Geometric Topology and Further Applications of Algebraic Topology

Geometric topology primarily studies manifolds and their embeddings in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, which makes a study of mathematical knots. This chapter gives a brief study of geometric topology by communicating the concepts of knots and knot groups. It also gives further applications of topological concepts and results discussed in earlier chapters with a view to understand the beauty, power and scope of the subject topology. Moreover, it provides alternative proofs of some results proved in the previous chapters such as Brouwer–Poincaré theorem, Van Kampen theorem, Borsuk–Ulam theorem for any finite dimension. It proves Ham Sandwich theorem and Lusternik–Schnirelmann theorem.