Mathematics 5863
Name (please print)
Examination II
April 7, 2005
Instructions: Give brief, clear answers, avoiding excessive details (especially when the instruction is Verify). For
example, if F : p is a path homotopy, you need not verify that f F is a
Math 5863 homework
39. (4/5) Let X, Y , and Z be spaces, with Y locally compact Hausdor, and give all sets of
continuous functions the C-O topology. Dene the composition function C(f, g) : C(X, Y )
C(Y, Z) C(X, Z) by C(f, g) = g f . Prove that C is contin
Math 5863 homework
33. (3/22) For n 2, the dihedral group of order 2n is the group Dn consisting of all
pairs i j where i is an integer modulo n and j is an integer modulo 2, with the
j
multiplication rule that i j k = i+(1) k j+ (that is, i 1 = i ). Veri
Math 5863 homework
35. (3/22) Recall that if H is a subgroup of a group G, then gHg 1 is the subgroup
consisting of all elements ghg 1 for h H, and recall that H is called a normal
subgroup if gHg 1 = H for all g G.
1. Verify that every subgroup of an abe
Math 5863 homework
31. (3/22) Denote the automorphism group of a group G by Aut(G). Determine the
following automorphism groups:
1. Aut(Z). (Consider (1).)
2. Cn . (Write Cn as cfw_1, , 2 , . . . , n1 . Observe that a homomorphism : Cn
Cn is completely d
Mathematics 5863
Name (please print)
Examination II
April 7, 2005
Instructions: Give brief, clear answers, avoiding excessive details (especially when the instruction is Verify). For
example, if F : p is a path homotopy, you need not verify that f F is a
Math 5863 homework
26. (3/8) Let : I S 1 be a path. Let 1 and 2 be two lifts of to R. Prove that
for some N Z, 2 (t) = 1 (t) + N for all t I (let N = 2 (0) 1 (0) and dene
(r) = r + N , check that p = p, and use uniqueness of path lifting). Deduce that
1
Math 5863 homework
8. (2/1) Suppose that h0 , h1 : X Y are isotopic. Prove that if g : Y Z is a homeomorphism, then gh0 is isotopic to gh1 . Prove that if k : Z X is a homeomorphism,
then h0 k is isotopic to h1 k.
9. (2/1) An imbedding j : I I is called o
Math 5863 homework
1. (1/18) The Klein bottle K can be constructed from two annuli A1 and A2 by identifying their boundaries in a certain way. For each of the three descriptions of K discussed
in class (two Mbius bands with boundaries identied, the square
Math 5863 homework
19. (2/22) Prove that two paths , : I Rn are path-homotopic if and only if they have
the same starting point and the same ending point.
20. (2/22) Let X be a path-connected space. We say that X is simply-connected if every
two paths in
Math 5863 homework
14. (2/8) Use handle slides to simplify and identify each of these two surfaces:
15. (2/8) A space X is dened to be contractible if the identity map of X is homotopic to
a constant map. Prove the following:
1. If C is a convex subset of
Mathematics 5863
Name (please print)
Examination I
February 222 2005
Instructions: Give brief, clear answers.
I.
(6)
III.
(10)
VI.
(10)
Let M be a manifold with boundary. What is a collar of 6M? Draw a picture of a Mobius band, showing
a collar on its bou
Mathematics 5863
Name (please print)
Examination I
February 22, 2005
Instructions: Give brief, clear answers.
I.
(6)
Let M be a manifold with boundary. What is a collar of M ? Draw a picture of a Mbius band, showing
o
a collar on its boundary.
II.
(5)
Let
Math 5863 homework
44. (4/14) Let p : G G be a covering map, where G and G are compact 2-manifolds.
It is a fact that that (G) = k(G), where p is k-fold. Here is the idea: Start
with a triangulation of G. If necessary, it may be repeatedly subdivided so t