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A new invariant and parametric connected sum of embeddings. (English) Zbl 1145.57019
For a polyhedron $$N$$ one denotes by $$\widetilde N$$ the deleted product
$\widetilde N=\{(x, y)\in N\times N;\;x\not= y\};$
it is the configuration space of two distinct points in $$N$$. The symbol $$\pi^{m-1}_{\text{eq}}(\widetilde N)$$ denotes the set of all $$\mathbb Z_2$$-equivariant homotopy classes of equivariant continuous maps $$f: \widetilde N \to S^{m-1}$$ where the sphere $$S^{m-1}$$ is equipped with the antipodal involution and the involution $$T: \widetilde N\to \widetilde N$$ acts by permuting the points, $$T(x, y)=(y, x)$$. Every embedding $$N\to \mathbb R^m$$ determines a canonical Gauss map $$f: \widetilde N\to S^{m-1}$$; thus we obtain a map
$\alpha : \text{Emb}^m(N) \to \pi^{m-1}_{\text{eq}}(\widetilde N)$
called the $$\alpha$$-invariant. Well-known theorems of A. Haefliger and others give conditions under which the map $$\alpha$$ is bijective. In the paper under review the author proves several related results. One of the main theorems states that for a closed $$k$$-connected orientable $$n$$-manifold $$N$$, where $$k\leq n/3$$, the map
$\alpha : \text{Emb}^m(N) \to \pi^{m-1}_{\text{eq}}(\widetilde N) \quad \text{with}\quad m= (3/2)\cdot n - k/2+1$
is surjective and the preimage of any homotopy class is in 1-1 correspondence with a quotient of $$H_{k+1}(N)$$.

MSC:
 57Q35 Embeddings and immersions in PL-topology 57Q37 Isotopy in PL-topology 55S15 Symmetric products and cyclic products in algebraic topology 55Q91 Equivariant homotopy groups 57R40 Embeddings in differential topology
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