MATH 562- TOPOLOGY II-SPRING 2026- A. FREIRE

Syllabus, references and topics

Tu 1/20  Topologies on spaces of maps (ref: [Munkres, sect. 46]

Compactness, countability, function spaces
(review Ex. 1,2, 5, 6, 7, 8, 9 of handout)

Th 1/22  Function spaces: examples
Ex. 3, 4 and 10, 11 of the above handout

Tu 1/27 [snow day] u.o.c topology: metrizability, separability
(Ex 12,13,14 of handout)

Th 1/29: Arzela-Ascoli and applications (start)
Arzela-Ascoli notes

Problem Set 1

Tu Feb 3: Arzela-Ascoli: proof, applications (end)

Th Feb 5: Stone-Weierstrass Theorem (for algebras, lattices)

Stone-Weierstrass notes

Tu Feb 10: Weierstrass thm/ Baire spaces/ meager and generic sets, Baire's theorem
Notes on Baire Spaces
(Includes exercises)

Th Feb 12: discontinuity sets of pointwise limits are meager/nowhere-diff'ble fns generic (outline)/
diff'ble manifolds (start)

Problem Set 2

Tu Feb 17 Manifolds arising as quotients: Hausdorff, second countability. Examples:
Sn (stereographic coords), RPn (n+1 charts). Tangent space at a point.

Th Feb 19 Differential of a smooth map. Tangent bundle, vector bundles. Grassmannians
as quotient manifolds (of Stiefel manifolds)

Tu Feb 24 Smooth atlas for Grassmannians/Submanifolds/Local form of immersions.

Problems [GP-p. 18:]  6(a)(b), 8 (show that the map given is a proper map R--> R^2, and an injective immersion)

Notes on manifolds
Notes on manifolds: content

(Updated 6/16/26; currently 84 pages; includes exercises.)

Th Feb 26: Injective immersions vs. embeddings (examples); proper maps/ local form of submersions/
preimages of regular values are submanifolds

Problems [GP--p. 25:] 1, 2, 6, 8, 11, 12

Tu Mar 3 Orthogonal groups (example)/Transversality/connection with regular values

Problems [GP--p. 32]: 4, 5, 7, 8, 9, 10

Th Mar 5 Partitions of unity and applications

Tu Mar 10, Th Mar 12: SPRING BREAK

Tu Mar 17: Riemannian metrics, C^1 topology, openness of immersions.

Th Mar 19: Whitney C^1 topology/ Stability of embeddings, regular values and transversality

Problems [GP--p.38: 9, 11--for 11, use the results in the notes]

Tu Mar 24; nullsets on manifolds/invariance under C^1 maps/
application: maps to the sphere from a lower dimensional manifold./ Sard's theorem (statement)/ Whitney embedding (statement)

Notes on Sard's theorem

Th Mar 26: Whitney immersion and embedding via Sard's theorem/existence of proper embeddings to R^2m+1
Problems [GP; p.45: 3,5,7/ p.55: 8, 10, 15]

Tu Mar 31: Examples: Sard's theorem, Whitney embedding/ genericity (or not) of injective immersions or embeddings/manifolds with boundary

Th Apr 2 SPRING RECESS (no class)
Two reading suggestions:

John C. Oxtoby, Measure and Category (Springer GTM no.2)
Keith Kendig, Never a Dull Moment (MAA Press) (mathematical and personal bio of Hassler Whitney)

Tu Apr 7: manifolds with boundary/[G-P 2.1] classif. of 1-manifolds, Brouwer fixed-pt thm (smooth maps of the disk) [G-P 2.2]
(Lecture 11 of the manifolds notes deals with tubular neighborhoods.)

Th Apr 9: Brouwer f.p. for continuous maps/tubular neighborhoods/families transversality and homotopy transversality theorems
[ref: G-P, ch.2, section 3]
Problems: [G-P, p. 62: 5, 10, 11; p. 66: 2,3,7; p.74: 5, 6, 7, 20]

Tu Apr 14: smoothing of continuous maps/mod 2 intersection and mod 2 degree [G-P 2.4]
Problems: [G-P, p. 82]: 1, 2, 4, 5, 6, 9, 11 & 12

Th Apr 16: Applications: Winding numbers, Borsuk-Ulam theorem [G-P 2.5, 2.6]
Problems:[G-P, p.93]: 1, 2, 3

Tu Apr 21: Orientation (start): boundary orientation, induced orientation;
Problems: [G-P, p.103]: 3, 18, 24, 27

Th Apr 23: Applications of Borsuk-Ulam (examples)/ Orientation (examples)/Local Brouwer degree

Tu Apr 28 Brouwer degree: definition, applications: antipodal map, maps to the sphere and homotopy/vector fields on the sphere
Problems: [G-P], p. 116: 1, 2, 3, 8, 9, 11

Th Apr 30: Hopf's degree theorem on maps to the sphere/ Oriented double cover
Suggested problems from J. Milnor's Topology from the Differentiable viewpoint (p.52-54):
3, 4, 5, 6, 7, 13, 14, 15

Tu May 5 (last day): Universal cover (existence/uniqueness)/properly discontinuous group actions

FINAL EXAM: take-home; will be sent by email on May 8, due May 14 by 12 noon (handwritten solutions, placed in my dept mailbox.)
Final Exam

PRELIM REVIEW LINKS: (mainly lists of review problems from previous times I taught M561-562)

https://web.math.utk.edu/~afreire/teaching/TopReviewSu23/TopReviewSu23_index.html

https://web.math.utk.edu/~afreire/teaching/PrelimReviewSets2021/Topology_Prelim_Review_2021.html