# Algebraic Geometry Bucharest 1982. Proc. conf by L. Badescu, D. Popescu

By L. Badescu, D. Popescu

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The idea of this proof comes from [B16]. g. 7 Addendum. Let N , N ′ be simple Poincar´e (n + 1)-ads with ∂n−1 N = ∅, ∂n N = ∂n−1 N ′ , and ∂n−1 N ′ ∩ ∂n N ′ = ∅: suppose |N | ∩ |N ′ | = |∂n N |. Define N ′′ by ∂n−1 N ′′ = ∂n−1 N , ∂n N ′′ = ∂n N ′ , and for {n − 1, n} ⊂ α ⊂ {1, 2, . . , n}, N ′′ (α) = N (α) ∪ N ′ (α). Then N ′′ is a simple Poincar´e (n + 1)ad with ∂n−1 N ′′ ∩ ∂n N ′′ = ∅. 3)]: in the case of finite complexes, the proof given in [W21] does give a simple homotopy equivalence. 8.

Since our spheres are nullhomotopic in N , Kk (N ) is unaltered : in fact N is replaced (up to simple homotopy) by its bouquet with corresponding (k + 1)-spheres, so we acquire a free module Kk+1 (N ′ ). Dually, the exact sequence of the triple M ′ ⊂ M ′ ∪ U ⊂ N ′ reduces (using excision) to 0 → Kk+1 (N ′ , M ′ ) → Kk+1 (N, M ) → Kk (U, U ∩ M ′ ) → Kk (N ′ , M ′ ) → 0 . The module Kk (U, U ∩ M ′ ) is free, with one basis element corresponding to each handle (represented by the core of the dual handle).

Since S m−2 links S 1 once, the inclusion S m−2 ⊂ U2 induces an isomorphism of (m − 2)nd homology groups, from which it follows at once that this inclusion is a simple homotopy equivalence. In our case, condition (b) likewise reduces to the requirement that π2 (U2 − K) = 0, and hence (n 5) that π2 (C) = 0. 2 we may certainly have π2 (C) = 0, π3 (C) = 0, if n 7. Mazur’s proof of his theorem appeared in [M4]. It is a straightforward deduction from his ‘relative non-stable neighbourhood theorem’. We conclude that our example is a counterexample to this theorem also.