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Embedding, compression and fiberwise homotopy theory. (English) Zbl 1003.57028

This paper considers the problem of determining whether, given Poincaré duality spaces \(M\) and \(X\), a Poincaré embedding of \(M \times I\) in \(X\times I\) arises from an embedding of \(M\) in \(X\). The main result of the paper gives a necessary and sufficient condition in the meta-stable range. The methods are homotopy theoretic rather than geometric (unlike results in this direction from the 1970’s). An important corollary of the work is that for a simply-connected Poincaré duality space \(X\) the diagonal map \(X\rightarrow X\times X\) has an embedded thickening; having a \({\mathbb Z}_{2}\)-equivariant analog of this would give an unstable tangent bundle for Poincaré duality spaces.

MSC:

57P10 Poincaré duality spaces
57R40 Embeddings in differential topology
55R70 Fibrewise topology
57Q35 Embeddings and immersions in PL-topology
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References:

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