EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 12, January 19 2006

Solutions due on Thursday, 26th of January.

Exercise 12.1. Let n ⩾ 2 and let M_{n} be the Moore space with

H̃_{k}(M_{n}; ℤ) = \{ \begin{matrix}ℤ/2 & k = n, \\ 0 & k ≠ n, \end{matrix}

constructed as the homotopy cofibre of a map S^{n} → S^{n} of degree 2. Since M_{n} = ΣM_{n−1}, the set [M_{n}, M_{n}] of homotopy classes of pointed maps is a group. Show that in this group the order of the identity of M_{n} in strictly larger than 2. Hint: view the map 2id_{M_{n}} as the composition f∧id : S^{1}∧M_{n−1} → S^{1}∧M_{n−1}, where f has degree 2. The homotopy cofibre of this map is ℝP^{2} ∧ M_{n−1}. If 2id_{M_{n}} ≃ 0 we would have

ℝP^{2} ∧ M_{n−1} ≃ M_{n} ∨ ΣM_{n}.

Show that this is not possible by showing that Sq^{2} acts non-trivially on the cohomology of ℝP^{2} ∧ M_{n−1} (use the Cartan formula and Exercise 2.3, where Sq^{1} = β was computed for M_{n}). Remark: in fact [M_{n}, M_{n}] ≅ ℤ/4.

Exercise 12.2. Let u be a class in H^{2}(X, A; 𝔽_{2}) such that Sq^{1}(u) = 0. Show that for all k ⩾ 1 and n ⩾ 1 we have

Sq^{2n+1}(u^{k}) = 0 and Sq^{2n}(u^{k}) = \binom{k}{n}u^{k+n}.

Exercise 12.3. Let A(2) be the sub-algebra of the Steenrod algebra A_{2} generated by Sq^{0}, Sq^{1}, Sq^{2}. Show that A(2) has dimension 8 as 𝔽_{2}-vector space, and describe its multiplicative structure.

Exercise 12.4. Express Sq^{2^{i}}Sq^{2^{i}} as a linear combination of admissible monomials.

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