Bordism of elementary abelian groups via inessential Brown-Peterson homology (Q2826643)

From the author's introduction: ``Conner and Floyd introduced and studied bordism groups of free oriented \(G\)-manifolds for finite groups \(G\). This is equivalent to the oriented bordism of \(BG\), the classifying space of \(G\). Of fundamental interest are the elementary abelian groups \(G = (\mathbb Z/p)^n\), where \(p\) is a prime.NEWLINENEWLINERecall that \(\Omega ^{\mathrm{SO}}_* (B\mathbb Z/p)\), the oriented bordism of free \(\mathbb Z/p\)-manifolds, is generated as a module over \(\Omega ^{\mathrm{SO}}_*= \Omega ^{\mathrm{SO}}_* (pt.)\) by elements \(z_m \in \Omega ^{\mathrm{SO}}_{2m+1}(B \mathbb Z/p), m \geq 0\), represented by classifying maps \(L^{2m+1}\rightarrow B\mathbb Z/p\) of standard lens spaces \(L^{2m+1} = S^{2m+1}/(\mathbb Z/p)\). These correspond to spheres \(S^{2m+1}\) equipped with standard free \(\mathbb Z/p\)-actions of weight \((1,\dots, 1)\).NEWLINENEWLINEFor odd \(p\), the oriented bordism of classifying spaces of elementary abelian \(p\)-groups fits into Landweber's exact Künneth sequence.NEWLINENEWLINEDespite a number of related many contributions, it remained unclear how to construct a splitting of Landweber's exact sequence for all \(n\).''NEWLINENEWLINE In this paper the author settles this problem.NEWLINENEWLINEAlso he applies his results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.