# Transversal Intersection of More than 2 Surfaces

1. Jul 4, 2015

### WWGD

Hi,
There is a result that if two manifolds $M_1, M_2$ ( I don't know to what extent this generalizes to other topological spaces) intersect transversally, say in $\mathbb R^m$ , then the dimension of the intersecting set is given by m - $\Sigma Cod(M_i ) ; i=1,2$, where $Cod(M_i):= m-Dim(M_i)$, i.e., the dimension of the ambient space minus the dimension of the manifold. Is there any result for intersections of 3- or more manifolds, i.e., for the case where the intersecting set contains points of all 3 manifolds? Do we consider pairwise transversal intersection, etc.?
Thanks,
WWGD: What Would Gauss Do?

2. Jul 5, 2015

### mathwonk

perhaps he would use induction? yes codimensions of any finite number of tranversal submanifolds add, (and degrees, for algebraic subvarieties, multiply). there is a beautiful and authoritative research level book on this topic, at least for algebraic varieties, by William Fulton, called Intersection Theory. There is also a more elementary undergraduate level one for smooth manifolds called Diferential Topology by Guillemin and Pollack, and (at least for my taste) an even better but much briefer one by John Milnor, called Topology from the differentiable viewpoint.

Last edited: Jul 5, 2015