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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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