Enumerating embeddings of homologically (k-1)-connected n-manifolds in Euclidean (2n-k)-space
The main result of this paper, an extension of a 1962/63 result of Haefliger, is the following theorem. If \(2\leq k\leq (n-4)/2\) and M is a homologically (k-1)-connected (\~H\({}_ i(M; {\mathbb{Z}})=0\), \(i\leq k-1)\) closed, connected, differentiable n-manifold with (n-k)-th normal Stiefel-Whitney class zero then the set \([M\subset {\mathbb{R}}^{2n-k}]\) of isotopy classes of differentiable embeddings of M into Euclidean (2n-k)- space is given as follows: (i) If \(k=2\) and M is not a spin manifold, then \[ [M\subset {\mathbb{R}}^{2n-2}]=H^{n-3}(M; {\mathbb{Z}}_ 2)\quad if\quad n\equiv 0 (4);\quad =H^{n-3}(M; {\mathbb{Z}}_ 2)\times {\mathbb{Z}}_ 2\quad if\quad n\equiv 2 (4); \] \[ =H^{n-3}(M; {\mathbb{Z}})\times H^{n-2}(M; {\mathbb{Z}}_ 2)\quad if\quad n\equiv 1 (4),\quad w_ 3\neq 0; \] \[ =H^{n-3}(M; {\mathbb{Z}})\times H^{n-2}(M; {\mathbb{Z}}_ 2)\times {\mathbb{Z}}_ 2\quad if\quad n\equiv 1 (4),\quad w_ 3=0\quad or\quad n\equiv 3 (4). \] (ii) If \(k\geq 3\) or M is a spin manifold, then \[ [M\subset {\mathbb{R}}^{2n-k}]=H^{n-k- 1}(M; {\mathbb{Z}}_ 2)\quad if\quad n-k\equiv 0 (4); \] \[ =H^{n-k-1}(M; {\mathbb{Z}})\times H^{n-k}(M; {\mathbb{Z}}_ 2)\times H^{n-k}(M; {\mathbb{Z}}_ 2)/Sq^ 2\rho_ 2H^{n-k-2}(M; {\mathbb{Z}})\quad if\quad n-k\equiv 1 (4); \] \[ =H^{n-k-1}(M; {\mathbb{Z}}_ 2)\times H^{n-k}(M; {\mathbb{Z}}_ 2)/Sq^ 2H^{n-k-2}(M; {\mathbb{Z}}_ 2)\quad if\quad n-k\equiv 2 (4); \] \[ =H^{n-k-1}(M; {\mathbb{Z}})\times H^{n-k}(M; {\mathbb{Z}}_ 2)\times H^{n-k}(M; {\mathbb{Z}}_ 2)/(Sq^ 1H^{n-k-1}(M; {\mathbb{Z}}_ 2)+Sq^ 2\rho_ 2H^{n-k-2}(M; {\mathbb{Z}})\quad if\quad n-k\equiv 3 (4). \] The method of proof rests on first describing \([M\subset {\mathbb{R}}^{2n-k}]\) as a set of homotopy classes of maps (liftings) into \(M^*\), the reduced symmetric product of M, and then computing this set in terms of the cohomology of \(M^*\).
homologically (k-1)-connected differentiable n-manifold
isotopy classes of differentiable embeddings
scientific article; zbMATH DE number 3964996