Date | Material covered |
---|---|
4/4 | Introduction |
4/7 | Manifolds with boundary. Duality for compact manifolds with boundary. If M is a compact n-manifold and A,B ⊆∂M are (n-1)-manifolds, then we have an isomorphism H^{l}(M,A) → H_{n-l}(M,B). |
4/9 | Alexander duality: For K⊆ R^{n} compact and locally contractible, H^{l}(K) = H_{n-l}(R^{n} - K). More generally, for M orientable n-manifold (without boundary) and K⊆ M compact and locally contractible, H^{l}(K) = H_{n-l}(M - K). Applications: Generalized Jordan Curve theorem: If M⊆ R^{n} is a closed, connected (n-1)-manifold, then M is oriented and R^{n} has two path components. |
4/11 | Existence of collar neighborhoods. More applications of duality: Borsuk-Ulam (a few words), a construction of the Poincaré sphere. Introduction to intersection theory. |
4/14 | Introduction to smooth manifolds (M&T Ch. 8). Basic definitions: charts, atlasses, smooth structures, differentiability of functions f: M → N, diffeomorphisms, tangent space T_{x}M at a point, the tangent space TM (and the fact that it is again a smooth manifold), the differential Df: TM → TN of a smooth function f: M → N. |
4/16 | The alternating algebra (M&T Ch. 2). |
4/18 | Differential forms, exterior derivative. de Rham cohomology (M&T Ch 3,9) |
4/21 | Exterior derivative, de Rham cohomology. |
4/23 | Poincaré's lemma. Mayer-Vietoris for de Rham cohomology. Partition of unity. (M&T Ch. 5). |
4/25 | Smooth manifolds with boundary. Integration of a compactly supported n-form over an oriented n-manifold. Stokes' theorem. (M&T Ch. 10). |
4/28 | de Rham's theorem. |
4/30 | Induction on open sets (M&T 13.9).. Sard's theorem (M&T Ch. 11.6). Whitney's embedding theorem: We proved that any compact n-manifold can be embedded in R^{2n+1}, first following (M&T 8.11) and then applying Sard's theorem. |
5/2 | Embedding theorem for non-compact manifolds: M^{n} embeds as a submanifold, and closed subset, of R^{2n+1}. Normal bundle NM of a submanifold M ⊆ R^{k}. Tubular neighborhood theorem. Compact manifolds have finite dimensional homology and cohomology. |
5/5 | NM is a submanifold of M x R^{k} if M ⊆ R^{k}. Tubular neighborhood theorem for one manifold inside another. Application: Smooth approximation theorem. |
5/7 | Fiber bundles and vector bundles. (M&T Ch. 15). |
5/9 | Started proof of Thom's transversality theorem. |
5/12 | Introduction to Thom isomorphism by Daniel Mathews. |
5/14 | Finished proof of Thom transversality. Brief introduction to cobordism theory. |
5/16 | Thom isomorphism, given a Thom class, i.e. a class u in H^{n}(E,E^{'}). Here, E → B is a D^{n}-bundle. We gave a proof assuming B is a smooth manifold (we used induction on open sets), but the theorem is true for all topological spaces B (Hatcher has a proof of the more general theorem. It uses more homotopy theory than we currently know). |
5/19 | Existence of Thom classes. We basically followed Hatcher's construction of fundamental classes: For a given ring R and an n-disk bundle E → B there is a covering space E_{R} → B whose fiber over b is H^{n}(E_{b},E'_{b}). There is a natural map H^{n}(E,E') → Γ(E_{R} &rarr B). In the exact same was as the construction of fundamental classes, this map is proved to be an isomorphism. |
5/21 | The most interesting case of orientability is R=Z: In the homework you proved that a disk bundle E → B is R-orientable if and only if it is Z-orientable for char(R) &ne 2, and always orientable if char(R)=2. Multiple equivalent definitions of (Z)-orientability: (1) The double cover E_{Z}^{x} → B has a section, (2) The "first Stiefel-Whitney class" w_1(E) ∈ H^{1}(B;Z/2) vanishes. Inner products on vector bundles (existence and uniqueness). Using inner products, we can take "the" disk bundle of a vector bundle (well defined up to unique isomorphism), the sphere bundle, the Thom space, ... etc. The euler class of an oriented vector bundle (defined as pullback of Thom class along the zero section). Example: The canonical bundle over CP^{n}. Its Thom space is homeomorphic to CP^{n+1}. It is orientable (because H^{1}(CP^{n};Z/2) = 0) and its euler class is a generator of H^{2}(CP^{n}). |
5/23 | The intersection product H_{k}(M) x H_{l}(M) → H_{k+l-n}(M) on an oriented n-manifold M. Reinterpretations of naturality properties of cap products: H_{*}(X) is a module over H^{*}(X). f_{*}: H_{*}(X) → H_{*}(Y) is H^{*}(Y)-linear when f: X → Y is continuous. Cap product H_{*}(X) x H^{*}(X) → H_{*}(X) is H^{*}(X)-bilinear. Poincare isomorphism H^{*}(M) → H_{*}(M) is an isomorphism of H^{*}(M)-modules. |
5/28 | Reinterpretation of Thom isomorphism using the Poincare duality isomorphism for the base and the total space of a disk bundle (provided base and total spaces are oriented manifolds). Geometric interpretation of intersection product: "[P_{1}]•[P_{2}] = [P]" if P_{1} and P_{2} are smooth, oriented, closed, transverse submanifolds of M and P = P_{1} ∩ P_{2}. |
5/30 | An easy corollary of the geometric interpretation of the intersection product: If [P_{1}] • [P_{2}] ≠ 0, then the submanifolds are not disjoint, and cannot be made disjoint by isotopy. Geometric interpretation of the Euler class of a vector bundle V → B: It is the Poincare dual of [s^{-1}(Z)], where Z⊆ V is the (image of the) zero section and s: B → V is a section transverse to Z. |
6/2 | Relation between Euler class and Euler characteristic: e(TB)[B] = χ(B) for oriented closed manifolds B. Main tool was a formula for the Thom class of the normal bundle of the diagonal. |
6/4 | Last class: Application of intersection theory to "enumerative geometry". Given 4 lines (one-dimensional affine subspaces) in 3-space in general position, how many lines intersect all four of them. Modulo a lot of details, I used intersection theory to translate this into a question in the cohomology ring of a compact 8-manifold: The Grassmann manifold of 2-dimensional sub vector spaces of 4-space. It turns out H^{2} and H^{8} are both Z, and the counting question is equivalent to calculating the 4th power of a generator of H^{2}. Then I quoted a general calculation (see here) for the structure of the cohomology ring. |