Oriented bordism of codimension one immersions of 7-manifolds and relative Thom polynomials (Q455640)

The main result of this paper is to present a formula for the isomorphism \(\mathsf{SI}(7,1) \approx \mathbb{Z}_{240}\), where \(\mathsf{SI}(7,1)\) is the group of oriented bordism classes of codimension one immersions of \(7\)-manifolds in \(\mathbb{R}^8.\) This group can be identified with the stable homotopy group \(\pi_{7}^{S}\) of spheres. The oriented bordism class of an immersion \(f: M^7 \looparrowright \mathbb{R}^8\), from a closed oriented \(7\)-manifold \(M^7\) into \( \mathbb{R}^8,\) is determined by the stably framed cobordism class \([M^7, \pi_f]\) of \((M^7, \pi_f)\), where \(\pi_f\) is the stable framing of \(M^7\) induced by the standard trivialisation of \(T\mathbb{R}^8\). The author presents a formula to compute a relative Pontryagin number in terms of a singular Seifert surface for the immersion \(f\) and an ``isomorphism \(\Upsilon : \mathsf{SI}(7,1) \longrightarrow \mathbb{Z}_{240}\) is given by \[ \Upsilon([f: M^7 \looparrowright \mathbb{R}^8]) = \frac{15[{\Sigma}^2{(F)}]^2 + 4 \#\Sigma_{FR}(F)}{216} \,\,\, \,\, (mod \,\,240), \] where \(F:V^8 \longrightarrow \mathbb{R}^8\) is a singular Seifert surface for \(f\) from a spin manifold \(V^8\) whose spin structure restricted to the boundary coincides with the spin structure on \(M^7\) induced by \(f\), \(\#\Sigma_{FR}(F)\) is the algebraic number of the \(0\)-cycle \(\langle(x^2+y^3, xy^2)- 2 I V_4\rangle\) in \(V^8\), where \(I V_4 = (x^2+y^2,x^4)\) and \( [{\Sigma}^2{(F)}]^2\) is the self-intersection number of the cycle \({\Sigma}^2{(F)}\) in \(V^8\).''