MATH 314 Final, Spring 2005
American University of Beirut · Algebraic Topology I · 1 page
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Mathematics 314, Final Examination, June 4, 2005--—-——--——
Problem 1. 20 pts.
Describe how the fundamental group 7(X, 79) is defined.
Problem 2. 20 pts.
Let M and N be connected n-dimensional manifolds without boundary and let M#N
denote their connected sum.
Suppose that n > 3 and that m(M) and 7(N) are known.
What is 7(M#N)? Why? Why this would not work when n = 2 ?
Problem 3. 15 pts.
Remind the definition of a topological polyhedron.
Do not quote the definitions of a simplex or of a linearly independent set of points in R™.
Problem 4. 15 pts.
Show that the torus T = S! x S} is homeomorphic to a subset
of the 3-dimensional sphere S°.
Then prove that T is not a deformation retract of S°.
Problem 5. 15 pts.
Let (X,p) be a covering space of X and (Y,q) be a covering space of Y.
Show that X x ¥ isa covering space for X x Y.
Problem 6. 15 pts.
Find the fundamental group of the following space/figure
Time: 120 minutes. Good Luck!