Recall that in algebraic topology, we construct homotopy invariants, e.g. homology. Homology measures “global” topology, but its not very sensitive to local structure.

However, Poincare noticed that if X is a closed oriented manifold of real dimension n, then .

**Poincare Duality**: Let X be an orientable manifold of dimension n. Then there exists natural isomorphisms

where [X] is the fundamental class.

**Theorem**: Let X be a smooth complex affine variety of complex dimension n. Then X has homotopy type of a CW complex of real dimension n. In particular, for k>n.

Proof idea: Morse Theory

Definition: Let X be a manifold. A smooth function is **Morse** if

- for any real number s, is compact,
- every critical point of f is non-degenerate

Recall, a point x in X is a critical point of f if , and a critical point is non-degenerate if the Hessian (symmetric bilinear form on defined by second partials) is non-degenerate. The index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian.

**Example**: consider the “height” function on an upright torus,

it has four critical points, depicted below

Note that d has index 0, c has index 1, b has index 1, and a has index 2.

**Theorem**: Let be a Morse function. Then X has the homotopy type of a CW complex with one cell for each critical point, and each cell has dimension equal to the index of the corresponding critical point.

So in the torus example, you think of building the CW complex by starting at the bottom, you start with a 0-cell. As you scan up, you have a disk, which is still homotopic to a point. Then when you reach c, you have to introduce a 1-cell, and it something like a basket with a handle. Then when you reach b, you have to add another 1-cell, and at a you add the 2-cell. This will give you a CW complex that looks like the standard way to build a torus by identifying sides of a square.

**Lemma**: Let be a manifold. For y in , the function given by sending a point x to is Morse.

**proof**: Note For any tangent vector u in . So x is a critical point of iff x-y is perpendicular to . Let . Let and be the projection maps. Note: p realizes Z as the orthogonal complement to the bundle TX in . Then dim Z = n + N – n = N. Now consider , and notice the domain and codomain have the same dimension. q is submersive at (x,y) iff q is immersive at (x,y).

**claim**: q is immersive at (x,y) in Z iff the critical point x of is non-degenerate.

**proof**: q is immersive at (x,y) iff for any , . We can express Z as the zero-set of

In local coordinates {x_i} on X near x,

Let be coordinates on $\mathbb R^N$. For and , let , and .

which gives

, and this somehow gives what we wanted.

**Lemma**: For smooth algebraic variety and defined by the standard Euclidean scalar product on , the index of is less than or equal to n at every critical point.

**proof**: Let x be a critical point of , let . Consider extending to vector fields on X near x.

If we could diagonalize in an orthonormal basis with respect to <,>, then we can calculate eigenvalues.

## 2 thoughts on “Algebraic Analysis notes Lecture 2 (9 Jan 2019)”