MAST90029 Di?erential Topology and Geometry 2015
Assignment 3
Due: 5pm Wednesday May 27
Please aim for clear, complete but concise explanations in all questions.
Some marks will be allocated for overall presentation and clarity of solutions.
1. Let M be a compact smooth n-manifold. Recall that two maps f0, f1 : M ! S1 are
smoothly homotopic if there exists a smooth map F : [0, 1] ? M ! S1 such that
F(0, x) = f0(x) and F(1, x) = f1(x) for all x 2 M. Write f0 ? f1.
(a) Prove that
f0 ? f1 ) f0 + g ? f1 + g
for any g : M ! S1. (Use the group structure on S1 induced from S1 ?= R/Z.)
(b) Hence prove that the set of homotopy classes of maps
[M,S1
] := {f | f : M ! S1
}/ ?
is a group, i.e. a Z-module.
(c) Define H1(M; Z) to consist of all ? 2 H1(M) satisfying R
N ? 2 Z for any
smooth compact 1-dimensional submanifold N ? M. (So N is di?eomorphic
to a disjoint union of circles. ) Prove that H1(M, Z) is a group, i.e. a Z-module.
(d) Prove that the map f 7! df defines an isomorphism of groups
[M,S1
] ?= H1
(M, Z)
where df uses the (local) lift of f to R ! R/Z = S1.
2. Consider the upper half-plane U = {(x1, x2) : x2 0} with the Poincar´e metric:
g = 1
x2
2 (dx1
2 + dx2
2
).
(This is one of the standard models for 2-dimensional hyperbolic geometry.)
(a) Show that each of the following maps is an isometry from U onto U.
(i) f(x) = ax + b where a 0, b = (b1, 0) 2 R ? {0}.
(ii) f(x) = x/||x||2 where ||x||2 = x2
1 + x2
2.
(iii) f(x1, x2)=(#x1, x2).
Conclude that there is an isometry from U to U taking any point to any other
point.
(b) Find the length L(c) in the Poincar´e metric of the curve c : [a, b] ! U defined
by c(t) = (0, t), where 0 a b.
(c) Show that the length L(¯c) % L(c) where ¯c is any other curve from (0, a) to
(0, b). Hence find the distance from (0, a) to (0, b) in the Poincar´e metric.