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Poincaré duality embeddings and fiberwise homotopy theory. (English) Zbl 0928.57028
A map $$f:K\to X$$ from a finite complex $$K$$ into an $$n$$-dimensional Poincaré duality space $$X$$ is said to Poincaré embed if $$f$$ extends to a homotopy equivalence $$K\cup_A C\cong X$$ such that each piece of the decomposition satisfies Poincaré duality. This means that $$(K,A)$$ and $$(C,A)$$ are Poincaré pairs having the same dimension. Moreover, the fundamental class in each case is included from the fundamental class of $$X$$. Thus, to specify the Poincaré embedding we have to find the complement $$C$$ and the way in which it is glued to $$K$$ to give $$X$$. For such a decomposition of $$X$$ to exist it is necessary that the homology of $$K$$ vanishes in degrees greater than $$n$$. We write $$\text{hodim} K\leq k$$ if $$K$$ is homotopy equivalent to a CW complex of dimension $$\leq k$$. The author works with the codimension greater than 3 and $$k \leq n-3$$. The author poses a question: Given a map $$f:\kappa\to X$$, when does it Poincaré embed? He gives a partial answer by proving that if $$f:\kappa\to X$$ is $$r$$-connected, then $$f$$ Poincaré embeds provided that $$k\leq n$$ and $$r\geq 2k-n+2$$.

##### MSC:
 57Q35 Embeddings and immersions in PL-topology 57P10 Poincaré duality spaces
Poincaré embed
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