Cutting and pasting of families of submanifolds modeled on Z\(_2\)-manifolds (Q1429090)

Let \(P\) and \(Q\) be \(m\)-dimensional compact manifolds with boundary \(\partial P\) and \(\partial Q\), respectively, and \(\varphi : \partial P \to \partial Q\) be a diffeomorphism. Pasting \(P\) and \(Q\) along the boundary by \(\varphi\) yields a closed manifold \(P\cup_{\varphi}Q\). Another diffeomorphism \(\psi : \partial P \to \partial Q\) gives another closed manifold \(P\cup_{\psi}Q\). The manifolds \(P\cup_{\varphi}Q\) and \(P\cup_{\psi}Q\) are said to be obtained from each other by cutting and pasting (Schneiden und Kleben). Two \(m\)-dimensional closed manifolds \(M\) and \(N\) are said to be \(SK\)-equivalent to each other if there is an \(m\)-dimensional closed manifold \(L\) such that the disjoint union \(M+L\) is obtained from \(N+L\) by a finite sequence of cuttings and pastings. Taking \(\mathbb{Z}_2\)-equivariant diffeomorphisms as pasting diffeomorphisms, it is possible to perform \(\mathbb{Z}_2\)-equivariant cuttings and pastings into the set \({\mathcal M}_m^{\mathbb{Z}_2}\) of \(m\)-dimensional closed \(\mathbb{Z}_2\)-manifolds and define an \(SK\)-equivalence relation on it. The fixed point set \(M^{\mathbb{Z}_2}\) of a \(\mathbb{Z}_2\)-manifold \(M\) is a submanifold of \(M\), whose components form a family of submanifolds of \(M\), denoted by \((M;M_m^{\mathbb{Z}_2}, M_{m-1}^{\mathbb{Z}_2}, \dots, M_0^{\mathbb{Z}_2})\), where \(M_i^{\mathbb{Z}_2}\) is the \(i\)-dimensional component of \(M^{\mathbb{Z}_2}\), for \(0\leq i\leq m=\)dim \(M\). An equivariant cutting and pasting on \(M\) induces a cutting and pasting on each \(M_i^{\mathbb{Z}_2}\). These results made the author suggest the definition of an \textit{\(m\)-dimensional family} \(\widetilde P=(P;P_m, P_{m-1}, \dots,P_0)\), where \(P\) is an \(m\)-dimensional compact manifold and \(P_i\) an \(i\)-dimensional submanifold of \(P\) such that \(\partial P_i=P_i \cap \partial P\) and \(P_i \cap P_j=\emptyset\) if \(i \neq j\). The definition of cutting and pasting and the \(SK\)-equivalence relation can be extended to the set \({\mathcal M}_m^{\mathcal F}\) of \(m\)-dimensional families of submanifolds of closed manifolds. Then it is proved that two families \(\widetilde M=(M;M_m, \dots,M_0)\) and \(\widetilde N=(N;N_m, \dots,N_0)\) are \(SK\)-equivalent in \({\mathcal M}_m^{\mathcal F}\) if and only if \(\chi(M)=\chi (N)\) and \(\chi(M_i)=\chi (N_i)\) for any \(i\), \(0\leq i\leq m\). The natural correspondence \({\mathcal M}_m^{\mathbb{Z}_2}\to {\mathcal M}_m^{\mathcal F}\) which assigns to a \(\mathbb{Z}_2\)-manifold \(M\) the family \((M;M_m^{\mathbb{Z}_2}, \dots, M_0^{\mathbb{Z}_2})\) induces a homomorphism \(\eta : SK_m^{\mathbb{Z}_2} \to SK_m^{\mathcal F}\) between the corresponding \(SK\)-groups; here the author proves that the homomorphism \(\eta\) is injective and that an element \(x \in SK_m^{\mathcal F}\) is in the image of \(\eta\) if and only if \(\chi(x)\equiv \sum_{i=0}^{m}\chi_i(x)\bmod 2\).