High-codimensional knots spun about manifolds (Q2464757)

This paper studies the generators of the Haefliger knot group \(C^q_n\), that is the group of isotopy classes of smooth embeddings of \(S^n\) in \(S^{n+q}\), for \(n=q=3\). In particular, the authors give a new description of a generator of \(C^3_3\), different from that given by \textit{R. Budney} in [``A family of embedding spaces.'' Iwase, Norio (ed.) et al., Proceedings of the conference on groups, homotopy and configuration spaces, University of Tokyo, Japan, July 5--11, 2005 in honor of the 60th birthday of Fred Cohen. Coventry: Geometry \& Topology Publications. Geometry and Topology Monographs 13, 41--83 (2008; Zbl 1158.57035)]. The main tool in this construction is the notion of \textit{spinning about a submanifold}, introduced in [\textit{D. Roseman}, Topology Appl. 31, 225--241 (1989; Zbl 0683.57010)]. Indeed, the authors start taking the \(2\)-torus \(T^2\) standardly embedded in \(S^3\), which is further standardly embedded in \(S^6\). Moreover, at each point of \(T^2\), they consider the normal \(4\)-disk in \(S^6\) and the normal \(1\)-disk in \(S^3\). Then they deform-spin \(S^3\subseteq S^6\) about \(T^2\) using a certain map \(\Phi\) from \(T^2\) to the space of long knots in \(4\)-space (smooth embeddings of \(\mathbb{R}^1\) in \(\mathbb{R}^4\) which are standard outside \(D^1\)), considered by Budney. To this aim, at each point \(\tau\in T^2\), the standard disk pair \((D^4,D^1)\) is replaced by a new disk pair \((D^4,\Phi(\tau)(D^1))\). The main result is that the so obtained \(3\)-sphere smoothly embedded in \(S^6\) represents a generator of \(C^3_3\). The authors point out that the same arguments work for higher-dimensional Haefliger knot groups \(C^{2k+1}_{4k-1}\), \(k\geq 2\).