On isotopy classification of embeddings of \(n\)-manifolds in \(\mathbb{RP}^{2n}\) (Q1313916)

The isotopy group of embeddings, in the metastable range, of an orientable closed manifold \(M\) in the real projective space of even dimension, is computed in terms of the homotopy groups of \(M\). The computation is based on the approach to embedding problems in normal bordism framework, which converts the calculations into those of singular homology groups of a certain elaborate space pair with twisted integers as coefficients. It forms part of a project which applies the normal bordism method to the existence and isotopy classification of differential embeddings of manifolds in manifolds in the metastable range. An example of the type of results obtained is the following Theorem: Suppose that \(n \geq 4\). Let \(M^ n\) be a closed connected manifold and let \(f : M^ n \to \mathbb{RP}^{2n}\) be a map which induces an epimorphism \(f_ * : \pi_ 1 (M) \to \pi_ 1 (\mathbb{RP}^ 2n) = \mathbb Z_ 2\). Assume that there is no element in \(\ker f_ *\) which is represented by orientation reversing loops (e.g. when \(M\) is orientable). Then, \[ [M^ n \subset \mathbb{RP}^{2m}]_ f = \begin{cases} \bigl( \ker f_ */ [\ker f_ *, \ker f_ *] \bigr) \otimes \mathbb Z_ 2 \quad & \text{if}\;n \;\text{is even}, \\ \ker\;f_ * / [\ker\;f_ *,\ker\;f_ *]G, \quad & \text{if}\;n\;\text{is odd}, \end{cases} \] where \(G\) stands for the subgroup of \(\ker\;f_ *\) generated by the squares of elements which are not in \(\ker\;f_ *\), and where \([M \subset N]_ f\) denotes \(\pi_ 1 (N^ M, E,f)\) for \(E\) the space of differential embeddings and \(f : M \to N\).