Transversality

Transversality describes how spaces can intersect, and can be seen as "opposite" of tangency. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the dimensions of linearization of spaces of intersection.

Transversality depends on ambient space. The two curves defining boundaries of two sets in a typical Venn diagram are transverse when considered as embedded in the plane, but not if we consider them as embedded in a plane in three-dimensional space.

# Definition

Notation for transverse intersection of two submanifolds $L_1$ and $L_2$ of $M$ is $L_{{1}}\pitchfork L_{{2}}$.

We define transversality of two submanifolds in terms of their set theoretic intersection $L_1 \cap L_2$ and the tangent space of $M$ at points $p \in L_1 \cap L_2$ as follows:

$$L_{1}\pitchfork L_{2}\iff \forall p\in L_{1}\cap L_{2},\; T_{p}M=T_{p}L_{1}+T_{p}L_{2}$$

## Transversality of Maps

A central theme of topology is the lifting of concepts from spaces to maps between spaces. Two smooth maps $f : V \to M$ and $g : W \to M$ are transverse as follows:

$$f \pitchfork g \iff \forall f(v) =g(w) = p,\; Df_v(T_vV) + Dg_w(T_wW) = T_p M$$

Note that submanifolds $L_1$ and $L_2$ are transverse in $M$ if and only if inclusions $\iota_{1} : L_1 \hookrightarrow M$ and $\iota_{2}: L_2 \hookrightarrow M$ satisfy $\iota_{1} \pitchfork \iota_{2}$ in $M$.

# Genericity

Consider examples of following generic features of smooth manifolds and mappings:

• Two intersecting curves in $\mathbb{R}^2$? generically intersect in a discrete set of points.
• Three curves in $\mathbb{R}^3$? generically do not have a point of mutual intersection.
• Two curves in $\mathbb{R}^n$ generically do not intersect for $n > 2$.
• Two intersecting surfaces in $\mathbb{R}^3$ generically intersect along curves.
• A real square matrix is generically invertible.
• Critical points of a real-valued function on a manifold are generically discrete.
• The roots of a polynomial are generically non-repeating.
• The fixed points of a vector field are generically discrete.
• The configuration space of a planar linkage is generically a manifold.
• A generic map of a surface into $\mathbb{R}^5$ is injective.

Some of these seem obviously true; others less obviously so. All are provably true with precise meaning using the theory of transversality.

## Degrees of Freedom

Recall transversality of maps: when $f \pitchfork g$ this means degrees of freedom in the intersection of images of $f$ and $g$ span the full degrees of freedom in $M$. Transversality helps manage singular behavior. A point $q \in N$ is a regular value of $f : M \to N$ if $f \pitchfork \{q\}$. This is equivalent to the statement that for each $p \in f^{-1}(q)$, the derivative is a surjection, that is the matrix of $Df_p$ is of rank at least $\dim N$.

### Preimage Theorem

Extending the Rank-Nullity Theorem for linear transformations between finite dimensional spaces is the Preimage theorem:

 Preimage Theorem Theorem. Consider a differentiable map $f : M \to N$ between smooth manifolds. If $f \pitchfork M$ for submanifold $W \subset N$, then $f^{-1}(W)$ is a submanifold of $M$ of dimension $$\dim f^{-1}(W) = \dim M - \dim N + \dim W$$

This provides an effective means of constructing manifolds without the need for explicit charts: it is often used in the context of a regular value of a map.

• Sphere $\mathbb{S}^n$ is the inverse image of $1$ under $f : \mathbb{R}^{n+1} \to \mathbb{R}$ given by $f(x) = \| x \|$. It is a manifold of dimension $n = (n+1)-1+0$
• Torus $\mathbb{T}^n$ is the inverse image $f^{-1}(1,\cdots,1)$ of $f : \mathbb{C}^n \to \mathbb{R}^n$ given by $f(z) = (\| z_1 \|, \cdots, \| z_n \|)$. It's dimension is $n = 2n-n+0$.
• The matrix group $\mathrm{O}(n)$, of rigid rotations of $\mathbb{R}^n$ (both orientation preserving and reversing) can be realized as the inverse image $f^{-1}(\mathrm{Id})$ of the identity under the map from $n \times n$ real-valued matrices to symmetric $n \times n$ real-valued matrices given by $f(A) = AA^\mathrm{T}$. The dimension of $\mathrm{O}(n)$ is: $$\dim \mathrm{O}(n) = n^2 - \frac{1}{2} n(n+1) + 0 = \frac{1}{2} n(n-1)$$ This fact is relevant in the choice of stalk in sheaf neural networks.
• The determinant map restricted to $\mathrm{O}(n)$ is in fact a smooth map to $\mathbb{S}^0 = \{\pm 1\}$ under this restricted $\det$ it is a manifold of the same dimension as $\mathrm{O}(n)$; thus, $\mathrm{O}(n)$ is a disjoint union of two copies of $\mathrm{SO}(n)$.

Transversality condition is checked by showing that the mapping $f$ has a derivative of full rank at the appropriate (regular) value. Such regularity seems to fail rarely, for special singular values. This intuition is the driving force behind the Transversality Theorem.

A subset of a topological space is said to be residual if it contains a countable intersection of open, dense subsets. A property dependent upon a parameter $\lambda \in \Lambda$ is said to be a generic property if it holds for $\lambda$ in a residual subset of $\Lambda$ — even when that subspace is not explicitly given. For reasonable (e.g., Baire) spaces, residual sets are dense, and hence form a decent notion of topological typicality.

### Transversality Theorem

 Transversality Theorem Theorem. For $M$ and $N$ smooth manifolds and $W \subset N$ a submanifold, the set of smooth maps $f : M \to N$ such that $f \pitchfork W$ is residual in $\mathcal{C}^\infty(M,N)$ - the space of all smooth maps from $M$ to $N$. If $W$ is closed, then this set of transverse maps is both open and dense. INCOMPLETE PAGE This page or section is incomplete, you can help by expanding it.