An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number (Q2902456)

In this paper, the author introduces the notion of an \(e\)-manifold and a rational-valued invariant \(\sigma\) for \(6\)-dimensional closed \(e\)-manifolds. An \(n\)-dimensional (closed) \(e\)-manifold is a triple \((Z,X,e)\), where \(Z \supset X\) are smooth oriented compact (closed) manifolds of dimensions \(n\) and \(n-3\) such that \(X\) is properly embedded in \(Z\), and \(e\), called an \(e\)-class, is a cohomology class in \(H^2(Z-X, \mathbb{Q})\) such that \(e|_{S(\nu_X)}=e(F_X)\), where \(S(\nu_X)\) is the total space of the unit sphere bundle \(\rho_X: S(\nu_X) \to X\) associated with the normal bundle \(\nu_X\) of \(X\), and \(e(F_X)\) is the Euler class of the vertical tangent subbundle of \(S(\nu_X)\) with respect to \(\rho_X\). The main result is the existence and the uniqueness of the invariant \(\sigma\). The existence of \(\sigma\) is shown by calculating the cobordism group of \(6\)-dimensional \(e\)-manifolds. The uniqueness is shown by defining \(\sigma\) in terms of cobordisms of \(e\)-manifolds and the signature of \(4\)-manifolds; actually, \(\sigma\) is uniquely characterized by the following two axioms: (Axiom 1) the invariant \(\sigma\) is additive, i.e. for 6-dimensional closed \(e\)-manifolds \(\alpha\) and \(\alpha^\prime\), \(\sigma(-\alpha)=-\sigma(\alpha)\) and \(\sigma(\alpha \coprod \alpha^\prime)=\sigma(\alpha)+\sigma(\alpha^\prime)\), where we define \(-\alpha=(-W, -V, e)\) for \(\alpha=(W,V,e)\), and (Axiom 2) if a 6-dimensional closed \(e\)-manifold \((W,V,e)\) bounds a \(7\)-dimensional \(e\)-manifold \((Z,X,e^\prime)\), i.e. \(\partial Z=W\), \(\partial X=V\) and \(e^\prime|_{W-V}=e\), then \(\sigma=\mathrm{Sign}\,X\).NEWLINENEWLINEThe author compares the invariant \(\sigma (W,V,e)\) with other invariants of embeddings of codimension three. When \(W=S^6\) and \(V\) is a smoothly embedded \(3\)-sphere, \((S^6, V)\) admits a unique \(e\)-class where \(e/2\) is the Poincaré dual of a Seifert surface of \(V\), and the invariant \(\sigma\) equals \(-8\) times Haefliger's invariant of \((S^6, V)\); (see \textit{A. Haefliger} [Ann. Math. (2) 83, 402--436 (1966; Zbl 0151.32502)]). Haefliger's invariant has been generalized to the invariant for embeddings of integral homology 3-spheres in \(S^6\) by \textit{M. Takase} [Int. J. Math. 17, No. 8, 869--885 (2006; Zbl 1113.57013)]. The invariant \(\sigma\) recovers Takase's invariant: When \(W=S^6\) and \(V\) is a smoothly embedded integral homology \(3\)-sphere, \((S^6, V)\) admits a unique \(e\)-class and the invariant \(\sigma\) equals \(-8\) times Takase's invariant of \((S^6, V)\). The invariant \(\sigma\) also recovers Milnor's triple linking number of oriented algebraically split \(3\)-component links in \(\mathbb{R}^3\), where we adopt the definition of Milnor's triple linking number given by perturbative Chern-Simons theory, cf. \textit{D. Altschuler} and \textit{L. Freidel} [Commun. Math. Phys. 187, No. 2, 261--287 (1997; Zbl 0949.57012)].