Some new immersions and nonimmersions of \(2^r\)-torsion lens spaces
Let \(L^{2n+1}(2^r)\) be the \((2n+1)\)-dimensional \(2^r\)-torsion lens space, for positive integers \(n\) and \(r\). This work deals with the immersion of these spaces in Euclidean spaces. One relevant number associated to the immersion problem of a manifold of dimension \(n\) in Euclidean spaces is \(\alpha(n)\), which is the number of \(1's\) in the binary expansion of \(n\). The author obtains some results for small values of \(\alpha(n)\), namely for \(\alpha(n)=2\) and \(\alpha(n)=3\), where most of them are optimal. We exemplify with two such results: Theorem (1.1) If \(n\) is odd and \(\alpha(n)=2\), then \(L^{2n+1}(4)\) immerses in \(\mathbb{R}^{4n-2\alpha(n)+1}\) and, for \(r\geq 3\), \(L^{2n+1}(2^r)\) does not immerse in \(\mathbb{R}^{4n-2\alpha(n)+1}\). Theorem(1.4) If \(n\equiv 2 \bmod 4\), \(\alpha(n)=3\) and \(r\geq 2\), then \(L^{2n+1}(2^r)\) does not immerse in \(\mathbb{R}^{4n-2\alpha(n)}\). Modified Postnikov towers have already been used for this type of problem with success. In order to prove the results he uses a modified version of the Postnikov tower of the map \(\widetilde{BU}(2n-3) \to BU\), where this map is the induced fibration from the \(BO(2n-3) \to BO\) map by the retraction \(r:BU \to BO\). This modified Postnikov tower is obtained using Adams spectral sequence (ASS) of the stunted real projective space. One should decide if a particular map lifts or not to a certain stage of this modified Postnikov tower. To solve this problem the author uses complex connective \(K\)-theory.
scientific article; zbMATH DE number 2063649