METU Department of Mathematics Math 422 Geometric Topology Spring 2014 S. Finashin

Midterm I (April 10, 2014) 60 points in 4 problems

Name and the student number: Problem 1. 15pts (a) What does mean “topological invariant” and “homotopy invariant” ? Give examples.

(b) What does mean “simply connected topological space” ?

(c) What is a regular homotopy between curves ?

(d) What is the difference between the Jordan and the Schönflies theorems ?

(e) What does mean that two spaces are homotopy equivalent ?

(f) Give a definition of mapping cylinder.

(g) Give a definition of connected sum of two n-dimensional manifolds.

Problem 2. 15pts Which of the following spaces are homotopy equivalent and which ones are homeomorphic ? (Sketch.)

(1) S^{2} ∨ S^{2};

(2) S^{2}/E, where E ⊂ S^{2} is the equator;

(3) D^{2} ∪_{f} S^{2}, where f: ∂D^{2} → E is a homeomorphism;

(4) the mapping cone C_{f}, where f: {0, 1} → S^{2} ⊔ S^{2} such that f(0) and f(1) lie in different spheres;

(5) the mapping cone C_{f}, where f: S^{1} → S^{2} is an embedding, f(S^{1}) = E;

(6) (S^{2} ⊔S^{2})∪_{g} [0, 1], where g: {N_{1}, N_{2}} → {0, 1} identifies the north poles N_{1}, N_{2} of the two spheres with the endpoints of the interval;

(7) S^{2} ∪_{h} S^{2}, where h is a homeomorphism between the upper hemispheres of the two spheres;

(8) the mapping torus M_{ϕ} of a constant map ϕ: S^{1} → S^{1};

(9) the complement of a pair of points in ℝ^{3}.

Problem 3. 15pts Consider the gluing data abcdec^{−1}, ba^{−1}fge^{−1}, chdf^{−1}gh^{−1}. (a) Determine the topological type (F_{g} or N_{k}) of the surface F obtained by gluing.

(b) Are curves a and f closed on F ? Do they intersect ?

(c) Which kind of surface (F_{g,ℓ} or N_{k,ℓ}, find g or k and ℓ) is obtained from F by cutting it along a and f ?

Problem 4. 15pts Consider surfaces X_{1} and X_{2} represented by words respectively w_{1} = abca^{−1}dbc and w_{2} = xyxy^{−1}ztzt.

(a) Sketch the polygons with the corresponding gluing data for their edges. Are these surfaces with boundary ?

(b) Represent connected sum X = X_{1}#X_{2} by a word and the corresponding polygon without analysis of the topological type of X_{1} and X_{2}.

(c) Sketch polygons obtained cutting of the first polygon according to the substitution d = abc. Is this cut separating ? Sketch a polygon and give a word describing pasting along edge c of the polygons obtained after cutting.

(d) Find a separating cut of surface X_{2} giving a representation X_{2} = X′_{2}#X′′_{2} as a connecting sum. What are the topological types of the summands ? Using this connected sum representation, find a surfaces F_{g} or N_{k} homeomorphic to X_{2} (find g or k).