Simplicial Structures in Topology

VI.3 Homotopy Groups 231
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4. Let f ∈ Top ∗((A,a0),(B,b0)) be a given map; construct the space of based functions
PB = {λ : I → B | λ(0)=b0}
(space of paths beginning at b0) andthemap
¯g
¯f
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g: PB −→ B , λ ↦→ λ(1).
Then, construct the pullback diagram determined by f and g to obtain the space
Cf = {(a,λ) ∈ A × PB | f (a)=g(λ )}
with the maps ¯f and ¯g. Prove that for every n the sequence of homotopy groups
is exact in πn(A,a0).
πn(Cf ,∗) πn( ¯g)
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πn(A,a0) πn( f )
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πn(B,b0)
5. Amap f : A → B is a fibration if,
(∀X ∈ Top)(∀g ∈ Top(X,A))(∀H ∈ Top(X × I,B)) | Hi0 = fg,
there exists G: X ×I → A such that G(−,0)=g and fG= H. Prove that a projection
map f : X ×Y → X is a fibration. Moreover, prove that the map p: PB → B of
Exercise 4 above is a fibration.
6. Prove that, if the map f : (A,a0) → (B,b0) of Exercise 4 above is a fibration, then
the space Cf is of the same homotopy type as the fiber f −1 (b0) over b0.
7. Prove that, if f : (A,a0) → (B,b0) is a fibration, there exists an (left) infinite exact
sequence of homotopy groups
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πn( f −1 (b0),a0)
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πn−1( f −1 (b0),a0)
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πn(A,a0)
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πn−1(A,a0)
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πn(B,b0)
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