Instructor: Prof. Bob Haslhofer ( roberth(at)math(dot)toronto(dot)edu )
TA: Saeyon Mylvaganam ( saeyon.mylvaganam(at)mail(dot)utoronto(dot)ca )
Website: http://www.math.toronto.edu/roberth/difftopo.html
Lectures: Tuesday 10am--12noon and Friday 11am--12noon in BA6183
Office Hours: Tuesday 2pm--3pm in lounge (Saeyon) and Friday 9am--10am in BA6208 (Bob)
Grading Scheme: Assignments 20%, Midterm 30%, Final exam 50%
There will be 5 homework assignments. Your lowest homework score will be dropped.Assignments: HW1, HW2, HW3, HW4, HW5
Midterm Exam: Oct 24 from 10am--12noon in BA6183
Final Exam: Dec 11 from 10am--1pm in BA6183
Main References:
V. Guillemin, A. Pollack: Differential topology, AMS, 2010
J. Lee: Introduction to smooth manifolds, Springer, 2013
Further References:
J. Milnor: Topology from the differentiable viewpoint, Princeton University Press, 1997 (for week 6)
R. Bott, L. Tu: Differential forms in algebraic topology, Springer, 1982 (for week 8--11)
J.-P. Demailly: Complex Analytic and Differential Geometry, 2012 (for week 11)
J. Milnor: Morse theory, Princeton University Press, 1963 (for week 12)
M. Gualtieri: Lecture notes on differential topology, 2018 (for most of the semester)
A. Kupers: Lecture notes on differential topology, 2020 (for most of the semester)
Topics to be covered:
topological and smooth manifolds, manifolds with boundary, smooth maps and their differential, tangent vectors and tangent bundle, vector fields and their flows, embeddings, immersions, submersions, partitions of unity, Whitney's embedding theorem, transversality, stability, Sard's theorem and applications, degree theory and applications, orientations, vector bundles, differential forms, integration, exterior derivative, Cartan's magic formula, Stokes theorem, Poincare lemma, de Rham cohomology, Mayer-Vietoris theorem, Hodge decomposition, Poincare duality, Morse theory and applications
Weekly schedule:
Week 1 (notes1)
Sep 12: topological manifolds, examples
Sep 15: smooth manifolds, manifolds with boundary
Week 2 (notes2)
Sep 19: smooth maps, tangent vectors, differential
Sep 22: computations in local coordinates
Week 3 (notes3)
Sep 26: tangent bundle, vector fields
Sep 29: integral curves and flows
Week 4 (notes4)
Oct 3: embeddings, immersions, submersions
Oct 6: partitions of unity, Whitney's embedding theorem
Week 5 (notes5)
Oct 10: transversality, stability, Sard's theorem
Oct 13: applications of Sard's theorem
Week 6 (notes6)
Oct 17: mod 2 degree and applications
Oct 20: orientations, integer degree
Week 7 (notes7)
Oct 24: midterm
Oct 27: vector bundles
Week 8 (notes8)
Oct 31: differential forms, review of multilinear algebra
Nov 3: integration
Week 9 (notes9)
Nov 14: exterior derivative, Stokes theorem
Nov 17: applications of Stokes theorem
Week 10 (notes10)
Nov 21: de Rham cohomology, Poincare lemma
Nov 24: algebra of differential complexes
Week 11 (notes11)
Nov 28: Mayer-Vietoris, elliptic operators on vector bundles
Dec 1: Hodge decomposition, Poincare duality
Week 12 (notes12)
Dec 5: Morse theory and applications