Download Elements of Homotopy Fall 2008 Prof. Kathryn Hess Series 13 Let B

Elements of Homotopy
Prof. Kathryn Hess
Fall 2008
Series 13
Let B be a path connected based space and recall that if p : E−→B is a
fibration (or locally trivial bundle) with nonempty fiber F , then the sequence
F −→E−→B fits into a long exact sequence of the form
···
πk (F )
πk (E)
πk (B)
···
πk−1 (F )
π0 (E)
Exercise 1. Let p be prime to q. Recall the following fibrations (or locally trivial bundles) p : E−→B with fiber F , which we display in the form
F −→E−→B:
Z−→R−→S 1 ,
Z2 −→R2 −→T,
Z/2Z−→S n −→RP n ,
1
3
(n = 1),
n
(n ≥ 1),
S −→S −→S ,
1
S −→S
2n+1
(n ≥ 1),
2
−→CP ,
3
Z/pZ−→S −→L(p, q).
Use the associated long exact sequence above together with earlier results
to prove the following:
(a) π1 (S 1 ) ∼
= Z and πk (S 1 ) = 0 for k 6= 1.
∼
(b) π1 (T ) = Z2 and πk (T ) = 0 for k 6= 1.
(c) If n ≥ 2, then π1 (RP n ) ∼
= Z/2Z and πk (RP n ) ∼
= πk (S n ) for k 6= 1.
3
2
(d) πk (S ) ∼
= πk (S ) for k ≥ 3.
(e) There is a short exact sequence
0
π2 (S 3 )
π2 (S 2 )
0
Z
of abelian groups.
(f) πk (CP n ) ∼
= πk (S 2n+1 ) for k ≥ 3.
(g) There is a short exact sequence
0
π2 (S 2n+1 )
π2 (CP n )
Z
0
of abelian groups.
(h) π1 L(p, q) ∼
= πk (S 3 ) for k 6= 1.
= Z/pZ and πk L(p, q) ∼
Exercise 2. Let B, E, X be spaces. Prove the following:
(a) The map p : ∅−→B is a fibration.
(b) The map p : E−→∗ is a fibration.
(c) The map i : ∅−→X is a cofibration.
Note that the empty space ∅ has no path components; i.e. π0 (∅) = ∅.
Exercise 3. Let p : E−→B be a fibration. Prove the following:
1
∗.
(a) If A ⊆ E is a path component of E, then p(A) ⊆ B is a path component of B.
(b) If B is path connected and E is nonempty, then p is a surjective
map.
Exercise 4. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E
the fiber over b0 . Assume F is nonempty. Denote by i : F −→E the inclusion
map. Prove the following:
(a) If B is path connected, then the induced map π0 (i) : π0 (F )−→π0 (E)
on path components is a surjection.
The following will be useful for studying the first several maps in the long
exact sequence associated to a fibration (or locally trivial bundle).
Exercise 5. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E
the fiber over b0 . Assume F is nonempty. A right action of π1 (B, b0 ) on the
set π0 (F ) is defined by lifting loops as follows:
be (1)].
(1)
π0 (F ) × π1 (B, b0 )−→π0 (F ), ([e0 ], [λ]) 7−→ [e0 ][λ] := [λ
0
be : [0, 1]−→E is a lift of the loop λ : [0, 1]−→B such that λ
be (0) = e0 ;
Here, λ
0
0
b
in other words, λe0 is a map which makes the diagram
{0}
⊆
[0, 1]
e0
E
∃
p
be
λ
0
λ
B
commute.
(a) Prove that (1) is a well-defined function.
(b) Prove that (1) defines a right π1 (B, b0 )-action on the set π0 (F ) of
path components of F .
(c) If B is path connected, prove that the induced map π0 (F )−→π0 (E) is
surjective and partitions the set π0 (F ) into π1 (B, b0 )-orbits; conclude
that π0 (E) is the orbit space π0 (F )/π1 (B, b0 ).
(d) Prove that the isotropy subgroup of [e0 ] ∈ π0 (F ) is the image of
π1 (p) : π1 (E, e0 )−→π1 (B, b0 ).
Consider solid commutative diagrams of the form
∼
=
I × {0} ∪ {0} × I ∪ I × {1}
I × {0}
⊆
⊆
I ×I
I ×I
ξ
E
p
B
in Top, and use existence of a lift ξ.
Here are some references for this material: [1, Chapter 11], [2, Chapter 9],
[3, Chapter 2.3], [4, Section 3.2].
References
[1] Brayton Gray. Homotopy theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975. An introduction to algebraic topology, Pure and Applied
Mathematics, Vol. 64.
[2] J. P. May. A concise course in algebraic topology. Chicago Lectures in
Mathematics. University of Chicago Press, Chicago, IL, 1999. Available at:
http://www.math.uchicago.edu/ may/ .
[3] Edwin H. Spanier. Algebraic topology. Springer-Verlag, New York, 1981. Corrected
reprint.
[4] Tammo tom Dieck. Algebraic topology. European Mathematical Society, Zurich, 2008.