MathDB

9

Part of 2010 Miklós Schweitzer

Problems(1)

M dimensional closed set

Source: Miklós Schweitzer 2010, P9

9/9/2020
For each MM m-dimensional closed C∞C^{\infty} set , assign a G(m)G(m) in some euclidean space Rq\mathbb{R}^{q}. Denote by RPq\mathbb{R} \mathbb{P}^{q} a qq-dimensional real projecive space. AG(M)⊆×RPqG(M) \subseteq \times \mathbb{R} \mathbb{P}^{q}. The set consists of (x,e)(x,e) pairs for which x∈M⊆Pqx \in M \subseteq \mathbb {P}^{q} and e⊆Rq+1=Rq×Re \subseteq \mathbb {R}^{q+1}= \mathbb{R}^{q} \times \mathbb{R} and a(0,…,0,1)∈Rq+1\mathrm{a} (0, \ldots,0,1) \in \mathbb{R}^{q+1} in a stretched (m+1)(m+1)-dimensional linear subspace. Prove that if NN is a nn-dimensional closed set C∞C^{\infty}, then P=G(M×M)P=G(M \times M) and Q=G(M)×G(N)Q=G(M) \times G(N) are cobordant , that is, there exists a (2m+2n+1)(2m+2n+1)-dimensional compact , flanged set C∞C^{\infty} with a disjoint union of PP and QQ.
topologyprojective space