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Poincaré duality embeddings and fibrewise homotopy theory. II. (English) Zbl 1030.57036

Suppose that a finite complex \(K\) is obtained form the finite complex \(L\) by adding cells of dimension at most \(k\) and that \((X, \partial X)\) is a Poincaré duality pair of formal dimension \(n\). Suppose that a map \(f: K \rightarrow X\) is such that \(f \mid L : L \rightarrow \partial X\) is the given Poincaré embedding and \(k \leq n-3\) and \(r \geq 2k - n + 2\). This paper uses purely homotopic methods to show that \(f\) is homotopic to a Poincaré embedding. This improves the reviewer’s earlier result on general position in the Poincaré duality category [Invent. Math. 24, 311-334 (1974; Zbl 0263.57002)]. In particular the author is able to show that every two-connected Poincaré complex has a handle decomposition.

MSC:

57P10 Poincaré duality spaces
55R70 Fibrewise topology

Citations:

Zbl 0263.57002
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