Prerequisites:

This course is aimed at students who are in their second year of Master's program and who have done courses on linear algebra, real analysis, complex analysis, abstract algebra up to and including Sylow theory. Presumably a student of this course would be concurrently doing a course on multi-variable calculus leading to differential forms and Stokes' theorem. We shall freely use ideas from linear algebra and some elementary complex analysis such as properties of the exponential map and Möbius maps as a source of examples. Notions such as orthogonal matrices and the spectral theorem are ubiquitous in all of mathematics and this course is no exception. A student who is uneasy with these notions is advised to brush up these concepts before embarking upon a study of algebraic topology. We shall not use Jordan canonical forms in this course. In algebra we expect the students to be familiar with group actions, isomorphism theorems and notions such as inner automorphisms, center of a group and commutator subgroup.

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Chapter - I (Introduction)

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General topology, a language for communicating ideas of continuous geometry, provides us useful tools for studying local properties of space. Notions of compactness and connectedness though important, are quite inadequate for obtaining a reasonable understanding of the global geometry of space. For example, the sphere and the torus are not homeomorphic although they are both compact, path-connected, locally connected metric spaces.

Algebraic topology is a powerful tool in global analysis - the study involving the global geometry of space. It is difficult to define precisely at this point what global analysis is. Perhaps the few examples discussed in the following paragraphs may help in understanding this. The most basic example comes from advanced calculus in connection with Stokes' theorem where a student encounters the notion of orientability of a two dimensional surface in $\mathbb{R}^3$. A sphere is easily seen to be orientable inasmuch as it has ``two sides''. Small pieces of a surface obviously have ``two sides'' but the Möbius band ``has only one side''. How would one formulate a precise notion of an orientable surface and prove that the Möbius band is non-orientable? Is non-orientability an intrinsic property of the surface or does it depend on the way the surface is presented in $\mathbb{R}^3$?

Frequently one also sees an interplay between local and global analysis. The powerful algebraic techniques that we shall develop streamlines the process of piecing together local information (which is often trivial) to provide non-trivial information on the global geometry of space. A good example illustrating this ``piecing of local information'' is provided by the proof of the famous theorem in complex analysis asserting the impossibility of a continuous branch of the argument function on the punctured plane $\mathbb{C} - \{0\}$. Although formal use of algebraic topology can be avoided for this specific case, it is less obvious that the function $\sqrt{1-z^2}$ is holomorphic on $\mathbb{C} - [-1, 1]$. Analogous problems in several dimensions would be practically intractable without the use of algebraic topology or some other equally powerful tool in global analysis.

The first example in our list is provided by the famous Jordan curve theorem which also arose in connections with complex analysis.