| 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
Hl(M,A) → Hn-l(M,B).
|
| 4/9 | Alexander duality: For K⊆
Rn compact and locally contractible,
Hl(K) = Hn-l(Rn -
K). More generally, for M orientable
n-manifold (without boundary) and K⊆
M compact and locally contractible,
Hl(K) = Hn-l(M - K).
Applications: Generalized Jordan Curve theorem: If
M⊆ Rn is a closed, connected
(n-1)-manifold, then M is oriented and
Rn 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
TxM 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
R2n+1, first following (M&T 8.11)
and then applying Sard's theorem.
|
| 5/2 | Embedding theorem for non-compact
manifolds: Mn embeds as a submanifold,
and closed subset, of R2n+1. Normal
bundle NM of a submanifold M ⊆
Rk. Tubular neighborhood theorem.
Compact manifolds have finite dimensional homology and
cohomology.
|
| 5/5 | NM is a submanifold of M x
Rk if M ⊆ Rk.
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
Hn(E,E'). Here, E
→ B is a Dn-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 ER → B whose fiber
over b is
Hn(Eb,E'b).
There is a natural map Hn(E,E') →
Γ(ER &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 EZx → B
has a section, (2) The "first Stiefel-Whitney class"
w_1(E) ∈ H1(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 CPn. Its Thom space is
homeomorphic to CPn+1. It is orientable
(because H1(CPn;Z/2) = 0)
and its euler class is a generator of
H2(CPn).
|
| 5/23 | The intersection product
Hk(M) x Hl(M) →
Hk+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:
"[P1]•[P2] = [P]" if
P1 and P2 are smooth, oriented,
closed, transverse submanifolds of M and P = P1
∩ P2.
|
| 5/30 | An easy corollary of the geometric
interpretation of the intersection product: If
[P1] • [P2] ≠ 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
H2 and H8 are both
Z, and the counting question is equivalent to
calculating the 4th power of a generator of
H2. Then I quoted a general calculation
(see here)
for the structure of the cohomology ring.
|