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MATH IN MOSCOW TOPOLOGY II
ASSIGNMENT 2
Problem 2.1. Let C be the union of all circles of center ( 1
n , 0) in the xy­plane and radius 1
n .
Prove that C is not homeomorphic to a CW­space.
Problem 2.2. a) Find a space satisfying (W) but not (C).
b) Find a space satisfying (C) but not (W).
c) Is the closure of cell necessarily a subspace ?
Problem 2.3. Find mimimal CW­complex structure on CP n , RP n , # q
i=1 T 2
i , # q
i=1 RP 2
i .
Problem 2.4. Define S # , RP # , CP # and supply them with CW­structure.
Problem 2.5. Prove that S # is contractible.
Problem 2.6. Give an example of a nontriangulable two­dimensional CW­complex.
Problem 2.7. Prove that # 1 X lives in X (2) (the 2­skeleton of X), i. e., # 1 (X) # # 1 (X (2) ).
Problem 2.8. Prove that a CW­complex is connected i# its 1­skeleton X (1) is connected.
Problem 2.9. Prove that a CW­complex is connected i# it is path­connected.
Problem 2.10. Prove that # k S n = 0 for all k < n.
Problem 2.11. Show that the Cartesian product, the cone the suspension, the join of CW­
complexes have natural CW­complexes structures.
Problem 2.12. Any finite CW­complex X n can be embedded in R N , N = (n+1)(n+2)
2 .
Problem 2.13. Show that # n (S 1 ) are trivial for all n > 1. (Hint : Consider the bundle
p : R 1
# S 1 whose projection is given by formula p(x) = exp(xi) )
Problem 2.14. Prove that a locally trivial fibration p : S n
# B whose base B consists of more
than one point is not homotopic to a constant map.