On properties of cobordism groups of skew-framed immersions (Q2202796)

The authors relate the cobordism group of skew framed immersions of \(3\)-manifolds into \({\mathbb R}^{8}\) to the obstruction group for immersions of \({\mathbb R}P^{10}\) into \({\mathbb R}P^{15}\). The argument depends on the use of quaternions to define operations of twisting and untwisting for skew framings in the dimensions at hand. These methods are used to give a geometrical proof that \({\mathbb R}P^{10}\) does not immerse in \({\mathbb R}P^{15}\). There are numerous low dimensional examples given to illustrate the ideas of the paper.