D Chatterjee - Collected Quotes

"A main problem of topology is the classification of topological spaces: Given two spaces X and Y, are they homeomorphic? This is usually a very difficult question to answer without employing some fairly sophisticated machinery, and the idea of algebric topology is that in which one should transform such topological problems into algebraic problems in order to have a better chance of solution. It turns out, however, that the algebric techniques are usually not delicate enough to classify spaces up to homeomorphism. Hence we shall introduce the notion of homotopy, in order to achieve a somewhat coarser classification." (D Chatterjee, "Topology: General & Algebraic", 2003)

"One of the greatest successes of the combinatorial topology has been the extension of Homology Theory to general topological spaces. In what discussed above it is clear that Homology groups can be defined for a special kind of space, namely, compact polyhedron and the complexes obtained there were finite althrough. Singular Homology theory extends the notion of Homology groups for general topological spaces by associating with each space a chain complex. A continuous map induces homology homomorphisms in an obvious way and as a consequence it follows that homotopic maps induce the same homomorphisms. There is a natural homomorphism also from homology groups to singular homology groups. In the following lines we give a sketch of notions relevant in singular homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"Some areas of human knowledge ever since its origin had shaken our understanding of the universe from time to time. While this is more true about physics, it is true about mathematics as well. The birth of topology as analysis situs meaning rubbersheet geometry had a similar impact on our traditional knowledge of analysis. Indeed, topology had enough energy and vigour to give birth to a new culture of mathematical approach. Algebraic topology added a new dimension to that. Because quantum physicists and applied mathematicians had noted wonderful interpretations of many physical phenomena through algebraic topology, they took immense interest in the study of topology in the twentieth century." (D Chatterjee, "Topology: General & Algebraic", 2003)

"The concept of path-connectedness, in which it is required that it be possible to reach any point in the space from any other point along a continuous path is necessary for the notion of fundamental group. This approach is especially useful in studying connectivity properties from an algebraic point of view, e.g., via homotopy theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"The exquisite world of algebraic topology came into existence out of our attempts to solve topological problems by the use of algebraic tools and this revealed to us the nice interplay between algebra and topology which causes each to reinforce interpretations of the other there by breaking down the often artificial subdivision of mathematics into different branches and emphasizing the essential unity of all mathematics. The homology theory is the main branch of algebraic topology and plays the main role in the classification problems of topological spaces. There are various approaches to the study of this theory such as geometric approach, abstract approach and axiomatic approach. Since geometric approach appeals easily to our intuition, we shall start with geometric approach, the relevant development being called simplicial homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"The study of fibre bundles makes an important component of algebraic topology for many reasons. On one hand it helps classification of the topological spaces and on the other gives remarkable results in physics, differential geometry and many other areas so far as applications are concerned." (D Chatterjee, "Topology: General & Algebraic", 2003)