Math 434/Math 542
Final Exam
Carter
Fall 2012
Certain things in life don’t commute. Thus we put on our socks before our shoes, and
check for the keys before we close the door.
1. Consider the quotient space of the decagon that is illustrated with sides identified as
indicated. Determine the Euler characteristic (V − E + F ). Then determine the genus
of the surface. Show the steps to put the surface into standard position.
c
d
b
a
e
c
b
d
a
e
2. Let X and Y denote a pair of (Hausdorff) topological spaces. Let A ⊂ X and B ⊂ Y .
3. This problem has to do with closures and interiors. If you don’t remember the definitions, please ask (with a deduction in credit).
(a) Define the closure of A. Let clA denote the closure.
(b) Define the interior of B. Let intB denote the interior.
(c) Let ∂A = cl(A)\intA. Of course, ∂(B) is defined similarly. Show that ∂(A×B) =
(∂A) × B ∪ A × (∂B).
P
P
2
n+1
2
4. Let S n = {(x1 , . . . , xn+1 ) : n+1
= {(x1 , . . . , xn+1 ) : n+1
i=1 xi = 1}, and let D
i=1 xi ≤
n+1
n
1}. Show that ∂D
=S .
5. Give a convincing proof that D1 × D1 is homeomorphic to D2 . More generally, show
that Dk × Dn+1−k is homeomorphic to Dn+1 .
6. Use these homeomorphisms to give the “tropical” decompositions of the 1, 2, and
3-dimensional spheres.
7. Show that S 2 is not homeomorphic to S 1 × S 1 .