hw3s_132sp17.pdf - Math 132 Topology II Smooth Manifolds...

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Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 3 SOLUTION Due in class by 1.15pm, Wednesday February 15. Late homework will not be accepted. Submit solutions to the following problems: 1. Let T 2 = S 1 × S 1 R 4 be the torus (a) Show that, for any α R , the smooth map f R T 2 , t ( cos t, sin t, cos αt, sin αt ) is an immersion. (b) For which α is f injective? (You must prove your answer.) (c) Does there exist α R such that f is an embedding? If yes, give an example; if no, give a proof. Solution: (a) Compute that ( df ) t R T t ( R ) T f ( t ) ( T 2 ) R 2 , Jac t ( f ) = - sin t cos t - α sin αt α cos αt . Since sin t = cos t = 0 does not occur for any t R , we conclude Jac t ( f ) has non-zero entries. Hence Jac t ( f ) has rank at least 1, which means it has full column rank and so ( df ) t is injective. Hence f is an immersion for all t . (b) We claim that f is injective if and only if α is irrational. First, suppose that f is not injective. Then there exist s t with f ( s ) = f ( t ) , i.e., the relations cos s = cos t, sin s = sin t, cos αs = cos αt, sin αs = sin αt all concurrently hold. These imply s - t = 2 πn and α ( s - t ) = 2 πm for some m,n Z with n non-zero, which together give α = 2 πm s - t = 2 πm 2 πn = m n , thus showing α must be rational. Conversely, if α = m n is rational, one checks that injectivity fails by taking any s,t such that s - t = 2 πn . (c) No such α exists. An embedding is a proper injective immersion, but properness fails for all f , since T 2 is itself a compact set (it is the finite product of compact S 1 ) but its preimage R = f - 1 ( T 2 ) is not compact (recall that for subsets of Euclidean space, compact is equivalent to closed and bounded). 2. (a) Let L R m R n be a linear map, rank L = r . Show that there exists linear isomorphisms T R m R m and S R n R n such that S L T ( x 1 ,...,x m ) = ( x 1 ,...,x r , 0 ,..., 0 ) . Deduce that there is a surjective linear map P R m R r and an injective linear map J R r R n such that L = J P . 1
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(b) A smooth map f X Y has constant rank r if rank ( df ) x = r , for any x X . Let f X Y be a map of constant rank r , where dim X = k , dim Y = l . Show that, for x X , there are local parameterisations R k V 1 U 1 X x U 1 , φ 1 ( 0 ) = x, φ 1 ψ 1 R l V 2 U 2 X y = f ( x ) U 2 , φ 2 ( 0 ) = y, φ 2 ψ 2 such that ( ψ 2 f φ 1 )( x 1 ,...,x k ) = ( x 1 ,...,x r , 0 ,..., 0 ) , ( x 1 ,...,x k ) V 1 . ( Hint: consider the proofs of the local immersion/submersion theorems, and part (a). ) (c) ( Generalised preimage theorem ) Let f X Y be of constant rank r , y f ( X ) . Prove that f - 1 ( y ) = { x X f ( x ) = y } is a submanifold of X , dim f - 1 ( y ) = dim X - r . Solution: (a) Let { y 1 ,...,y r } R n be a basis of im L , and extend to a basis { y 1 ,...,y r ,z 1 ,...,z n - r } of R n . Let { v 1 ,...,v r } R m be elements such that L ( v i ) = y i . Then, { v 1 ,...,v r } is linearly independent: if c 1 v 1 + ... + c r v r = 0 , c 1 ,...,c r R , then 0 = L ( c 1 v 1 + ... + c r v r ) = c 1 y 1 + ... + c r y r c 1 = ... = c r = 0 .
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