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M dimensional closed set

Source: Miklós Schweitzer 2010, P9

September 9, 2020

topologyprojective space

Problem Statement

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

.

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