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Contact topology studies global properties of contact structures on odd-dimensional smooth manifolds. Locally such a structure is a hyperplane distribution $\xi =\ker \alpha$ with $\alpha \wedge (d\alpha {)}^n$ nowhere zero in dimension $2n+1$. Multiplying $\alpha$ by a nowhere-zero function leaves the hyperplane distribution unchanged: the structure is the field of hyperplanes, not a preferred one-form.^[ref]

In ${\mathbb{R}}^3$, take $\alpha =dz-ydx$. Then $d\alpha =dx\wedge dy$ and $\alpha \wedge d\alpha =dz\wedge dx\wedge dy\ne 0$. The spanning vector fields ${\partial }_x+y{\partial }_z$ and ${\partial }_y$ have Lie bracket $-{\partial }_z$, outside the distribution. Thus the planes cannot be tangent to a foliation by surfaces. The local normal form is universal, while global topology distinguishes contact structures and constrains their relation to symplectic geometry.

Robert E. Gompf studies four-dimensional topology, symplectic topology and contact topology. His University of Texas research profile also identifies Stein surfaces as a focus.^[ref]

His symplectic sum construction glues symplectic manifolds along suitable codimension-two submanifolds. Among its consequences, every finitely presented group occurs as the fundamental group of a compact symplectic manifold in each even dimension at least four. This shows how flexible symplectic geometry can be compared with the additional restrictions imposed by a Kähler manifold.^[ref]