By Mahima Ranjan Adhikari

ISBN-10: 8132228413

ISBN-13: 9788132228417

ISBN-10: 813222843X

ISBN-13: 9788132228431

This booklet presents an available advent to algebraic topology, a ﬁeld on the intersection of topology, geometry and algebra, including its purposes. in addition, it covers numerous similar themes which are actually vital within the total scheme of algebraic topology. Comprising eighteen chapters and appendices, the booklet integrates quite a few recommendations of algebraic topology, supported through examples, workouts, functions and ancient notes. essentially meant as a textbook, the e-book oﬀers a useful source for undergraduate, postgraduate and complicated arithmetic scholars alike.

Focusing extra at the geometric than on algebraic facets of the topic, in addition to its traditional improvement, the e-book conveys the fundamental language of contemporary algebraic topology by means of exploring homotopy, homology and cohomology theories, and examines quite a few areas: spheres, projective areas, classical teams and their quotient areas, functionality areas, polyhedra, topological teams, Lie teams and mobile complexes, and so on. The publication experiences various maps, that are non-stop features among areas. It additionally unearths the significance of algebraic topology in modern arithmetic, theoretical physics, machine technological know-how, chemistry, economics, and the organic and scientific sciences, and encourages scholars to have interaction in additional study.

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**Extra info for Basic Algebraic Topology and its Applications**

**Example text**

27 A nonempty set X is said to have a metric or a distance function f : X × X → R if for every pair of elements x, y in X (i) (ii) (iii) d(x, y) ≥ 0, equality holds iff x = y; d(x, y) = d(y, x); d(x, y) + d(y, z) ≥ d(x, z) for all z ∈ X . d(x, y) is called the distance between x and y and the pair (X, d) is called a metric space or X is said to be metrized by d. A metric space X can be made into a topological space in a natural way by defining as open sets all unions of the open balls β (x) = {y ∈ X : d(x, y) < }, for x ∈ X and > 0.

Sometimes we write the topological space as (X, τ ) to avoid any confusion regarding the topology τ . 2 (i) (Trivial or indiscrete topology) The two subsets ∅ and the whole set X constitute a topology of X , called trivial topology. (ii) (Discrete topology) The family of all subsets of X constitutes a topology of X , called the discrete topology of X . This topology is different from trivial topology, if X has more than one element. 3 Let (X, τ ) be a given topological space. A family of open sets is said to form an open base (basis) of the topology τ if every open set (relative to the topology τ ) is expressible as the union of some sets belonging to .

2 Let M and N be two R-modules. Given an R-module T and an R-bilinear map f : M × N → T, there exists a unique R-homomorphism g : M × N → T such that the diagram in Fig. , g ◦ ψ = f. 3 Let M, N and T be R-modules. Then there exist isomorphisms such that (a) (b) (c) (d) M⊗N ∼ = N ⊗ M; (M ⊗ N ) ⊗ T ∼ = M ⊗ (N ⊗ T ); R⊗M ∼ = M; (M ⊕ N ) ⊗ T ∼ = M ⊗ T ⊕ N ⊗ T. 4 (Structure theorem for finitely generated modules over a principal ideal domain) Let R be a principal ideal domain and M be a finitely generated R-module.

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