Homotopy classification of singular links of type \((1,1m;3)\) with \(m>1\) (Q1608106)

A singular link of type \((m_1, m_2, \dots, m_r; n)\) is a collection of continuous maps \(f_i: S^{m_i}\to S^n\) (where \(i=1, \dots, r\)) with mutually disjoint images. Two singular links are homotopic if there exists a homotopy connecting them such that at any moment different link components stay disjoint. The paper suggests a very nice explicit homotopy classification of singular links of type \((1,1,m;3)\) where \(m>1\). Let \({\mathfrak H}_m\) denote the set of the isotopy classes. The linking number between the first and the second component gives a map \(\lambda: {\mathfrak H}_m \to \mathbb Z\). The author proves: (1) the isotopy types of singular links having nonzero linking number are determined uniquely by the linking number; (2) the isotopy classes of singular links having zero linking number are in one-to-one correspondence with the set of free homotopy classes of continuous maps \(S^m \to S^1\vee S^1\vee S^2\).