Differential topology Study Guide

📖 Core Concepts Smooth manifold – a space that locally looks like ℝⁿ and has a differentiable‐class structure (no metric needed). Diffeomorphism – smooth bijection with smooth inverse; the notion of “sameness” in differential topology. Smooth vs. geometric properties – smooth properties are preserved by diffeomorphisms (e.g., number of holes); geometric properties depend on a chosen metric (e.g., curvature, volume). Immersion / Submersion – a smooth map whose differential is respectively injective (no folding) or surjective (covers all directions) at every point. Transversality – generic way submanifolds intersect; they meet “as cleanly as possible.” Algebraic invariants – quantities unchanged by diffeomorphisms (de Rham cohomology, intersection form, Euler characteristic). Exotic smooth structures – manifolds that are homeomorphic but not diffeomorphic (e.g., exotic ℝ⁴, Milnor’s exotic spheres). --- 📌 Must Remember Whitney Embedding: Any smooth $n$‑manifold embeds in $\mathbb{R}^{2n}$. Hairy Ball Theorem: No nowhere‑zero continuous tangent vector field exists on even‑dimensional spheres $S^{2k}$. Poincaré–Hopf: $\displaystyle \sum{\text{zeros}} \operatorname{index}(X)=\chi(M)$ (Euler characteristic). Classification by dimension 1‑D: $S^1$, $\mathbb{R}$, $[0,1]$, $[0,1)$. 2‑D closed surfaces: determined by genus, Euler characteristic, orientability. 3‑D: Simply‑connected closed manifolds ≅ $S^3$ (Poincaré conjecture). Smooth 4‑D Poincaré Conjecture: Open – asks if a homotopy 4‑sphere is diffeomorphic to $S^4$. Exotic spheres exist in dimensions $7$ and higher (Milnor). Intersection form: symmetric bilinear form $Q:H2(M;\mathbb{Z})\times H2(M;\mathbb{Z})\to\mathbb{Z}$; crucial for simply‑connected 4‑manifolds. --- 🔄 Key Processes Using Morse Theory to read topology Choose a smooth function $f:M\to\mathbb{R}$. Identify critical points (where $df=0$). The index (number of negative eigenvalues of Hessian) tells how many handles are attached. Attach a $k$‑handle for each critical point of index $k$ → reconstruct $M$ step‑by‑step. Cobordism argument To compare $M0$ and $M1$, build $W^{n+1}$ with $\partial W = M0 \sqcup M1$. If $W$ exists, $M0$ and $M1$ are cobordant → share many smooth invariants (e.g., Stiefel‑Whitney numbers). Computing de Rham cohomology Form the complex $0\to \Omega^0 \xrightarrow{d}\Omega^1 \xrightarrow{d}\cdots\xrightarrow{d}\Omega^n\to0$. $H{\text{dR}}^k(M)=\ker d:\Omega^k\to\Omega^{k+1}\big/ \operatorname{im} d:\Omega^{k-1}\to\Omega^{k}$. Use closed/exact forms, Mayer‑Vietoris, or the Hodge theorem (harmonic forms) for calculations. --- 🔍 Key Comparisons Immersion vs. Submersion Immersion: $dfp$ injective → locally like embedding without self‑intersection. Submersion: $dfp$ surjective → locally looks like a projection onto a lower‑dimensional space. Differential Topology vs. Differential Geometry Topology: cares only about global smooth structure (holes, diffeomorphism class). Geometry: studies metric‑dependent data (curvature, lengths). Exotic sphere vs. Standard sphere Standard: unique smooth structure up to diffeomorphism. Exotic: same underlying topological sphere, but different smooth structure (no diffeomorphism to the standard one). --- ⚠️ Common Misunderstandings “All manifolds of the same dimension are diffeomorphic.” False: e.g., $S^2$ vs. $T^2$ (different genus). “Homeomorphic ⇒ diffeomorphic.” Wrong in dimension ≥ 4 (exotic $\mathbb{R}^4$, exotic spheres). “Whitney embedding dimension $2n$ is minimal.” Not always; many manifolds embed in lower $\mathbb{R}^m$ (e.g., $S^2$ embeds in $\mathbb{R}^3$). --- 🧠 Mental Models / Intuition Handle decomposition: Visualize building a shape by attaching “thickened” disks (handles) of increasing index—Morse critical points are the “blueprints” for where handles go. Transversality = “misses by a hair’s breadth.” Think of two sheets intersecting at a clean line rather than tangentially sticking together. Exotic smoothness: Imagine the same clay sculpture (topology) molded with a different smoothing process; it looks the same to the naked eye (homeomorphism) but the “smoothness” of the surface differs. --- 🚩 Exceptions & Edge Cases Dimension 4: Homeomorphic 4‑manifolds may have infinitely many distinct smooth structures (e.g., $\mathbb{R}^4$). Hairy Ball: Holds only for even spheres; odd spheres $S^{2k+1}$ do admit nowhere‑zero tangent fields. Whitney embedding: While $2n$ always works, some manifolds embed in $\mathbb{R}^{2n-1}$ (e.g., $S^n$ in $\mathbb{R}^{n+1}$). --- 📍 When to Use Which Classifying a closed surface → compute genus and Euler characteristic; use orientability to pick the right model (sphere, torus, higher‑genus). Distinguishing 4‑manifolds → compute intersection form; if forms differ, manifolds are not diffeomorphic. Proving two manifolds are the same up to diffeomorphism → try to build a cobordism with trivial cobordism invariants or exhibit a Morse function with identical handle decompositions. Showing a map cannot exist → apply hairy ball (no non‑vanishing vector field) or degree/Hopf theorem (degree must be integer). --- 👀 Patterns to Recognize Euler characteristic = sum of indices → whenever a problem mentions vector‑field singularities, think Poincaré–Hopf. Critical point index sequence → look for “+1, -1, …” patterns that hint at handle cancellations (Morse cancellations). Fundamental group complexity → in dimension 4+, any finitely presented group can appear; expect “classification impossible” statements when the problem mentions arbitrary $\pi1$. Even‑dimensional sphere statements → automatically trigger hairy‑ball or degree‑parity considerations. --- 🗂️ Exam Traps “All 4‑manifolds are classified by their fundamental group.” Distractor: ignores exotic smooth structures and intersection forms. “Whitney embedding theorem guarantees an embedding in $\mathbb{R}^{n+1}$.” Tempting but false; the theorem guarantees $\mathbb{R}^{2n}$, not $n+1$. “If two manifolds are homeomorphic they have the same smooth invariants.” Wrong in dimensions ≥ 4 (exotic examples). “Hairy ball theorem works on $S^3$.” Misleading; $S^3$ (odd) does admit a nowhere‑zero vector field. “Morse theory only counts critical points, not their indices.” Incomplete; the index determines the type of handle attached, crucial for topology. ---