Involutions fixing the disjoint union of 3-real projective space with Dold manifold (Q1591513)

This paper determines the cobordism classification of smooth involutions \((M,T)\) having fixed point set the union of the projective space \(\mathbb{R} P^3\) and a Dold manifold \(P(m,n)\). It is assumed that the normal bundle of \(\mathbb{R} P^3\) in \(M\) does not bound and that \(m\) and \(n\) are positive, so that the fixed set is genuinely such a union. The main results are that one must have \(m=3\) and \(n\) even. For each of the possible choices of the normal bundle of \(\mathbb{R} P^3\) there is exactly one possible stable normal bundle for \(P(3,n)\), and the range of dimensions for \(M\) is determined.