Normal bundles for an embedded \({\mathbb{R}}P^ 2\) in a positive definite 4- manifold
This paper is concerned with a determination of the possible normal bundles which can occur for an embedding of the real projective plane into a positive definite 4-manifold M. The prototypical result of this type was \textit{W. Massey}'s proof of the Whitney conjecture [Pac. J. Math. 31, 143-156 (1969; Zbl 0198.567)], which said that in the case of the 4- sphere, one has normal Euler number \(e=\pm 2\). Using \textit{V. A. Rokhlin}'s generalization of Massey's work [Funkts. Anal. Prilozh. 5, No.1, 48-60 (1971; Zbl 0268.57019)] one can get an estimate of the form \(-2\leq e\leq 2+4r\), where r is the rank of the second homology group as well as a congruence \(e\equiv 2 mod 4\) if \(H_ 1=0\) and the embedding is nullhomotopic. If the embedding is characteristic, then the nonorientable Rochlin theorem of \textit{L. Guillou} and \textit{A. Marin} [C. R. Acad. Sci., Paris, Ser. A 285, 95-98 (1977; Zbl 0361.57018)] will apply to give restrictions mod 16 on the normal Euler number. Suppose M has the same homology as the connected sum of k copies of the complex projective plane and the image of the generator of \(H_ 2({\mathbb{R}}P^ 2,{\mathbb{Z}}/2)\) in \(H_ 2({\mathbb{C}}P^ 2,{\mathbb{Z}}/2)\) is the sum of \(\ell\) generators. Conjecture. \(e\equiv -2+\ell mod 4\) and \(- 2+\ell \leq e\leq 2+\ell +4(k-1)\). The main theorem of this paper proves the congruence above as well as one half of the inequality \(-2+\ell \leq e\) when \(\ell <8\). The techniques involved are adaptations of those used by \textit{R. Fintushel} and \textit{R. Stern} in their work on pseudofree orbifolds [Ann. Math., II. Ser. (to appear)] to the context of V- manifolds. From the embedding there is an associated V-manifold consisting of the complement of a tubular neighborhood union with the cone on its boundary. One studies the moduli space of self-dual V- connections on a certain SO(3) V-bundle over this manifold.
embedding of the real projective plane into a positive definite 4-manifold
scientific article; zbMATH DE number 3913278