Equivariant surgery on incompressible tori and Klein bottles in 3- manifolds with respect to involutions (Q760054)

The following is the main theorem: Let M be an irreducible 3-manifold, \(M\neq S^ 1\times P^ 2\), \(P^ 2\) the projective plane, and let \(\iota\) : \(M\to M\) be an involution with at most isolated fixed points. Suppose M contains an incompressible torus \(T_ 0\). Then one of the following two properties holds. (I) There is a 2-sided incompressible torus or Klein bottle \(F\subset int M\) with either \(F\cap \iota F=\emptyset\) or \(\iota F=F\). In the latter case there are no fixed points of \(\iota\) on F. Furthermore if O is a neighborhood of \(T_ 0\cup \iota T_ 0\) then we assume that \(F\subset O.\) (II) M is orientable and \(M=V_ 1\cup V_ 2\cup V_ 3\cup V_ 4\), where \(V_ 1,...,V_ 4\) are solid tori with \(\partial V_ 1=A\cup \iota A\), A is an annulus with \(A\cap \iota A=\partial A=\partial \iota A\) and with \(V_ 1\cap V_ 3=A\), \(V_ 1\cap V_ 4=\iota A\), \(V_ 2\cap V_ 3=\overline{\partial V_ 3-A}\), \(V_ 2\cap V_ 4=\overline{\partial V_ 4-\iota A}\), and \(V_ 1\cap V_ 2=\partial A\). In particular, M has a Seifert fibration with four exceptional fibres and with Seifert surface a 2-sphere. The tori \(A\cup \overline{\partial V_ 4-\iota A}\), \(\iota\) \(A\cup \overline{\partial V_ 3-A}\) are incompressible. The involution \(\iota\) : \(M\to M\) is orientation preserving and satisfies \(\iota V_ i=V_ i\), \(i=1,2\), and \(\iota V_ 3=V_ 4\). \(M/\iota =(V_ 1/\iota)\cup (V_ 2/\iota)\cup V_ 3\), where \(V_ 1/\iota\), \(V_ 2/\iota\), \(V_ 3\) are solid tori. M/\(\iota\) has a Seifert fibration with three exceptional fibres and with Seifert surface a 2-sphere. There are various corollaries. We obtain also this theorem: Let M be a 3- manifold and let \(\iota\) : \(M\to M\) be an involution with at most isolated fixed points. Suppose M contains a 2-sided incompressible Klein bottle. Then there is a 2-sided incompressible torus or Klein bottle \(F\subset int M\) with either \(F\cup \iota F=\emptyset\) or \(\iota F=F\). In the latter case there are no fixed points of \(\iota\) on F.