By Satya Deo
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Extra resources for Algebraic Topology: A Primer (Texts and Readings in Mathematics)
Then the free group FG∞ identifies with the set of all freely reduced words equipped with the operation u · v = red(uv). 15. (sets S(x)) For x in X ∪ X −1 , we define S(x) to be the subset of FG∞ consisting of all reduced words that finish with the letter x. ±1 Let us investigate the image of the set S(x−1 1 ) under the automorphism σi . 16. For every f in Aut(FG∞ ), sh(f ) maps S(x−1 1 ) into itself. −1 / Proof. Let us consider an arbitrary element of S(x−1 1 ), say wx1 , with w ∈ −1 ). Assume S(xi ).
15. (sets S(x)) For x in X ∪ X −1 , we define S(x) to be the subset of FG∞ consisting of all reduced words that finish with the letter x. ±1 Let us investigate the image of the set S(x−1 1 ) under the automorphism σi . 16. For every f in Aut(FG∞ ), sh(f ) maps S(x−1 1 ) into itself. −1 / Proof. Let us consider an arbitrary element of S(x−1 1 ), say wx1 , with w ∈ −1 ). Assume S(xi ). By construction, we have sh(f )(wx1 ) = red(sh(f )(w) x−1 1 −1 −1 sh(f )(wx−1 / S(x−1 1 ) ∈ 1 ). Then the final letter x1 in sh(f )(w)x1 is cancelled by some letter x1 occurring in sh(f )(w).
5) hold. We prove now that (Q, ∨) is an LD-system. Let a, b, c be arbitrary elements of Q. 7) a ∧ (a ∨ b) ∧ (a ∨ b ∨ c) = (a ∧ a ∨ b) ∧ (a ∧ a ∨ b ∨ c) = b ∧ (b ∨ c) = c, a ∧ (a ∨ b) ∧ ((a ∨ b) ∨ (a ∨ c)) = a ∧ a ∨ c = c, which implies a∨(b∨c) = (a∨b)∨(a∨c) as ad∧a∨b and ad∧a are injective. We prove finally that ∧ is left distributive with respect to ∨. Indeed, we have a ∨ (a ∧ b) ∨ (a ∧ (b ∨ c)) = (a ∨ a ∧ b) ∨ (a ∨ a ∧ b ∨ c) = b ∧ (b ∨ c) = c, a ∨ (a ∧ b) ∨ ((a ∧ b) ∧ (a ∧ c)) = a ∨ a ∧ c = c, which implies a∧(b∨c) = (a∧b)∨(a∧c) as ad∨a∧b and ad∨a are injective.