Middle East Technical University Spring 1998 Math 422 Elementary Topology and Geometry

HOMEWORK III (MAY, 1999)

Problem 1. Determine the boundary homomorphisms for the complexes X. Find the Betti numbers b_{i}(X) and χ(X).

(1) abcabca^{−1}b^{−1}c^{−1}

(2) ababcb, abcbab

Problem 2. Find the Betti numbers of the following sets (1) F_{g},

(2) N_{k},

(3) F_{g} with a hole,

(4) S^{2} × S^{3},

(5) S^{2} × S^{2} × S^{2}

Problem 3. Classify the letters (A, B, C, etc.) up to homotopy equivalence. Determine their Betti numbers b_{0} and b_{1}. Which letters are contractible ?

Problem 4. Determine the number, f_{k}, of k-dimensional faces of X, where k = 0, 1, 2, … and

(1) X is 5-dimensional symplex.

(2) X is 4-dimensional cube. Determine the Euler characteristic of the 2-skeleton X^{(2)}.

Problem 5. Consider a group G acting freely on X. What can be the order of G ? Can it be infinite ?

(1) X = S^{2} × S^{2},

(2) X = F_{3},

(3) X = ℝP^{2},

(4) X = S^{1} × S^{2}.

Problem 6. f: X → X is an involution with the fixed point set F. Determine χ(F) if

(1) X = S^{3} × S^{1},

(2) X = F_{2},

(3) X = ℝP^{2}.

Problem 7. Prove that any action of ℤ/3 on F_{2} and on F_{3} has a fixed point. Describe a fixed point free action of ℤ/3 on a torus and on F_{4}.

Problem 8. X → F_{g} is a double covering branched at m points. Determine χ(X) and find the genus of X.

Problem 9. Prove that a Möbius band cannot be retracted to its boundary.

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