# Transversal Intersection of More than 2 Surfaces

• WWGD
In summary, there is a result that states if two manifolds intersect transversally 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)##. It is unclear if this generalizes to other topological spaces. It is also unknown if there is a similar result for intersections of 3 or more manifolds. The concept of pairwise transversal intersection is also mentioned. Some possible resources for further research on this topic are the books Intersection Theory by William Fulton, Differential Topology by Guillemin and Pollack, and Topology
WWGD
Gold Member
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?

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:
WWGD

## 1. What is the definition of transversal intersection?

Transversal intersection refers to the point or line at which two or more surfaces intersect, forming perpendicular angles.

## 2. How is transversal intersection different from regular intersection?

The main difference between transversal intersection and regular intersection is that transversal intersection involves the intersection of more than two surfaces, while regular intersection involves the intersection of only two surfaces.

## 3. What factors affect the transversal intersection of surfaces?

The transversal intersection of surfaces is affected by the orientation, shape, and position of the surfaces in relation to each other. Additionally, the number of surfaces and their dimensions also play a role in determining the transversal intersection.

## 4. What are some real-world applications of transversal intersection?

Transversal intersection has several applications in fields such as architecture, engineering, and computer graphics. For example, in architecture, transversal intersection is used to determine the angles at which walls intersect, while in computer graphics, it is used to create complex 3D models.

## 5. How is transversal intersection calculated?

The calculation of transversal intersection involves finding the points or lines where the surfaces intersect and determining the angles at which they intersect. This can be done using mathematical equations and geometric principles, such as the Pythagorean theorem and trigonometric functions.

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