# Is the Transversal Intersection of Manifolds a Manifold?

by WWGD
Tags: intersection, manifold, manifolds, transversal
 P: 391 Hi, All: Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally, so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection a manifold? Thanks.
 Sci Advisor HW Helper PF Gold P: 4,768 Yes, according to the "canonical form theorem" for a transverse intersection, if X,Y are submanifolds of the n-manifold M that intersect transversally "of dimension k", and if p is a point of intersection, there is a coordinate nbhd of p in M such that X n Y corresponds to R^k in R^n under the coordinate map.
 P: 391 Thanks, Quasar; any chance you have a ref?
HW Helper
PF Gold
P: 4,768

## Is the Transversal Intersection of Manifolds a Manifold?

See the book Differential manifolds by Antoni Kosinski where the stronger result is proved that actually, there is a coordinate chart around p in which X corresponds to R^r x {0} while Y corresponds to {0} x R^s, so what X n Y, of course, corresponds to {0} x R^k x {0} (where r=dim(X), s=dim(Y), and k=(r+s)-).

But surely the theorem can also be found in Differential Topology by Guillemin & Pollack and possibly in the book of the same name by M. Hirsh
 P: 662 WWGD: Not to quibble too much, but I have seen two main definitions of transversality used, and I wondered which one you are using: 1) First and weaker of the two, states that if submanifolds S,S' of ambient M intersect transversely at p, then the (vector space) sum of the tangent spaces TpS and TpS' equals TpM , i.e., the sum spans the tangent space of the ambient manifold. 2) The second and stronger one (stronger in that it excludes some cases of 1) , is that each point p of intersection has a neighborhood Up with Phi(Up)= (x1,x2,..,xn,0,0,..,0) (Phi is a chart map). This excludes, e.g., Sin(1/x) and the x-axis. BTW, the Mathworld entry "http://mathworld.wolfram.com/HomologyIntersection.html" seems to agree in a weaker sense, in that it states that cycles (homology classes) that intersect transversely are also cycles ,and it gives the example of y=x^2 and the x-axis as an example of a non-transverse intersection in the sense 2) above, showing how the intersection is unstable, in that a small perturbation --e.g., moving y=x^2 upwards changes the (algebraic) intersection number .

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