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Topology/Geometry Qualifying Exam
(January 2012)
Remark: In all solutions you must write a justification for your answers. Please start each problem in a new page.
Problem 1. Given P 2 R^2 let BĻµ(P ) and SĻµ(P ) denote, respectively, the open Euclidean ball and the circle of radius Ļµ centered at P. Let X = B 1 (0) [ S 1 (0) denote the closed ball of radius 1 centered at the origin 0 2 R^2. Given P 2 S 1 (0) and 0 < Ļµ < 1 =2, de ne
NĻµ(P ) = fP g [ BĻµ ((1 Ļµ)P ) ;
where (1 Ļµ)P denotes the multiplication of the vector P 2 R^2 by the scalar (1 Ļµ). Note that BĻµ((1 Ļµ)P ) is tangent to S 1 (0) at P and that NĻµ(P ) = [ 0 <r<Ļµ Sr((1 r)P ). (See picture below.) De ne
B = fNĻµ(P ) j P 2 S 1 (0); 0 < Ļµ < 1 = 2 g [ fB(Q) j Q 2 B; 0 < < 1 ā„Qā„g
where ā„Qā„ denotes the Euclidean norm, and observe that B forms a basis for a topology on X.
a. Show that this topology is 1st countable and separable, but neither 2nd countable nor LindelĀØof.
b. Describe the closure of NĻµ(P ) and prove that X is a regular space.
c. Fix P 2 S 1 (0) and 0 < Ļµ < 1 =2. De ne f : X! [0; 1] by:
f (x) =
1 ; if x 2 X NĻµ(P ) 0 ; if x = 0 r=Ļµ; if x 2 Sr(P ) fP g and 0 < r < Ļµ:
Show that f is continuous.
d. Prove that X is a completely regular space.
b
b
O
P
(1 Ļµ)P
b
Ļµ Qb
BĻµ(P )^ NĻµ(P^ )
Ļµ
Problem 2. Let X denote the following subspace of the Euclidean plane R^2 :
X = f(x; y) j xy =
n
; for some n 2 Ng [ Q f 0 g [ f 0 g Q:
a. Is X locally-connected?
b. Describe the connected components and the quasicomponents of X.
Problem 3. Let X be a second countable, locally compact, Hausdor space.
a. Show that the one-point compacti cation X+ of X is second countable.
b. Show that every subspace A X is paracompact.
Problem 4. Let f : X! Y be a function from a space X into a locally compact, Hausdor space Y. Show that f is continuous if and only if the following holds: whenever Y^ is a compact Hausdorff space containing Y as a subspace, then the graph of f is a closed subspace of X Y^.
Problem 5. Consider R^3 with the Euclidean metric. Let S R^3 be a smooth oriented surface and C S a regular, oriented curve. Let SO(3) be the set of oriented orthonormal framings (or bases) of R^3. Construct a smooth moving frame e = (e 1 ; e 2 ; e 3 ) : C! SO(3) as follows: given x 2 C, let e 1 (x) 2 TxC be the oriented unit tangent vector to C; let e 3 (x) 2 NxS be the oriented unit normal vector to S; set e 2 (x) = e 3 (x) e 1 (x) 2 TxS.
a. Show that there exist 1-forms ; ; 2 Ī©^1 (C) on C such that
de 1 = e 2 + e 3 ; de 2 = e 1 + e 3 and de 3 = e 1 e 2 :
b. Show that C is a geodesic if and only if = 0.
c. Compute the curvature of C in terms of ; ;.
d. Let II denote the second fundamental form of S. Compute II(e 1 ; e 1 ) in terms of ; ;.
Problem 6. Let M be a smooth n{dimensional manifold. Let X 1 ; : : : ; Xk be point-wise linearly independent vector elds on M , and let!^1 ; : : : ; !nāk^ be point-wise linearly independent 1-forms such that Annf!^1 ; : : : ; !nākg = spanCā(M )fX 1 ; : : : ; Xkg ;
where Ann denotes the annihilator in the dual space. Prove that [Xa; Xb] 0 modulo X 1 ; : : : ; Xk, for all 1 a; b k, if and only if d!s^ 0 modulo!^1 ; : : : ; !nāk^ for all 1 s n k.
Problem 7. Let f : R^3! R be given by f (x; y; z) = (x 1)^2 yz. For which t 2 R is f ā^1 (t) an embedded 2-dimensional submanifold of R^3?
Problem 8. Let f : R^2! R be a smooth function such that f (x; y) = 0 for all (x; y) outside the unit disk, i.e., for all (x; y) with x^2 + y^2 1. Consider the surface in R^3 given by the graph of f. What can you say about the average Gauss curvature of the surface?
Problem 9. Consider the tangent bundle to the 3-sphere. Is it a smooth manifold? If so, what is its dimension? Is it compact?
Problem 10. Either prove the projective plane RP^2 is not orientable, or nd an orientation for it. (Hint: construct RP^2 as the 2-sphere quotiented by the antipodal map.)
