greg mcshane

Topology from a differential viewpoint

Course Schedule

Date	Name	Title
January 20, 2026	Greg Mc	introduction
January 27, 2026	Louve + Emanuel	espace tangent, d’alembert gauss
February 3, 2026	Louis et Théo	1 manifolds
February 10, 2026	Emanuel et Maryam	Sard
February 17, 2026		Vacances

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Week 3 files

• exposé)
• long line, second countablity

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Week 2 files

Louve + Emanuel

• exposé
• SL(2,R) as a (sub)manifold
• S0(3) as a (sub)manifold

• D’Alembert Gauss via homotopy

Week 1 files

• Philosophy
• Example: the 2 sphere
• big three theorems
• 1-forms and the example the torus

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Summary

The book’s “highlight” is its ability to prove profound topological results -like the Brouwer Fixed Point Theorem - using almost nothing but the concept of smooth maps and regular values.

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Bibliography the “Trio”

• text in pdf
• Guilleman Spivak
• Spivak Calculus on Manifolds

Supplementary References

• Comment on Hirsch’s proof of Brouwer’s theorem

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Summary of the “Trio”

Book	Role in your Study
Milnor	The Vision: Tells you the deep truths (What).
G&P	The Intuition: Shows you the pictures and examples (How).
Spivak	The Rigors: Proves the underlying calculus (Why).

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Key Summary Table

Concept	Significance
Regular Values	The tool used to “slice” manifolds and see their internal structure.
Homotopy	Proving that “deforming” a map doesn’t change its fundamental properties (like degree).
Orientability	How “left-handed” and “right-handed” systems affect the way we count preimages.
Euler Characteristic	The single number that captures a manifold’s most basic shape (holes, etc.).