From even to odd: why contact geometry exists
Through the first four guides we lived on a symplectic manifold (M, omega): an even-dimensional space carrying a closed, non-degenerate 2-form. Non-degeneracy forced even dimension, because a skew-symmetric form on an odd-dimensional vector space always has a kernel. So the obvious question is: what is the right structure on an odd-dimensional manifold? The answer is contact geometry, and it is not a poor cousin — it is where the flow of time, the energy hypersurface of a Hamiltonian system, and the boundary of a symplectic region naturally live.
Here is the picture in words. On a (2n+1)-dimensional manifold M we choose at each point p a 2n-dimensional hyperplane xi_p inside T_p M — a smoothly varying field of hyperplanes. The defining demand is that this field be as far from being tangent to any hypersurface as possible: it should twist so violently that no surface can stay tangent to it on an open set. That maximal twisting is what 'contact' means, and the cleanest way to encode it is with a 1-form.
The contact condition: maximal non-integrability
Suppose the hyperplane field is the kernel of a 1-form alpha, so xi = ker(alpha). When is xi a contact structure? The condition is that alpha ^ (d alpha)^n be nowhere zero on M, where (d alpha)^n is the n-fold wedge of d alpha with itself. Recall from Guide 1 of this rung that Frobenius integrability said a hyperplane field is tangent to a foliation exactly when d alpha ^ alpha = 0. The contact condition is the loud opposite: alpha ^ (d alpha)^n is a volume form, so the field is maximally non-integrable. No piece of any hypersurface can be tangent to xi.
Symplectic (even, dim 2n): omega closed, omega^n != 0 (volume form) Contact (odd, dim 2n+1): alpha ^ (d alpha)^n != 0 (volume form) Local model (Darboux): symplectic: omega = dp_1 ^ dq_1 + ... + dp_n ^ dq_n contact: alpha = dz - p_1 dq_1 - ... - p_n dq_n
There is a contact analogue of the Darboux theorem you met in Guide 1: every contact form looks locally like the standard model alpha = dz - sum p_i dq_i on R^(2n+1). So, just as symplectic manifolds have no local invariants, contact manifolds are all locally identical too. Every interesting question — like which contact structures on a fixed manifold are genuinely different — is global, exactly as in the symplectic world.
The Reeb field: a canonical flow on an odd manifold
Choosing a contact form alpha (not just the hyperplane field) hands you a free gift: a distinguished vector field. The Reeb vector field R is the unique field solving two equations at once — d alpha(R, .) = 0 and alpha(R) = 1. The first says R lies in the kernel of d alpha; the second normalizes it transverse to the contact planes xi. Because alpha ^ (d alpha)^n is a volume form, these two conditions have exactly one solution, so R exists and is unique once alpha is fixed.
Compare this to the Hamiltonian vector field of Guide 2. There, given energy H, you solved omega(X_H, .) = dH to get the flow. Here d alpha plays the role of omega on the planes xi, and the normalization alpha(R) = 1 plays the role of fixing the energy level. The Reeb flow is the natural 'clock' on a contact manifold; on the unit cotangent bundle of a Riemannian manifold, where the canonical contact form restricts the Liouville form, the Reeb flow is exactly the geodesic flow. That single example is worth memorizing: Reeb orbits generalize closed geodesics.
Inside a contact manifold there is also a distinguished class of submanifolds, the odd-dimensional cousins of Lagrangian submanifolds. A Legendrian submanifold is an n-dimensional submanifold L of the (2n+1)-dimensional M that is everywhere tangent to the contact planes — that is, alpha restricted to L vanishes. Legendrian knots in the standard contact R^3 are the playground where contact topology became visibly rich: two Legendrian knots can be smoothly isotopic yet Legendrian-distinct, detected by invariants like the Thurston-Bennequin number.
Symplectization and contact type: gluing the two worlds
The even and odd worlds are not separate countries; one is the boundary of the other. Given a contact manifold (M, alpha), the product R x M with the 2-form omega = d(e^s alpha), where s is the coordinate on R, is a genuine symplectic manifold — the symplectization of M. Conversely, the boundary of many symplectic manifolds inherits a contact structure: a hypersurface in a symplectic manifold is of 'contact type' when there is a transverse vector field Y with Lie derivative L_Y omega = omega, and then alpha = (iota_Y omega) restricted to the hypersurface is a contact form. This is exactly the energy hypersurface picture from Guide 2 made structural.
Why care? Because closed Reeb orbits on a contact-type hypersurface are precisely periodic orbits of the Hamiltonian system at fixed energy. The Weinstein conjecture — that every closed contact manifold has at least one closed Reeb orbit — is the modern, geometric incarnation of the centuries-old hunt for periodic orbits in mechanics. It was proved in dimension three by Taubes using Seiberg-Witten theory, and remains open in general. That honest gap is typical of this field: the conjectures are clean, the proofs draw on the heaviest machinery available, and 'open in general' is the normal state of affairs.
A glimpse of symplectic topology: non-squeezing
We close the whole rung with the result that turned symplectic geometry into symplectic topology. From Guide 2 you know symplectomorphisms preserve the Liouville volume omega^n. A naive guess: maybe volume-preservation is the only constraint, so anything you can do to a region with a volume-preserving map you can also do symplectically. Gromov's 1985 theorem shatters that guess.
Gromov's non-squeezing theorem says: in standard R^(2n) with coordinates (q_1, p_1, ..., q_n, p_n), a ball of radius r can be symplectically embedded into the cylinder { q_1^2 + p_1^2 < R^2 } only if r is at most R. The cylinder has infinite volume, so a volume argument permits any r — yet symplectically the ball cannot be squeezed past radius R in the single conjugate plane (q_1, p_1). This is genuine rigidity that volume cannot see. The proof introduced pseudo-holomorphic curves, the technique behind Gromov-Witten invariants and Floer theory; here we state and motivate it, we do not prove it — a real proof is a course, not a paragraph.
Where this rung leaves you
Step back and see the arc of the five guides. Darboux gave us no local invariants; Hamiltonian flows and the Poisson bracket turned classical mechanics into geometry; Lagrangian submanifolds and Weinstein's neighborhood theorem said the symplectic world is organized around its Lagrangians; moment maps and reduction extracted symmetry; and now contact geometry and non-squeezing show the odd-dimensional boundary and the surprising rigidity hiding inside.
Honest next steps, with prerequisites stated. To go further into symplectic topology you need pseudo-holomorphic curves, which lean on elliptic PDE and the analysis of moduli spaces — McDuff and Salamon's books are the standard climb. For the algebraic flavor, Floer homology connects to the index theory you may meet elsewhere in this ladder. None of that is light; all of it builds directly on what these five guides made concrete. You now hold the vocabulary and the pictures; the theorems are the next mountain.