Download Math 148: Homework 2

Math 148: Homework 2
Notes:
- We identify C with R2 when necessary. In particular, we view S 1 as a
subspace of C.
- Under this identification, 1 ∈ C is (1, 0) ∈ R2 . Thus we have the point
1 = (1, 0) ∈ S 1 .
Definition (local homeomorphism). Recall that a map f : X → Y is a
homeomorphism if each x ∈ X has a neighbourhood U in X such that f (U )
is open in Y , and f |U : U → f (U ) is a homeomorphism.
1. Local homeomorphisms and covering spaces.
a) Give an example of a local homeomorphism which is not a covering map.
b) Assume that f : X → Y is a local homeomorphism such that:
- X is Hausdorff,
- for each y ∈ Y the fibre f −1 (y) is finite, and
- for any y, z ∈ Y the fibres f −1 (y) and f −1 (z) have the same size.
Show that f is a covering map.
c) Assume that X, Y are Hausdorff spaces. Show that a local homeomorphism
f : X → Y is a covering map if X is compact.
2. Examples of covering spaces.
a) Show that
p : R → S1
t 7 → e2πit
is a covering map.
b) Show that
pn : S 1 → S 1
z 7 → zn
is a covering map for each n ∈ Z \ {0}.
c) Show that the projection q : S n → RP n is a covering map for each n ∈ N.
What does this map give when n = 1?
[Note: Recall that RP n is the quotient of S n by the equivalence relation
generated by x ∼ −x for x ∈ S n .]
3. Fibres of covering spaces.
a) Let p : E → B be a covering map with B path connected. Show that all
the fibres of f have the same cardinality (i.e. for any points x, y ∈ B, the
cardinalities of f −1 (x) and f −1 (y) coincide).
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b) Show that the above conclusion still holds assuming that B is only connected (instead of path connected).
4. Change of basepoint in π1 and effect of a map. Let f : X → Y be
a continuous map. Let α be a path in X from x0 to x1 . Furthermore, let
β = f ◦ α, y0 = f (x0 ), and y1 = f (x1 ). Consider the pointed maps
fx0 : (X, x0 ) −→ (Y, y0 )
fx1 : (X, x1 ) −→ (Y, y1 )
induced by f . Show that
βb ◦ (fx0 )∗ = (fx1 )∗ ◦ α
b
i.e. the following diagram commutes
π1 (X, x0 )
α
b
↓
π1 (X, x1 )
(fx0 )∗
→ π1 (Y, y0 )
βb
↓
→ π1 (Y, y1 )
(fx1 )∗
5. Maps on S 1 and π1 . Consider the map
pn : S 1 → S 1
z 7 → zn
for n ∈ Z. Compute the induced homomorphism (pn )∗ from the infinite cyclic
group π1 (S 1 , 1) to itself.
6. Degree of maps S 1 → S 1 . Let f : S 1 → S 1 be a continuous map.
Consider the loop
α : I → S1
t 7 → e2πit
Let g be a lift of f ◦ α to R (for the covering map p : R → S 1 , p(t) = e2πit ).
Define the degree of f to be
deg f := g(1) − g(0)
a)
b)
c)
d)
Show
Show
Show
Let f
that the degree deg f is independent of the choice of lift g.
that deg f is an integer.
that two homotopic maps from S 1 to S 1 have the same degree.
: (S 1 , 1) → (S 1 , 1) be a pointed map. Show that the homomorphism
f∗ : π1 (S 1 , 1) −→ π1 (S 1 , 1)
is given by multiplication by deg f . Compare with the previous question 5,
and calculate deg(pn ) for n ∈ Z (pn is as defined in question 5).
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e) Show that two maps from S 1 to S 1 which have the same degree are
homotopic.
f) Show that deg(f 0 ◦ f ) = (deg f 0 ) · (deg f ).
[Hint: One method starts by using homotopy invariance of the degree to
reduce to the case in which f (1) = 1. Another method simply replaces f ,
f 0 with appropriate pn ’s.]
7. Different proof of Brouwer fixed point theorem.
a) Assume that f : S 1 → S 1 is a map such that f (x) 6= −x for all x ∈ S 1 .
Give a homotopy between f and the identity idS 1 . Conclude that f is not
null-homotopic.
b) Assume that f : S 1 → S 1 is a map without fixed points. Use part (a) to
show that f is not null-homotopic.
c) Use part (b) to reprove Brouwer’s fixed point theorem.
[Hint: Given a map g : D2 → D2 without fixed points, consider the map
f : S 1 → S 1 given by f (x) = (g(x) − x)/kg(x) − xk. Try to find a fixed
point of f .]
Definition (action of a group). Let G be a group (with unit e), and S a
set. An action of G on S is a function
G×S →S
(g, s) 7 → g · s
such that
e·s=s
(gh) · s = g · (h · s)
for any g, h ∈ G and s ∈ S.
Definition (category associated to a group). Recall that to a group G
we associated a category CG with a single object ∗ and such that:
- the morphisms of CG are CG (∗, ∗) = G;
- the composition of two morphisms in CG is given by multiplication in the
group: g ◦ h = h · g.
[Note that we reversed the order of the multiplication in the definition of
composition in class, but not here.]
8. Categories and representations of groups. Let G be a group.
a) Observe that the data for an action of G on a set is the same as the data
for a functor CG → Set (Set denotes the category of sets). Construct a
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bijective correspondence between sets equipped with an action of G and
functors from CG to Set.
b) Let VectR denote the category of real vector spaces (whose objects are
real vector spaces, and whose morphisms are linear maps between vector
spaces). Construct a bijective correspondence between real representations
of G (in arbitrary real vector spaces) and functors from CG to VectR .
Definition (covering functor). Let F : C → D be a functor between
categories. F is called a covering functor (or a discrete fibration) when the
following condition is verified: for any morphism f : x → y in D and any
e of C with F (e
x) = x, there exist a unique object ye and a unique
object x
e
morphism f : x
e → ye in C such that F (fe) = f .
9. Covering spaces and the fundamental groupoid. Let p : E → B be
a covering map.
a) Show that the functor
F = π≤1 (p) : π≤1 (E) −→ π≤1 (B)
is a covering functor.
b) Construct a functor (and show that it is a functor)
fibp : π≤1 (B) −→ Set
verifying:
- to each point x ∈ B, fibp associates the fibre of p over x:
fibp (x) = p−1 (x)
- for each path f in B from x to y, the induced function
fibp [f ] : p−1 (x) −→ p−1 (y)
verifies
(fibp [f ]) fe(0) = fe(1)
for any lift fe of f to E.
[Hint: Part (a) may be useful.]
c) How do we recover the lifting correspondence
φx : π1 (B, b) −→ p−1 (b)
(for b ∈ B and x ∈ E with p(x) = b) from the functor fibp ?
[Note: For a “nice” space B, we can actually reconstruct a covering space of
B from a functor π≤1 (B) → Set.]
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10. Fundamental group of S n and RP n . Assume n ≥ 2 is an integer. Fix
a point x ∈ S n .
a) Observe that S n \ {x} is homeomorphic to Rn . Conclude that any loop in
S n \ {x} is path homotopic to a constant loop.
b) Let γ be a loop in S n based at a point distinct from x. Show that γ is
path homotopic to a loop whose image does not contain x.
[Hint: Consider a small ball U around x. Deform the path γ only within
U so that the path goes around the ball U , instead of through it.
Note: There exist surjective continuous maps I → S n . These are sometimes
called space filling curves.]
c) Conclude that S n is simply connected.
d) Use the lifting correspondence to show that the fundamental group of RP n
is cyclic of order two.