On normal Euler numbers of embedding \({\mathbb{P}{}}{\mathbb{R}{}}^ 2\) in indefinite 4-manifolds
Let \(N\) be a closed, nonorientable surface, \(M\) be a simply connected 4- manifold. \(f:N\to M\) be an embedding with normal bundle \(v_ f\). The normal Euler class \(e(v_ f)\) of \(f\) is an element in \(H^ 2(N,k)\), where \(k\) is the local coefficient determined by \(w_ 1(v_ f)=w_ 1(N)\). It is very important to determine \(e(v_ f)[N]\) for all embeddings. It is closely related to whether a two-homology class can be represented by a smooth embedded sphere. Let \(M(p,q)\) be a smooth manifold diffeomorphic to \(p\mathbb{C} P^ 2\#q\overline{\mathbb{C} P}^ 2\), where \(\overline{\mathbb{C} P}^ 2\) is \(\mathbb{C} P^ 2\) with opposite orientation. The main results are as follows: Theorem 1. The possible normal Euler characteristics of embedding \(\mathbb{R} P^ 2\) in \(\mathbb{C} P^ 2\#\overline{\mathbb{C} P}^ 2\) are \(\pm 2\), \(\pm 6\) and \(2n-1\), \(n\in\mathbb{Z}\). Moreover, all these values are realizable by appropriate embeddings. Theorem 2. Let \(M\) be an indefinite 4-manifold, \(M=m(s^ 2Xs^ 2)\#M(p,q)\). \(\mathbb{R} P^ 2\) embeds in \(M\) with normal Euler number \(n\). i) If \(p=q=0\), then \(n\) can only attain all even numbers. ii) If either \(p\) or \(q\) is nonzero, and \(M\neq M(1,1)\), then \(n\) can be any integer.
scientific article; zbMATH DE number 123647