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Let $M$ be a compact manifold, and let $M_1,\ldots, M_k$ (k>2) be embedded submanifolds. Suppose that $p\in\cap_{i=1}^k M_k$ and that for any subset $S$ of $\{1,\ldots, k\}$ and any $j\notin S$ that $\cap_{i\in S}M_i$ intersects $M_j$ transversally at $p$.

I believe that in this case the fact that $\cap_{i=1}^k M_k$ is nonempty is stable (still true after homotoping each $M_i$ a little bit). Does anyone have a reference for this fact?

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This is an induction argument with the $k=2$ case implying all the $k>2$ cases. For $k=2$ see Guillemin and Pollack. – Ryan Budney Jul 26 '10 at 20:07
up vote 3 down vote accepted

The matter being local, we can restrict to a nbd $U$ of $p$ and think that $M_i$ is the zero set of some local submersion $g_i:U\to\mathbb{R}^{n_i}$. If I'm not wrong your transversality assumption then translates into the surjectivity of the differential at $p$ of the map $g:=(g_1,\dots,g_k):U\to\mathbb{R}^m$ (here $m:={n_1+\dots+n_k}$). So $0\in\mathbb{R}^m$ is a regular value for $g$, which implies your thesis. Note that the compactness assumption on $M$ plays no role.

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If I take M to be the filled torus $D^2\times S^1$ and $M_1$ and $M_2$ two circles (the two transversal generators of the torus $S^1\times S^1$) then when you can contract one of them a little so that they no longer intersect.

Uff, manyfolds. Then replace $D^2\times S^1$ by $S^3$ and keep the two circles.

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You've verified Alex's hypothesis in the vacuous case -- when all transverse intersections are empty. – Ryan Budney Jul 26 '10 at 22:16
? The two circles were intersecting at the beginning. Just two intersecting circles (not in the same plane) in $\mathbb{R}^3$ – Mlazhinka Shung Gronzalez LeWy Jul 27 '10 at 0:32
The theorem need some condition on the dimensions and the number of sub-manifolds as in Pietro's post. – Mlazhinka Shung Gronzalez LeWy Jul 27 '10 at 0:34
In manifold theory, a transverse intersection means that at any point of intersection, the tangent spaces of the two submanifolds must span the tangent space of the ambient manifold. Since $1+1=2<3$, the only way two 1-manifolds can intersect transversely in a 3-manifold is for them to be disjoint. – Ryan Budney Jul 27 '10 at 1:58
Franklin, please check the definition of transversality (for example, The sum of the tangent spaces of the two submanifolds has to be equal to the tangent space of the ambient space. This is not the case in your example. – Deane Yang Jul 27 '10 at 1:59

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