Problem Set 3
Due Thursday, April 7
Problem 1.
(a) Let X and Y be pointed spaces, with X well pointed and Y path-connected. Show that
the map [X, Y ]∗ → [X, Y ] from pointed to unpointed classes induces an isomorphism
[X, Y ]∗ /π1 (Y ) ∼
= [X, Y ], for an appropriate action of π1 (Y ) on [X, Y ]∗ .
(b) Let G be a topological group. If X is a pointed connected CW complex, we know that
[X, BG]∗ ∼
= π0 BunG,∗ (X). Deduce that, for an arbitrary CW complex X, [X, BG] ∼
=
π0 BunG (X).
(c) Let G be a discrete group, so that BG ' K(G, 1). In Problem Set 2, we saw that π1
induces a bijection [BG, BG]∗ ∼
= Hom(G, G). What is the action of π1 (BG) = G on
Hom(G, G)?
Problem 2. Let n ≥ 0 and let E be a finite-dimensional (real or complex) vector space.
Denote by Vn (E) ⊂ E n the space of n-tuples of linearly independent vectors in E, and by
Grn (E) = Vn (E)/GLn the Grassmannian of n-planes in E. Show that the quotient map
Vn (E) → Grn (E) is a principal GLn -bundle.
Problem 3. Let Z̃ be the sign representation of π1 (RP 2 ) ∼
= Z/2. Compute H∗ (RP 2 , Z̃).
Problem 4. (Parallel transport) Let p : X → B be a fibration. Construct a functor
Π1 (B)op → hTop,
b 7→ p−1 (b).
Here, hTop denotes the category whose objects are topological spaces and whose morphisms
are homotopy classes of maps.
Problem 5. (Convergence) Let F∗ C be a filtered chain complex
· · · ⊂ Ft−1 C ⊂ Ft C ⊂ Ft+1 C ⊂ · · ·
and set C =
S
t
Ft C. Consider the associated spectral sequence with
r
r
r
Es,t
= Zs,t
/Bs,t
,
∂
r
Zs,t
= kernel of Hs (Ft /Ft−1 ) → Hs−1 (Ft−1 /Ft−r ),
∂
r
Bs,t
= image of Hs+1 (Ft+r−1 /Ft ) → Hs (Ft /Ft−1 ).
Let Ft Hs (C) denote the image of Hs (Ft C) → Hs (C).
(a) Show that the map π : Hs (Ft C) → Hs (Ft C/Ft−1 C) induces a monomorphism
∞
φ : Grt Hs (C) ,→ Es,t
.
1
(b) Suppose that, for every s, there exists t0 such that Hs (Ft C) = 0 for t ≤ t0 . Show that
φ is an isomorphism.
Problem 6. Compute all pages and differentials of the homological Serre spectral sequence
for the Hopf fibration S 3 → S 2 .
Problem 7. Let B be a CW complex and p : X → B a fibration.
(a) Suppose that the fibers of p are connected. Show that the composition
2 ∼
∞
,→ Es,s
Hs (X) Hs (X)/Fs−1 Hs (X) ∼
= Hs (B)
= Es,s
is p∗ : Hs (X) → Hs (B).
(b) Suppose that B is simply connected and let i : F ,→ X be a chosen fiber of p. Show
that the composition
2
∞ ∼
Hs (F ) ∼
Es,0
= Es,0
= F0 Hs (X) ,→ Hs (X)
is i∗ : Hs (F ) → Hs (X).
Problem 8. SLet F∗ C be a filtered chain complex satisfying the assumption of Problem 5
(b), let C = t Ft C, and let n ≥ 1. Suppose that the E n page is everywhere zero, except
on two parallel lines of finite integral slope e < n in the (s, t)-plane, separated vertically by
n − e units. Draw a picture of the situation and note that the differential dn maps the upper
line to the lower line.
This happens for example if the filtration F∗ C only jumps twice, in which case e = 0 and
n is the distance between the jumps.
(a) Construct a long exact sequence of the form
d
n
n
n
· · · → Hs+1 (C) → Es+1,f
→
Es,f
(s)+n −
(s) → Hs (C) → · · · ,
where t = f (s) is the equation of the lower line.
In the rest of this problem, we look at two special cases of this phenomenon to be found in
the Serre spectral sequence. Let p : X → B be a fibration with chosen fiber i : F ,→ X.
(b) Suppose that B ' S n for some n ≥ 2. Construct a long exact sequence
i
∗
· · · → Hs+1 (X) → Hs+1−n (F ) → Hs (F ) −
→
Hs (X) → · · · .
This is called the Wang sequence.
(c) Suppose that F ' S n for some n ≥ 1 and that B is simply connected. Construct a
long exact sequence
p∗
· · · → Hs+1 (X) −
→ Hs+1 (B) → Hs−n (B) → Hs (X) → · · · .
This is called the Gysin sequence.
(d) Optional. What changes in (b) when n = 1 and in (c) when B is not simply connected?
2