Consider the standard quotient map q: R Sx S(~ R/Z) induced by the equivalence relation on...

Transcribed Image Text:

Consider the standard quotient map q: R Sx S(~ R/Z) induced by the equivalence relation on R2 defined by (1, 1)~ (2, 2)(22-21, 92- y) Z. Consider the multiplicative matrix group a b SL(2,Z) = {(c) | (a,b,c,d) Z and ad bc = 1 c=1}. d For a fixed matrix A = R given by a b 1) SL(2, Z), define a map MA : R d a b by - ( 2 ) ( ;) = (a + b), d dy MA((x, y)) = for any (r, y) = R. (a) Show that MA induces a map M: S x S S x S, which is a homeomorphism. (b) Show that for a fixed A SL(2, Z), we have q o MA = M q. Consider the standard quotient map q: R Sx S(~ R/Z) induced by the equivalence relation on R2 defined by (1, 1)~ (2, 2)(22-21, 92- y) Z. Consider the multiplicative matrix group a b SL(2,Z) = {(c) | (a,b,c,d) Z and ad bc = 1 c=1}. d For a fixed matrix A = R given by a b 1) SL(2, Z), define a map MA : R d a b by - ( 2 ) ( ;) = (a + b), d dy MA((x, y)) = for any (r, y) = R. (a) Show that MA induces a map M: S x S S x S, which is a homeomorphism. (b) Show that for a fixed A SL(2, Z), we have q o MA = M q.

Answer rating: 100% (QA)

The detailed ... View the full answer