# Normal intersections of submanifolds

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?

• 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

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.
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.
• ? The two circles were intersecting at the beginning. Just two intersecting circles (not in the same plane) in $\mathbb{R}^3$ – O.R. Jul 27 '10 at 0:32
• 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