incompressible surfaces found in Haken manifolds. Heegaard splitting A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory. Taut foliation A taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Taut foliations were brought to prominence by the work of William Thurston and David Gabai. Foundational results Some results are named as conjectures as a result of historical artifacts. We begin with the purely topological: Moise's theorem In geometric topology, Moise's theorem, proved by Edwin E. Moise in, states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure. As corollary, every compact 3-manifold has a Heegaard splitting. Prime decomposition theorem The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension. Kneser–Haken finiteness Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements. Loop and Sphere theorems The loop theorem is a generalization of Dehn's lemma and should more properly be called the "disk theorem". It was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if there is a map with not nullhomotopic in , then there is an embedding with the same property. The sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented