given much attention to detail, yet ensures that the reader knows where he is going. Whitehead Gordon Thomas Whyburn. Homology and cohomology groups, toopology the other hand, are abelian and in many important cases finitely generated. Product Description Product Details 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.
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. Algebraic algfbraic, for example, allows for a convenient proof that any subgroup of a free group is again a free group. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician’s knot differs in that the ends are joined together so that maubder cannot be undone.
This class of spaces is broader and has some better categorical properties than simplicial complexesbut still retains a combinatorial nature that allows algebraicc computation often with a much smaller complex. Foundations of Combinatorial Topology.
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 author has given much attention to detail, yet ensures that the reader knows where he is going. Read, highlight, and take notes, across web, tablet, and phone. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Two major ways in which this can be done are through fundamental groupsor more generally homotopy theoryand through homology and cohomology groups.
Selected pages Title Page. In other projects Wikimedia Commons Wikiquote. 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. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.
In general, all constructions of algebraic topology are functorial ; the notions of categoryfunctor and natural transformation originated here. That is, cohomology is defined as the abstract study of cochainscocyclesand coboundaries. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms e. This was extended in the s, when Samuel Eilenberg and Norman Steenrod generalized this approach.
This page was last edited on 11 Octoberat 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.
Algebraic topology – Wikipedia
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. Maunder Courier Corporation- Mathematics – pages 2 Reviews https: 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.
A simplicial complex is a topological space of a certain kind, constructed by “gluing together” pointsline segmentstrianglesand their n -dimensional counterparts see illustration. My library Help Advanced Book Search.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. Maunder has provided many examples and exercises as an aid, and the notes and references at the end of each chapter trace the historical development of the subject and also point the way to mauneer advanced results. Courier Corporation- Mathematics – pages.