In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.
Definition
Two submanifolds of a given finite-dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point. Manifolds that do not intersect are vacuously transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the ambient space), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a submanifold whose codimension is equal to the sums of the codimensions of the two manifolds. In the absence of the transversality condition the intersection may fail to be a submanifold, having some sort of singular point. In particular, this means that transverse submanifolds of complementary dimension intersect in isolated points (i.e., a 0-manifold). If both submanifolds and the ambient manifold are oriented, their intersection is oriented. When the intersection is zero-dimensional, the orientation is simply a plus or minus for each point. One notation for the transverse intersection of two submanifolds L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} of a given manifold M {\displaystyle M} is L 1 ⋔ L 2 {\displaystyle L_{1}\pitchfork L_{2}} . This notation can be read in two ways: either as “ L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} intersect transversally” or as an alternative notation for the set-theoretic intersection L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} of L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} when that intersection is transverse. In this notation, the definition of transversality reads
L 1 ⋔ L 2 ⟺ ∀ p ∈ L 1 ∩ L 2 , T p M = T p L 1 ⊕ T p L 2 . {\displaystyle L_{1}\pitchfork L_{2}\iff \forall p\in L_{1}\cap L_{2},T_{p}M=T_{p}L_{1}\oplus T_{p}L_{2}.}
Transversality of maps The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. If the maps are embeddings, this is equivalent to transversality of submanifolds.
Meaning of transversality for different dimensions
Suppose we have transverse maps f 1 : L 1 → M {\displaystyle f_{1}:L_{1}\to M} and f 2 : L 2 → M {\displaystyle f_{2}:L_{2}\to M} where L 1 , L 2 {\displaystyle L_{1},L_{2}} and M {\displaystyle M} are manifolds with dimensions ℓ 1 , ℓ 2 {\displaystyle \ell _{1},\ell _{2}} and m {\displaystyle m} respectively. The meaning of transversality differs a lot depending on the relative dimensions of M , L 1 {\displaystyle M,L_{1}} and L 2 {\displaystyle L_{2}} . The relationship between transversality and tangency is clearest when ℓ 1 + ℓ 2 = m {\displaystyle \ell _{1}+\ell _{2}=m} . We can consider three separate cases:
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