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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.