Differential Topology

Assignment 5

Exercise 1: Classification of smooth 1-manifolds Let M be a compact, smooth and connected 1−manifold with boundary. Show that M is either diffeomorphic to a circle or to a closed interval [a, b] ⊂ R. This then shows that every compact smooth 1−manifold is build up from connected components each diffeomorphic to a circle or a closed interval. To prove the result, show the following steps: 1. Consider a Morse function f : M → R. Let B denote the set of boundary points of M, and C the set of critical points of f. Show that M \ (B ∪ C) consists of a finite number of connected 1−manifolds, which we will call L1,...,LN ⊂ M.

2. Prove that fk = f|Lk : Lk → f(Lk) is a diffeomorphism from Lk to an open interval f(Lk) ⊂ R. 3. Let L ⊂ M be a subset of a smooth 1−dimensional manifold M. Show that if L is diffeomorphic to an open interval in R, then there are at most two points a, b ∈ L such that a, b∈ / L. Here, L is the closure of L.

4. Consider a sequence (Li1 ,...,Lik ) ∈ {L1,...,LN }. Such a sequence is called a chain if for

every j ∈ {1, . . . , k − 1} the closures Lij and Lij+1 have a common boundary point.

Show that a chain of maximal length m contains all of the 1−manifolds L1,...,LN defined above.

5. To finish the proof, we will need a technical lemma. Let g :[a, b] → R be a smooth function with positive derivative on [a, b] \{c} for c ∈ (a, b). Then there exists a smooth function g˜ :[a, b] → R with positive derivative on the whole of [a, b] such that g =g ˜ on a neighborhood of a and on a neighborhood of b.

6. Finally use the previous lemma to show that if L1 and LN , contained in the maximal chain, have a common boundary point, then M is diffeomorphic to a circle. Otherwise, M is diffeo- morphic to a closed interval.

Exercise 2: Brouwer’s fixed point theorem In the following, we use the continous version of Brouwer’s fixed-point theorem: every continuous ¯ ¯ ¯ n function f : Bn → Bn has a fixed point. Here Bn is the closed unit ball in R . In this exercise, we show the Perron-Frobenius theorem: n×n A matrix A ∈ R with Ai,j ≥ 0 for all i, j ∈ {1, . . . , n} has a non-negative eigenvalue. In the following, think of A as a linear map A : Rn → Rn (expressed in the standard basis). 1. Show that n−1 n−1 S+ = {x = (x1, . . . , xn) ∈ S | xj ≥ 0 for all j = 1, . . . , n} ¯ is homeomorphic to Bn−1. 2. Consider the map f(x) = Ax/kAxk to prove the theorem. Let X be a topological space and A ⊂ X a subspace. Then A is called a retract of X if there is a retraction from X to A, that is, a continuous map r : X → A whose restriction to A is the identity map. ¯ 3. Show that if A is a retract of Bn and f : A → A is continuous, then f has a fixed point.

1 Differential Topology

Exercise 3: mod 2-degree and “one half” the fundamental theorem of algebra Let M be a smooth, compact (n+1)−dimensional manifold with boundary, Y a smooth, connected n−dimensional manifold and f : ∂M → Y a smooth map.

1. Show that deg2(f) = 0 if there is a smooth extension F : M → Y of f to M.

Let M ⊂ R2 be a compact smooth submanifold with boundary. Let g : R2 → R2 be smooth 1 g(x) with g(x) 6= 0 for all x ∈ ∂M. Define f : ∂M → S by f(x) = kg(x)k .

2. Show: if deg2(f) 6= 0, then there is an x ∈ M with g(x) = 0.

In the following, we will identify R2 with C, and use complex variables to specify functions.

n 1 3. Compute deg2(g) for the case where g(z) = z , n ∈ N and M = S = N.

4. Deduce that every complex polynomial of odd degree has a root in C. (Hint: Define a homotopy to zm, where m is the degree of the polynomial.)

Exercise 4: Smooth no-retraction theorem and mod 2-degree 1. Show that the identity map id : M → M on a compact connected smooth manifold M is not smoothly homotopic to a constant map.

2. Use this result for M = Sn in order to prove the smooth no retraction theorem for the unit ball.

Exercise 5: mod 2-intersection theory Let M,N be smooth manifolds where M is compact, and Z ⊂ N a submanifold with M,N,Z boundaryless and dim M + dim Z = dim N. Assume f : M → N smooth with f t Z. Define

−1 I2(f, Z) := |f (Z)| mod 2 .

1. Show that this is well-defined. (Hint: look at exercise 4 of assignment 5.)

2. Show that I2(f, Z) is a smooth homotopy invariant among all functions f : M → N with f t Z. You may use the following result: if f t Z and g t Z are homotopic, then there is a homotopy H : M × [0, 1] → N with H t Z. The transversality homotopy theorem states that for any smooth map f : M → N, there is a smooth map g : M → N which is homotopic to f and satisfies g t Z and ∂g t Z. For an arbitrary smooth function f : M → N define

I2(f, Z) := I2(g, Z) , where g : M → N is homotopic to f and satisfies g t Z. 3. Show that this is well-defined and a homotopy invariant of functions f : M → N.

4. Show that I2(f, {q}) does not depend on q ∈ N and coincides with deg2(f).

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