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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
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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 $\mathbb{R}^3$ –  ABC 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. –  ABC Jul 27 '10 at 0:34
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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
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Franklin, please check the definition of transversality (for example, en.wikipedia.org/wiki/Transversality). 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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