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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Examples include the planethe sphereand the toruswhich 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. Views Read Edit View history.
An older name for the subject was combinatorial topologyimplying an emphasis on how a space X was constructed from simpler ones  the modern standard tool for such construction is the CW complex. Finitely generated abelian groups are completely classified and are particularly easy to work with. 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 more advanced results.
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A CW complex is a type of topological space introduced by J. The presentation of the homotopy theory and the account of duality in homology manifolds make tpology text ideal for a course on either homotopy or homology theory. In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.
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 more advanced results.
The basic goal is maundrr find algebraic invariants that classify topological spaces up to homeomorphismthough usually most classify up to homotopy equivalence.
Retrieved from ” 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. Simplicial complex and CW complex.
A manifold is a topological space that near each point resembles Euclidean space. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
Msunder Groups and CWComplexes. One of the first mathematicians to work with topklogy types of cohomology was Georges de Rham. Cohomology arises from the algebraic dualization of the construction of homology.
Algebraic Topology – C. R. F. Maunder – Google Books
Homotopy and Simplicial Complexes. The idea of algebraic topology is alebraic translate problems in topology into problems in algebra with the hope that they have a better chance of solution. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve maunxer problems is sometimes also possible.
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 it cannot be undone. 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 agebraic Rham cohomology.
Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism or more general homotopy of spaces. A simplicial complex is a topological space of a certain kind, constructed by “gluing together” pointsline segmentstrianglesand their n -dimensional counterparts see illustration. 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.
The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. Whitehead Gordon Thomas Whyburn.
Foundations of Combinatorial Topology. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. 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 maujder and duality theorems for manifolds.