EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 2, 28 October 2005

Solutions due on Monday, 7th of November.

Exercise 2.1. Show that for any pair of spaces (X, A) the diagram

\xymatrix{H^{n}(A; R) \ar[d]^{φ} \ar[r]^{∂^{n}} & H^{n+1}(X, A; R) \ar[d]^{φ} \\ hom_{R}(H_{n}(A; R), R) \ar[r]^{∂_{n}^{∗}} & hom_{R}(H_{n+1}(X, A; R), R)}

is commutative. Here ∂^{n} is the connecting homomorphism for cohomology, and ∂_{n} is the connecting homomorphism for homology.

Exercise 2.2. Let p be a prime number and consider the short exact sequence of ℤ-modules 0 → ℤ ⟶^{p} ℤ → 𝔽_{p} → 0. Applying S_{∗}(X; ℤ) ⊗_{ℤ} − to this sequence we obtain a short exact (why ?) sequence of chain complexes

0 → S_{∗}(X; ℤ) ⟶^{p} S_{∗}(X; ℤ) → S_{∗}(X; 𝔽_{p}) → 0.

Taking homology we obtain a long exact sequence

⋯ ⟶^{∂_{n+1}} H_{n}(X; ℤ) ⟶^{i_{n}} H_{n}(X; ℤ) ⟶^{k_{n}} H_{n}(X; 𝔽_{p}) ⟶^{∂_{n}} H_{n−1}(X; ℤ) ⟶^{i_{n−1}} · · · .

For all n ⩾ 1, we define the mod p Bockstein homomorphism

β_{n} : H_{n}(X; 𝔽_{p}) → H_{n−1}(X; 𝔽_{p})

in homology by β_{n} = k_{n−1} ∘ ∂_{n}. Show that

(a) i_{n} is multiplication by p on the ℤ-module H_{n}(X; ℤ),

(b) β_{n} is a homomorphism, and it is natural with respect to continuous maps,

(c) β_{n−1} ∘ β_{n} = 0. Compute the mod p Bockstein homomorphisms for the Moore space M(n, ℤ/p), where n ⩾ 2.

Exercise 2.3. Define the mod p Bockstein homomorphism

β^{n} : H^{n}(X; 𝔽_{p}) → H^{n+1}(X; 𝔽_{p})

in cohomology (this time apply the functor hom_{ℤ}(S_{∗}(X; ℤ), −) to the short exact sequence 0 → ℤ ⟶^{p} ℤ → 𝔽_{p} → 0). Phrase and prove its properties (dual to the properties of the homology Bockstein above), and compute it for the Moore space M(n, ℤ/p), n ⩾ 2.

Typeset by AMS-TEX