Adams spectral sequence and higher torsion in \(MSp\)
The symplectic cobordism ring \(MSp_*\) presents a remarkable computational challenge for which the most naive questions require deep insight. Through the efforts of the second author, \(MSp_n\) is known out to \(n=100\). As a ring, \(MSp_*\) has only 2-torsion -- however, the authors have shown [Can. J. Math. 46, 485-516 (1994; Zbl 0854.55005)] that there are nontrivial elements of all orders \(2^k\) in \(MSp_*\). Where there is torsion, there are Toda brackets. The authors' main results treat certain families of Toda brackets built on the Ray elements \(\phi_k \in MSp_{8k-3}\). In particular, they construct elements in \(MSp_{4L+1}\) of order at least \(2^{h(s)}\) where \(h(s) =[(s+1)/2]-2\), and \(s\geq 7\). The analysis is through the Adams spectral sequence (to obtain upper bounds on order) and the Adams-Novikov spectral sequence (to obtain lower bounds). Rich use is made of the companion bordism theories, \(MSp^{\Sigma_n}\) and \(MSp^{\widehat\Sigma_n}\) where \(\Sigma_n = (P_1,\dots,P_n)\) and \(\widehat\Sigma_n =(P_2,\dots, P_n)\) are sets of singularities; \(P_1 = \eta\). In \S 2 the May spectral sequence is used to compute \(MSp_*^{\Sigma_n}\) and \(MSp_*^{\widehat\Sigma_n}\). Explicit input for the Adams spectral sequence emerges that is applied in \S\S 3-5 to construct the higher torsion elements, and in \S 6 they are analyzed in \(MSp_*\) after applying the appropriate Bocksteins. As demanded by the study of symplectic cobordism, the authors display virtuosic means to this rich, computationally intense study. For example, the element of order 8 produced by their methods appears in the 16,377 stem. They conjecture that the elements they construct are indecomposable in certain cases, bringing the first element of order 8 into view in the 729 stem.
scientific article; zbMATH DE number 911932