The Kervaire invariant of immersions
The authors study E. H. Brown's generalized Kervaire invariant \(K: \Omega\) \({}_*^ G\to {\mathbb{Z}}/8\) where \(\Omega_*^ G\) is an appropriate cobordism theory of immersed submanifolds in Euclidean space. Let \(G=O(1)\) or SO(2). The main results are: (1) unless \(n=1\) or 3 and \(G=O(1)\), the Kervaire invariant of a class \(x\in \Omega^ G_{2n}\) is nonzero if and only if x is represented in the Adams spectral sequence by a class in \(Ext_ A(H^*MG,{\mathbb{Z}}/2)\) in the image of multiplication by \(h^ 2_ j\) and (2) let \(u_*: Ext_ A({\mathbb{Z}}/2,{\mathbb{Z}}/2)\to Ext_ A(H^*MSO(2),{\mathbb{Z}}/2)\) be induced by the Thom class \(u: S^ 0\to MSO(2)\). Then for every \(j\geq 1\), there exists an element \(\theta_ jSO(2)\) represented by \(u_*(h^ 2_ j)\); thus \(\theta_ jS0(2)\) has nonzero Kervaire invariant in this theory. The result given by (1) generalizes results of \textit{W. Browder} [Ann. Math., II. Ser. 90, 157-186 (1969; Zbl 0198.285)]
cobordism theory of immersed submanifolds
scientific article; zbMATH DE number 3921558
scientific article