110.420 TOPOLOGY OF SURFACES 14 May 1996

Final Examination

Three hours. Open book.

50 points, 10 per question.

Partial credit may be available, but only if you show your working.

Use only the officially provided blue books.

Begin each of the five questions on a fresh page and number it clearly in the margin.

1. Compute the homology groups of a torus with two points removed. (It does not matter which two points are removed.)

2. What connected compact surfaces can be covering spaces of:

(a) The torus T^{2}?

(b) The Klein bottle K^{2}?

(c) The Möbius band?

3. (a) Maps f: X → Y and g: Y → Z induce the homomorphisms of fundamental groups f_{∗}: π_{1}(X, x) → π_{1}(Y, f(x)) and g_{∗}: π_{1}(Y, f(x)) → π_{1}(Z, g(f(x))), where x is a point in X. Show that the composite map g ∘ f: X → Z induces the composite homomorphism g_{∗} ∘ f_{∗}: π_{1}(X, x) → π_{1}(Z, g(f(x))); in other words, that (g ∘ f)_{∗} = g_{∗} ∘ f_{∗}.

(b) Deduce from (a) that if f: X → Y is a homeomorphism, then f_{∗}: π_{1}(X, x) → π_{1}(Y, f(x)) is an isomorphism of groups.

4. (a) If X is a retract of Y and Y is a retract of Z, show that X is a retract of Z. (b) Show that the circle S^{1} is a retract of the torus T^{2}, but not a deformation

retract.

5. A surface with boundary has the following planar diagram. Classify the surface.

THIS QUESTION SHEET MAY BE RETAINED JMB