| 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 | |
| February 24, 2026 | Louve Grosjean–Ducateau | Brouwer |
| March 3, 2026 | Maryam | Brouwer |
| March 10, 2026 | PARTIELS | |
| March 17, 2026 | exercices | |
| March 24, 2026 | Théo | Varieté orientable |
| March 31, 2026 | Louis | Champs de vecteurs |
| April 13, 2026 | Emmanuel |
| Exercise # | Topic | Key Hint / Strategy | |
|---|---|---|---|
| Theo | 2 | Complex Polynomials | Use the Fundamental Theorem of Algebra. For a regular value $w$, $P(z) = w$ has $n$ roots. Since $P$ is holomorphic, the Jacobian determinant of the module of $P’(z)^2$ is always $\ge 0$, so each preimage counts as $+1$ toward the degree. |
| Theo | 6 | Brouwer Fixed Point | Use proof by contradiction. If $f(x) \neq x$ for all $x$, then $f$ is homotopic to the antipodal map $a(x) = -x$. Since $\text{deg}(a) = (-1)^{n+1}$, any map with a different degree must have a fixed point. |
| Maryam | 3 | Non-Antipodal Maps | The condition $|f(x) - g(x)| < 2$ means $f(x)$ and $g(x)$ are never opposite. Define a straight-line homotopy $H(t) = (1-t)f(x) + tg(x)$ and normalize it: $H/|H|$ to keep it on the sphere. |
| Louve | 4 | Smooth Approximation | Extend $f$ to a neighborhood and use the Stone-Weierstrass Theorem to find a smooth map $g$ such that $|f - g| < \epsilon$. If $\epsilon$ is small, $g(x)/|g(x)|$ is a smooth map to $S^p$ homotopic to $f$. |
| Louve | 9 | Graphs of Maps | Consider the map $\psi: M \to M \times N$ defined by $\psi(x) = (x, f(x))$. Show this is an embedding. The tangent space $T\Gamma$ is the image of the derivative $d\psi_x(v) = (v, df_x(v))$, which is exactly the graph of $df_x$. |
| Emanuel | 7 | Odd Degree Maps | If $f(x)$ never equals $-f(-x)$, then $f$ is homotopic to an even map. Even maps $S^n \to S^n$ have even degree (specifically $0$ for even $n$), which contradicts the “odd degree” premise. |
| Louis | 5 | Dimension Mismatch | By Sard’s Theorem, if $m < p$, the image $f(M)$ has measure zero in $S^p$. Therefore, $f$ is not surjective. A non-surjective map into $S^p$ maps into a contractible punctured sphere and is thus homotopic to a constant. |
–
Louve + Emanuel
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.
| 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). |
| 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.). |
louis.bissay@ens-lyon.fr, louve.grosjean–ducateau@ens-lyon.fr, maryam.kouhkan@ens-lyon.fr, theo.feijoo@ens-lyon.fr