Description:The goal of this course is to show how Combinatorial and topological ideas met and grew together into an important branch of mathematics. The course also conveys the fun and adventure that can be part of a mathematical investigation. Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations, computer science, graph theory. Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics â analysis, geometry, and algebra â working together amicably on important problems."
Combinatorial study of convex polytopes : (14/15 lectures)
Basics of Convex Polytopes, Combinatorial type of a polytope, Classification of convex polytopes up to homeomorphism, Polyhedral complex, The classical concept of Euler characteristic and a geometric proof of Eulerâs formula for convex polytopes, Classification of Regular 3-Polytopes upto combinatorial equivalence.
Triangulations of manifolds :(14/15 lectures)
Principal Fibrations, a brief recollection on Rn bundles, spherical fibrations and their classifying spaces, PL triangulation on Manifolds and Fundamental Theorem of PL triangulation, Statement of the Product Structure Theorem, Existence and uniqueness of triangulations and PL triangulations of topological manifolds.
Smoothings of piecewise linear manifolds : (14/15 lectures)
Piecewise differentiable, Whitehead triangulation of smooth manifolds, Statement of the Fundamental Theorem of Smoothing and its applications, existence and uniqueness of smooth structuresÂ of the underlying piecewise linear structure on topological manifolds.
1. Yuli Rudyak, Piecewise Linear Structures on Topological Manifolds, World Scientific, 2016.
2. Morris W. Hirsch, Barry Mazur, Smoothings of Piecewise Linear Manifolds, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 80
1. R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton Univ. Press, Princeton, N.J., 1977.
2. Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag,New York, 1977, Graduate Texts in Mathematics, Vol. 47. MR 0488059 (58 #7631)
3. C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. MR 0350744 (50 #3236)
Prerequisite:Topology (MA 5380)