MATH 561-TOPOLOGY I-FALL 2022-A. FREIRE
Syllabus
References and Outline
Th 8/25 topological manifolds (def.)
paracompactness: locally cpt Hausdorff 2nd ctble spaces are
sigma-compact and paracompact
Tu 8/30 Urysohn lemma, Urysohn metrization (start)
Problem Set 1
Th 9/1 (Zoom) Urysohn metrization (end), locally compact Hausdorff implies regular
regular 2nd countable implies normal
Notes (scanned pages)
Review in [Munkres Topology]: sections 31 to 35, section 41
Also in Hocking-Young, Topology: 2-11, Paracompact spaces
Tu 9/6 topological manifolds with boundary/ Invariance of domain theorem (statement)
inverse function theorem (statement)
C^r immersion and diffeomorphism (in R^n)/local form of immersions
C^r m-dimensional surfaces in R^n (def)
Reference: Munkres Elementary Differential Topology sections 1.1, 1.2
esp. Problem 1.10 (hw 2), thm. 1.11. Cor. 1.12 (read)
Th 9/8 local form of submersions, implicit fn. theorem
Tu 9/13 C^r atlas, C^r structure, C^r map between diff'ble manifolds
embeddings, submanifolds
Ref: Munkres EDT Ch. 1, Guillemin-Pollack ch.1
Th 9/15 Tangent bundle, differential of a C^r map between manifolds
Ref: Hirsch, ch. 1
Problem Set 2 (due 9/20)
Tu 9/20 Local forms of immersions and of submersions for maps of manifolds/submanifolds as
imagess of embeddings/ regular values, preimages are submanifolds
Transversality (def)
Th 9/22 preimage of submanifold under transversal map. Def. of tangent vector via curves
Examples of manifolds: surfaces in R^N
Problem set 3
Solutions
Tu 9/27: regular level sets, matrix groups. Differentiable partition of unity.
Th 9/29: applications: closed sets are level sets,
Differentiable Tietze extension
Diffferentiable manifolds: summary
Tu 10/4 embeddings compact manifolds in euclidean space/Riemannian metrics (existence)
Th 10/6 FALL BREAK
Tu 10/11 norms on spaces of linear maps/metric structure of Riem. manifolds
Th 10/13 Sets of measure zero and Sard's theorem (start)
Notes on Sard's theorem
HW 4: Exercise on p. 2 and Problem 1 on p. 5 of this handout (Notes on Sard's theorem.)
Tu 10/18 Sard's theorem (proof)
Th 10/20 Sard's therem (conclusion, application)
Tu 10/25 Applications of Sard
Th 10/27 Midterm
solutions
Tu 11/1 Midterm (cont)/ Whitney's embedding theorem/ Covering dimension (intro)
Th 11/3, Tu 11/8 Covering dimension: upper bounds and lower bounds
Ref: [Munkres 1], no. 50 and p.352
Th 11/10 Covering dimension: embedding theorems
Ref: [Munkres 1] no. 50, [Munkres 2], p.20--24.
Notes on covering dimension and embedding theorems
(In preparation: already includes 10 problems=HW 5, due 11/22.)
Tu 11/15 Baire property
Ref: [Munkres 1], no. 48
Th 11/17 Baire spaces, applications (cont.)
Notes on Baire spaces
Tu 11/22 Baire spaces: examples/ Arzela-Ascoli theorems: start
HW 6, due Th 12/1: Exercises 1 and 2 in Baire spaces notes and Exercises1-5 and 7 in Arzela-Ascoli notes
(Not the "problems" at the end.)
Th 11/24 Thanksgiving
Tu 11/29: Topologies on spaces of maps
Ref: [Munkres 1], no. 46
Notes on Arzela-Ascoli
Th 12/1: Arzela-Ascoli: proof, examples
Compactness, countability, function spaces: examples
Tu 12/6 Spaces of maps: Stone-Weierstrass, examples (last day)
Notes on Stone_Weierstrass
(course presentation by Ethan Kessinger, Math467 fall 2019)
Notes on the theorems of Weierstrass and Stone
Problems on function spaces and Stone-Weierstrass
FINAL: Friday 12/9, 11:30--12:45
Comprehensive, closed book/closed notes
Final exam
Solutions