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MATH IN MOSCOW TOPOLOGY II
ASSIGNMENT 1
Problem 1.1. Prove that any functor T : C # D assigns isomorphic objects to isomorphic
objects.
Problem 1.2. In an arbitrary category define the notion of right inverse and left inverse
of a morphism f : X # Y and show that if they both exist, then they coincide and f is an
isomorphism. Prove that any isomorphism has a unique inverse.
Problem 1.3. Define the inverse limit in an arbitrary category by using the category Dir
In order to solve the subsequent problems, you can assume known the homology groups of
D n , S n , RP n , CP n , T 2 , M 2
g .
Problem 1.4. Is there a retraction of S 5 on S 4 , where S 4 is the ''equator'' of S 5 , i.e. S 5 = #S 4
(here # denotes suspension) ?
Problem 1.5. Is there a retraction of RP 5 on RP 4 , where RP 4 ## RP 5 is the natural embedding
given by (x 1 ; . . . ; x 5 ) ## (x 1 ; . . . ; x 5 ; 0) ?
Problem 1.6. Is there a retraction of the solid torus S = S 1
в D 2 on its boundary dS = T 2 =
S 1
в S 1 ?
Problem 1.7. Given a map p : RP 2
# S 1
вD 17 and a homeomorphism h : S 1
вD 17
# S 1
вD 17 ,
is it possible to lift h to a map H : S 1
в D 17
# RP 2 ?
Problem 1.8. Prove that Euclidean spaces of di#erent dimensions are not homeomorhic.
Date: September 8, 2009.