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
           Problem: on a locally compact, second countable Hausdorff space: starting from a countable basis of open sets, remove
           those whose closure is not compact. Prove that the remaining ones still form a (countable) basis of the space.


Tu 8/30  Urysohn lemma, Urysohn metrization  (start)

Problem set 1: ([Munkres 1]), p.213: 4, 5; and the abpve problem

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

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 to Tu 12/6(3 lectures): Topologies on spaces of maps, Arzela-Ascoli, Stone-Weierstrass.
Ref: [Munkres 1], no. 46
Notes on Arzela-Ascoli

Notes on Stone_Weierstrasss
(course presentation by Ethan Kessinger, Math467 fall 2019)