In the fifth tutorial session, we shall discuss the following.
1. Consider the map f : S1 × S1 → S1 × S1 given by (z1 , z2 ) 7→ (z1 z2 , z2 ). Compute the induced homomorphism f# on the fundamental group.
2. Compute degree of any non-surjective map on S1 .
3. Let M be the Mobius band and B its boundary circle and let C be the central circle in M which is the
image of I × 21 under the standard quotient map I × I → M . Show that M deforms to C. Compute the
homomorphism induced by inclusion j : B ,→ M . Conclude that B is not retracted of M .
4. We have seen generator of π1 (S1 , 1) viz. the path homotopy class of the map t 7→ exp (2πit), t ∈ [0, 1].
Using this find one generator of π1 (S1 , x0 ) for any arbitrary point x0 ∈ S1 .
5. Consider any rotation of circle viz. consider the map, r : S1 → S1 given by exp (2πit) 7→ exp (2πi(t + t0 ))
for some t0 ∈ [0, 1]. Calculate its degree by computing its lift.
6. Show that S1 is not contractible.