3-manifold

A three-manifold is a topological manifold locally modeled on ${\mathbb{R}}^3$, or on a half-space at its boundary.

Relative to a triangulation, a normal surface meets every tetrahedron in disjoint normal disks: four types of triangles cutting off vertices and three types of quadrilaterals separating opposite pairs of edges. A normal disk meets any edge at most once; its face arcs join distinct edges. Two different quadrilateral types cannot coexist in a tetrahedron.

For $t$ tetrahedra, count the seven disk types in each tetrahedron. Across each glued face the counts of each of its three normal arc types must agree. These are homogeneous linear equations with integer coefficients. An admissible solution also has at most one nonzero quadrilateral type in each tetrahedron. Matching the arcs in their boundary order reconstructs the normal surface up to an isotopy preserving the tetrahedra.

The weight of a normal surface counts its intersections with the edges of the triangulation. It is a positive linear function of its coordinates: each corner of a normal disk contributes the reciprocal of the number of tetrahedral sectors around its edge. Thus a nontrivial decomposition into compatible coordinate vectors gives both summands strictly smaller weight.

The Haken sum of compatible normal surfaces is obtained by resolving their intersection curves in the way that preserves the normal disk types. It adds normal surface coordinates, Euler characteristic and mod-two homology classes. The alternative local resolution can create returning face arcs and need not be normal.

A fundamental normal surface has a nonzero admissible coordinate vector which is not the sum of two nonzero admissible vectors. Fixing the allowed quadrilateral type in every tetrahedron reduces the problem to the Hilbert basis of a rational cone. There are finitely many choices of quadrilateral types and finitely many indecomposable vectors for each choice.

A punctured three-sphere is $S^3$ with the interiors of finitely many disjoint piecewise linear balls removed. Capping its spherical boundary components with balls restores $S^3$. Removing another ball or gluing two such manifolds along one spherical boundary component preserves this class.

Suppose disjoint separating spheres cut a 3-manifold into pieces none of which is a punctured three-sphere. Compress one sphere along a disk disjoint from the others. The old sphere and the two new spheres enclose a three-times punctured sphere. At least one of the two pieces on the compressed side is not a punctured sphere; retain that new sphere and discard the other. The piece on its opposite side cannot become a punctured sphere, since capping its boundaries and applying the Three-dimensional Schoenflies theorem would make the original opposite piece a punctured sphere as well. Hence the number of spheres and the nontriviality condition can be preserved.

A three-ball is a three-manifold homeomorphic to the closed unit ball in ${\mathbb{R}}^3$. Its boundary is a two-sphere. Attaching a three-ball to a spherical boundary component is called capping that boundary; marked endpoints may be joined by a straight arc inside the cap.

A closed simply connected three-manifold is homeomorphic to the three-sphere. Now a theorem, this identifies every homotopy three-sphere with the standard sphere and removes homotopy-sphere ambiguities in three-dimensional connected-sum classification. Smooth structures are also unique in dimension three.

If a boundary surface inclusion has noninjective fundamental group map, there is a compression disk whose boundary represents a nontrivial element of that kernel. More generally the boundary can be required to avoid a prescribed normal subgroup if a singular disk with that property exists.

Start with a singular disk whose boundary represents a prescribed nontrivial class in a boundary subsurface. Repeatedly lift it through double covering spaces of regular neighbourhoods of its image. The number of distinct image simplexes strictly increases and is bounded by the number of domain simplexes, so the tower ends. At the top, vanishing first mod-two homology forces every boundary component to be a sphere. An embedded disk can then be constructed and projected down the tower. Resolving double curves preserves a boundary class outside a chosen normal subgroup, giving an embedded compression disk at the bottom.

A simple closed curve on the boundary of a three-manifold which is nullhomotopic in the manifold bounds a properly embedded disk. The boundary curve is retained; the Loop theorem can instead find a different simple essential curve in a nontrivial kernel.

If an orientable three-manifold has nonzero ${\pi }_2$, it contains an embedded essential two-sphere. Consequently a closed orientable irreducible three-manifold with infinite fundamental group is aspherical: its simply connected noncompact universal cover has zero second homotopy and top ordinary homology, hence is acyclic and contractible by the Hurewicz theorem and Whitehead theorem.

A complete hyperbolic three-manifold is a complete Riemannian manifold of dimension three with constant sectional curvature $-1$. Its universal cover is hyperbolic space ${\mathbb{H}}^3$ and its deck group acts freely and discretely by Riemannian isometries. In the orientable case it is a torsion-free Kleinian group.

For complete finite-volume hyperbolic space manifolds of dimension at least three, an isomorphism of fundamental groups is induced, up to the usual basepoint-path choice, by a unique hyperbolic isometry. The finite-volume noncompact case includes the Prasad extension. Thus complete hyperbolic metrics on a fixed finite-volume three-manifold have no deformation moduli, unlike metrics on closed hyperbolic surfaces.

