Why we need Sard: critical values are a measure-zero accident
In the previous guide the regular value theorem handed you a clean prize: if q in N is a regular value of a smooth f: M -> N, then the preimage f^(-1)(q) is a smooth submanifold of dimension dim M - dim N. The obvious worry is whether regular values even exist. A map can have whole arcs of critical points; what stops the critical values from filling up N and leaving you nothing to work with?
Sard's theorem is the answer, and it is almost shockingly strong. For any smooth f: M -> N, the set of critical values (images of points where df fails to be surjective) has measure zero in N. Note the asymmetry: the critical points can be huge — a constant map has every point critical — but their images are negligible. So Sard's theorem guarantees that regular values are not just present, they are the overwhelming majority: almost every q you could pick is regular.
Transversality: the right way for two submanifolds to meet
Regular values are the special case dim N = (a point's worth). The grown-up version is transversality. Two submanifolds A, B of M meet transversally at a point p (written A pitchfork B) when their tangent spaces together span everything: T_p A + T_p B = T_p M. This is not a direct sum — the spaces are allowed to overlap — you only demand that between them they fill the room.
The picture in R^3 is everything. Two surfaces (each dim 2) meeting transversally cross in a curve (dim 2 + 2 - 3 = 1); that is the generic, stable picture. Two curves (dim 1) in R^3 are transverse only when they miss each other entirely, because 1 + 1 < 3 means they cannot span unless they never touch — and indeed two random threads in space do not meet. When transversality holds, A intersect B is itself a submanifold, of dimension dim A + dim B - dim M.
transverse: dim(A cap B) = dim A + dim B - dim M surfaces in R^3: 2 + 2 - 3 = 1 (a curve) curves in R^3: 1 + 1 - 3 = -1 (so generically empty)
Why care? Because transversality is stable and generic. Stable: a small perturbation of a transverse intersection stays transverse (the spanning condition is open). Generic: by the Thom transversality theorem — a parametric upgrade of Sard's theorem — any map can be wiggled by an arbitrarily small amount until it becomes transverse to a fixed submanifold. So even when nature hands you a tangential, degenerate intersection, you may assume transversality after a harmless nudge. This is the differential-topology analogue of "in general position."
Degree mod 2: counting preimages honestly
Now make M compact and let f: M -> N be smooth between manifolds of the same dimension, with N connected. Pick a regular value q (Sard says you can). Then f^(-1)(q) is a compact 0-dimensional submanifold of M — a finite set of points. The simplest invariant just counts them mod 2: deg_2(f) = number of points in f^(-1)(q), taken modulo 2.
The miracle is that this number does not depend on which regular value q you chose, and it is invariant under smooth homotopy: if f and g are smoothly homotopic, deg_2(f) = deg_2(g). The engine behind both facts is the classification of compact 1-manifolds: the preimage of a regular value of a homotopy F: M x [0,1] -> N is a compact 1-manifold with boundary, and a compact 1-manifold has an even number of boundary points. That parity is the entire proof, and it is why "mod 2" appears before any orientation enters.
The Brouwer degree: a signed integer that remembers orientation
Suppose now M and N are oriented, M compact, N connected, both n-dimensional. At each preimage point p in f^(-1)(q) the differential df_p: T_p M -> T_q N is an isomorphism (q is regular), so it either preserves or reverses orientation. Assign sign(p) = +1 or -1 accordingly. The degree of f is the signed sum: deg(f) = sum over p in f^(-1)(q) of sign(p). This is the degree of a map, an honest integer that can be negative.
The canonical example is the circle map z -> z^k on S^1 (or its higher cousin on S^n). Every value has exactly k preimages, all with the same sign, so the degree is exactly k — the integer winds-around number you already feel in your bones. The reflection z -> conjugate(z) has degree -1: one preimage, orientation flipped. Degree is multiplicative under composition, deg(g . f) = deg(g) deg(f), and like deg_2 it is a homotopy invariant — but now it remembers direction, distinguishing the k from the -k that mod-2 degree could not.
Degree is the gateway to a stack of theorems. The fundamental theorem of algebra falls out (a degree-n polynomial map of the sphere has degree n, hence n roots counted with multiplicity). The hairy-ball theorem is the statement that a vector field on S^(2n) must vanish somewhere, read through the degree of the Gauss-type map. And via Poincaré-Hopf, the sum of indices of a vector field's zeros equals the Euler characteristic of M — degree connecting a local count to a global topological number.
Intersection number and a glance at Morse theory
Transversality and degree merge into the intersection number. If A and B are compact oriented submanifolds of a compact oriented M with complementary dimensions (dim A + dim B = dim M), and they meet transversally, then A intersect B is a finite signed set of points; their signed total I(A, B) is a homotopy invariant. This is the geometric heart of intersection multiplicity and, dually, of cup product in cohomology — algebraic topology counting the same crossings the geometry sees.
Here is the standard recipe when an intersection looks degenerate.
- Check transversality: at each common point p, test whether T_p A + T_p B = T_p M. If yes, you are done — list the points.
- If some intersection is tangential, perturb B by an arbitrarily small smooth isotopy; Thom's transversality theorem (Sard with parameters) makes the perturbed B transverse to A.
- At each transverse point, compare the orientation of T_p A (+) T_p B with the ambient orientation of T_p M to read off the sign +1 or -1.
- Sum the signs. Because the count is homotopy-invariant, the messy original and the clean perturbation give the same integer I(A, B).
All of this sets up the finale of this rung: Morse theory. A Morse function is a smooth f: M -> R whose critical points are non-degenerate — exactly the genericity that Sard makes typical. Each critical point carries an index (the number of downhill directions), and the indices reconstruct the topology of M, telling you its Betti numbers via the Morse inequalities. Transversality, regular values, and degree were the warm-up; Morse theory is where counting critical points starts handing back the whole shape of the manifold. We pursue it in the next guide.