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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That is, cohomology is defined as the abstract study of cochainscocyclesand coboundaries. 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.

The translation process is usually carried out by means of the homology or homotopy groups of a topological space.

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. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Cohomology and Duality Theorems. 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.

The author has given much attention to detail, yet ensures that the reader knows where he is going. One of the first mathematicians to work with different types of cohomology was Georges de Rham. 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.

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. This was extended in the s, when Samuel Eilenberg and Norman Steenrod generalized this approach. Maunder Snippet view – Maunder Courier Corporation- Mathematics – pages 2 Reviews https: The translation process is usually carried out by means of the homology or homotopy groups of a topological space.

By using this site, you agree to the Terms of Use and Privacy Policy. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. 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.

Account Options Sign in. Cohomology Operations and Applications in Homotopy Theory. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.

An mauncer name for the subject was combinatorial topologyimplying an emphasis on how a space X was constructed from simpler ones [2] the modern standard tool for such construction is algebraif CW complex.

The presentation of the homotopy theory and the account of duality in homology manifolds Wikimedia Commons has media related to Algebraic topology.

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. K-theory Lie algebroid Lie groupoid Important publications in algebraic topology Serre spectral sequence Sheaf Topological quantum field theory. The first and simplest homotopy group is the fundamental groupwhich records information about loops in a space.

In less abstract language, cochains in the fundamental sense should assign ‘quantities’ to the chains of homology theory.

### Algebraic Topology – C. R. F. Maunder – Google Books

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. Finitely generated abelian groups are completely classified and are particularly easy to work with. Whitehead Gordon Thomas Whyburn.

The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A manifold is a topological space that near each point resembles Euclidean space.

## Algebraic topology

In other projects Wikimedia Commons Wikiquote. Fundamental groups and homology and cohomology groups are not mxunder invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated wlgebraic 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.

This class of spaces is broader and has some better categorical properties than simplicial complexesbut still retains a combinatorial nature that allows for computation often with a much smaller complex. For the topology of pointwise convergence, see Algebraic topology object. Selected pages Title Page.

Simplicial complex and CW complex.