SCHWEITZER 2010 Problem 9

For each $M$ m-dimensional closed $C^{\infty}$ set , assign a $G(m)$ in some euclidean space $\mathbb{R}^{q}$ . Denote by $\mathbb{R} \mathbb{P}^{q}$ a $q$ -dimensional real projecive space. A $G(M) \subseteq \times \mathbb{R} \mathbb{P}^{q}$ . The set consists of $(x,e)$ pairs for which $x \in M \subseteq \mathbb {P}^{q}$ and $e \subseteq \mathbb {R}^{q+1}= \mathbb{R}^{q} \times \mathbb{R}$ and $\mathrm{a} (0, \ldots,0,1) \in \mathbb{R}^{q+1}$ in a stretched $(m+1)$ -dimensional linear subspace. Prove that if $N$ is a $n$ -dimensional closed set $C^{\infty}$ , then $P=G(M \times M)$ and $Q=G(M) \times G(N)$ are cobordant , that is, there exists a $(2m+2n+1)$ -dimensional compact , flanged set $C^{\infty}$ with a disjoint union of $P$ and $Q$ .