Middle East Technical University Fall 1998 Math 541 Differential Topology

MIDTERM II (TAKE-HOME EXAM)

DECEMBER 24

NO COLLABORATION IS ALLOWED !!!

Problem 1. (10 pts) Given a manifold, M, prove that there exists a smooth function, f: M → ℝ such that f^{−1}(0) = ∂M.

Problem 2. (10 pts) Assume that M is a manifold and Δ ∈ M × M is the diagonal. Prove that the normal bundle to Δ in M × M is isomorphic to the tangent bundle.

Problem 3. (10 pts) Assume that M and N are closed submanifolds of a manifold X, dim M +dim N = dim X, and M = ∂W for some compact submanifold W ⊂ X.

(1) Prove that the Z/2-valued intersection index, I_{2}(M, N), vanishes.

(2) Assuming that M, N and X are oriented, prove that the integer-valued intersection index, I(M, N), vanishes.

Problem 4. (10 pts) Assume that M ⊂ ℝ^{4} is a 3-dimensional submanifold. Prove that there exist a, b, c ∈ ℝ such that

(1) M_{a} = {(x_{1}, …, x_{4}) ∈ M | x_{1} = a} is a 2-dimensional submanifold of M (surface),

(2) M_{a,b} = {(x_{1}, …, x_{4}) ∈ M | x_{1} = a, x_{2} = b} is a 2-dimensional submanifold (curve),

(3) M_{a,b,c} = {(x_{1}, …, x_{4}) ∈ M | x_{1} = a, x_{2} = b, x_{3} = c} is a discrete set of points.

Question. (30 pts) In the definition of the intersection index (both ℤ/2-valued or integer-valued) for submanifolds M, N ⊂ X of the complementary dimensions we require that (A) M and N have no boundary, (B) at least one of the submanifolds, M or N, is compact, and another is properly embedded to X (i.e., is a closed subset of X).

(1) How do you understand the importance of these conditions ?

(2) Is it possible to omit any of these conditions if M and N intersect transversally ?

(3) Why it is possible to define integer-valued self-intersection index, I(M, M), for a non-orientable closed surface, M ⊂ X, in a 4-manifold X, although one can define only ℤ/2-valued index, I(M, N), for a pair of non-orientable closed surfaces ?