\({\mathbb{Z}}_ 2\)-smooth action on RP(2k) (Q1824250)

Let \(M^ n\) be a connected manifold with \(w_ 1^ n[M^ n]\neq 0\) and having a nontrivial involution T. Such an involution has fixed components of both even and odd dimension. Letting \(n_ e\) and \(n_ 0\) be the largest even and odd dimensions of such fixed components, this paper asserts that 1) \(n_ e+n_ 0\geq n-1,\) and 2) These fixed components \(F^{n_ i}\) have mod 2 cohomology containing subrings isomorphic to that of \({\mathbb{R}}P^{n_ i}.\) It is also asserted that for \(M^ n={\mathbb{R}}P^ n\), n even, the involution \((M^ n,T)\) is bordant to one of the standard linear involutions. (This is well-known from the usual cohomological methods.)