By Mahima Ranjan Adhikari

This e-book offers an available advent to algebraic topology, a ﬁeld on the intersection of topology, geometry and algebra, including its functions. in addition, it covers a number of comparable issues which are actually vital within the total scheme of algebraic topology. Comprising eighteen chapters and appendices, the ebook integrates a variety of techniques of algebraic topology, supported by way of examples, routines, purposes and ancient notes. basically 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 normal improvement, the ebook conveys the elemental language of recent algebraic topology by way 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 cellphone complexes, and so forth. The booklet reviews numerous maps, that are non-stop services among areas. It additionally unearths the significance of algebraic topology in modern arithmetic, theoretical physics, laptop technology, chemistry, economics, and the organic and clinical sciences, and encourages scholars to have interaction in extra study.

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In particular, if F is finitely generated, then the number of elements in a basis of F is the rank of F. 26 (i) Z is a free abelian group of rank 1, finitely generated by 1 (or −1). (ii) Z ⊕ Z is a free abelian group of rank 2. (iii) Z2 is finitely generated by 1 but not free, because 1 + 1 = 0 shows that 1 is not linearly independent. 27 An abelian group G has rank r (possibly infinite) if there exists a free abelian subgroup F of G such that (a) rank of F is r ; and (b) the quotient group G/F is of finite order.

5 (Circle group) The set S 1 = {z ∈ C : |z| = 1} forms a group under usual multiplication of complex numbers, called the circle group in C. An arbitrary subset of a group forming a group, called a subgroup contained in a larger group, sometimes creates interest and plays an important role in group theory and algebraic topology. 6 A nonempty subset H of a group G is a subgroup of G if and only if ab−1 ∈ H for all a, b ∈ H . 7 (i) The additive group of integers is a subgroup of the additive group of real numbers.

12 Let G and H be groups. Then a map f : G → H is said to be a homomorphism if f (x y) = f (x) f (y) for all x, y in G. 6 1 Prerequisite Concepts and Notations Particular homomorphisms carry special names having interesting properties. 13 Let f : G → H be a homomorphism of groups. Then (a) f is said to be (i) (ii) (iii) (iv) (v) an epimorphism if f is surjective; a monomorphism if f is injective; an isomorphism if f is bijective; an endomorphism if G = H ; an automorphism if G = H and f is an isomorphism.