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Transversal Intersection


TransversalIntersection

Two submanifolds X and Y in an ambient space M intersect transversally if, for all p in X intersection Y,

 TX_p+TY_p={v+w:v in TX_p,w in TY_p}=TM_p,

where the addition is in TM_p, and TX_p denotes the tangent map of X_p. If two submanifolds do not intersect, then they are automatically transversal. For example, two curves in R^3 are transversal only if they do not intersect at all. When X and Y meet transversally then X intersection Y is a smooth submanifold of the expected dimension dimX+dimY-dimM.

In some sense, two submanifolds "ought" to intersect transversally and, by Sard's theorem, any intersection can be perturbed to be transversal. Intersection in homology only makes sense because an intersection can be made to be transversal.

Transversal

Transversality is a sufficient condition for an intersection to be stable after a perturbation. For example, the lines y=x and y=0 intersect transversally, as do the perturbed lines y=x+t, and they intersect at only one point. However, y=x^2 does not intersect y=0 transversally. It intersects in one point, while y=x^2+t intersects in either none or two points, depending on whether t is positive or negative.

When dimX+dimY=dimM, then a transversal intersection is an isolated point. If the three spaces have an vector space orientation, then the transversal condition means it is possible to assign a sign to the intersection. If e_1,...,e_k are an oriented basis for TX_p and e_(k+1),...,e_n are an oriented basis for TY_p, then the intersection is +1 if e_1,...,e_n is oriented in M and -1 otherwise.

More generally, two smooth maps f:X->M and g:Y->M are transversal if whenever p=f(x)=g(y) then df(TX_x)+dg(TY_y)=TM_p.


See also

Homology, Homology Intersection, Sard's Theorem, Submersion, Vector Space Orientation

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Transversal Intersection." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TransversalIntersection.html

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