$\mathfrak {M}^3$ admitting a certain embedding of $P^2$ is a pseudo $P^3$

Abstract:

Let $M$ be a $3$-manifold and ${P^2}$ projective $2$-space. In this paper it is shown that if there exists an embedding $f:{P^2} \to M$ such that $f{ \ast _2}:{\pi _2}({P^2}) \to {\pi _2}(M)$ is trivial, then $M$ is, except for a fake cell, projective $3$-space.