A possible MO formulation of the \vee question
February-26-14
5:21 PM
Let $T=S^1\times S^1$ be an embedded 2-dimensional torus in $S^4$, and let $\gamma_1$ and $\gamma_2$ be $S^1\times\{1\}$ and $\{1\}\times S^1$ - both can be regarded as curves in $S^4$. Let $\Sigma_1$ and $\Sigma_2$ be two dimensional surface also embedded in $S^4$ and disjoint from $T$, such that the linking number of $\Sigma_i$ with $\gamma_i$ is $1$, for $i=1,2$ (note that in $S^4$ there is a well-defined notion of a linking number between a curve and a surface disjoint from it, similar to the standard linking number between two disjoint curves in $S^3$).
**Question.** Do $\Sigma_1$ and $\Sigma_2$ necessarily intersect?
**Example.** Thinking of $S^4$ as a compactified ${\mathbb R}^4={\mathbb R}^2\times{\mathbb R}^2$, let $T$ be the unit circle in the first copy of ${\mathbb R}^2$ times the unit circle in the second copy of ${\mathbb R}^2$, let $\Sigma_1=\{0\}\times{\mathbb R}^2$ and let $\Sigma_2={\mathbb R}^2\times\{0\}$. All conditions are satisfied and $\Sigma_1$ and $\Sigma_2$ intersect twice, once at $0$ and once at $\infty$.
Let T=S1×S1 be an embedded 2-dimensional torus in S4, and let γ1 and γ2 be S1×{1} and {1}×S1 - both can be regarded as curves in S4. Let Σ1 and Σ2 be two dimensional surface also embedded in S4 and disjoint from T, such that the linking number of Σi with γi is 1, for i=1,2 (note that in S4 there is a well-defined notion of a linking number between a curve and a surface disjoint from it, similar to the standard linking number between two disjoint curves in S3).
Question. Do Σ1 and Σ2 necessarily intersect?
Example. Thinking of S4 as a compactified ℝ4=ℝ2×ℝ2, let T be the unit circle in the first copy of ℝ2 times the unit circle in the second copy of ℝ2, let Σ1={0}×ℝ2 and let Σ2=ℝ2×{0}. All conditions are satisfied and Σ1 and Σ2 intersect twice, once at 0 and once at ∞.
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