A smooth, piecewise-linear or locally flat embedded two-sphere in the three-sphere has two complementary components, each with closure homeomorphic to a three-ball. Equivalently a tame embedded two-sphere in ${\mathbb{R}}^3$ bounds a ball on its compact side. Local flatness is essential; arbitrary wild sphere embeddings are excluded.

A closed orientable three-manifold is geometrizable if its prime summands can be split along finitely many disjoint essential tori into pieces whose interiors carry complete geometric structures modeled on the eight Thurston geometries. Prime decomposition and capping sphere boundaries precede the torus cuts; a single homogeneous geometry on the original connected sum is not required.

Every closed orientable three-manifold admits a prime and torus decomposition into pieces modeled on the eight Thurston geometries. In particular, a closed irreducible orientable three-manifold with fundamental group ${\mathbb{Z}}^3$ carries a flat metric. The Bieberbach theorem then identifies it with the three-torus because an abelian crystallographic group has trivial linear holonomy. Simply connected closed three-manifolds are three-spheres by the Poincaré conjecture.

The eight simply connected three-dimensional homogeneous model geometries are $S^3$, ${\mathbb{R}}^3$, ${\mathbb{H}}^3$, $S^2\times \mathbb{R}$, ${\mathbb{H}}^2\times \mathbb{R}$, ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$, $\mathrm{Nil}$ and $\mathrm{Sol}$. A geometric structure is a locally homogeneous structure obtained as a quotient of a model by a discrete freely acting isometry group, equivalently with compatible model charts.

A closed connected orientable smooth manifold of dimension three is prime when a decomposition as a connected sum forces at least one summand to be the three-sphere. Unlike an irreducible three-manifold, it may contain a nonseparating embedded two-sphere.

Every closed connected orientable three-manifold is a finite connected sum of prime three-manifolds, unique up to order and the insertion of three-sphere factors. The corresponding fundamental group is the free product of the summand groups. A prime orientable three-manifold is either irreducible or $S^2\times S^1$; this separates aspherical infinite-group cases from the infinite cyclic case.

In an orientable three-manifold, an embedded nonseparating two-sphere has a product neighborhood. Join its two boundary spheres by an arc outside that neighborhood and take a regular neighborhood of the union. This region is $S^2\times S^1$ with an open ball removed. Its boundary sphere exhibits the displayed connected sum. Consequently a prime but nonirreducible orientable three-manifold is $S^2\times S^1$.

A compact three-manifold is boundary-irreducible when there is no embedded disk with interior in the manifold and boundary an essential curve in its boundary. Thus its boundary has no compressing disk.

An orientable three-manifold is irreducible if every embedded two-sphere bounds a three-ball. This makes innermost-disk simplifications of embedded topological surfaces possible without changing the ambient three-manifold.

In the orientable case an excellent compact three-manifold is an irreducible three-manifold and a boundary-irreducible three-manifold, is not a ball, contains a two-sided properly embedded incompressible surface, and has every properly embedded incompressible zero-Euler characteristic surface a boundary-parallel surface. In particular it has no essential annulus or torus. These properties give lower bounds on the Thurston norm of classes with essential boundary.

In a compact connected orientable three-manifold with no spherical boundary components, every embedded knot is homotopic to a knot with an excellent three-manifold as exterior. In particular this can preserve the generator class in $S^1\times S^2$, or the winding number of a satellite pattern one class in a solid torus. Homotopy here is weaker than isotopy.

For a compact oriented three-manifold, the integral relative class $u$ has norm the minimum of ${\chi }_{-}(F)={\sum }_j\max (0,-\chi (F_j))$ over properly embedded oriented topological surfaces representing $u$. Homogeneity and continuity extend it to real relative homology. It is a seminorm, since spheres, disks, annuli and tori have zero cost.

For $Y=P\times S^1$ with $P$ a pair of pants and $h$ its circle fiber, a horizontal surface in a Seifert fibered space of degree $d$ has ${\chi }_{-}=|d|$; vertical surfaces in a Seifert fibered space have zero cost. Cutting and pasting these representatives, and classifying essential surfaces, proves $\Vert u{\Vert }_T=|u\cdot h|$. The dual Thurston polytope is therefore the segment between the two functionals $\pm h$ under the natural pairing.

In an irreducible three-manifold with incompressible boundary, a minimizing surface can be cut along a separating incompressible torus after removing inessential intersection circles. The essential pieces have no disk or sphere components, so their ${\chi }_{-}$ costs add. Minimizing independently in the pieces and gluing their matching boundary curves gives the corresponding upper bound. This allows Thurston norm calculations for satellite knots.

The dual polytope consists of the real linear functionals $\alpha$ on $H_2(Y,\partial Y;\mathbb{R})$ such that $|\alpha (u)|\le \Vert u{\Vert }_T$ for every $u$. Thus it is the polar of the unit ball of the Thurston norm and annihilates that seminorm's kernel.

An incompressible surface is a properly embedded two-sided topological surface with no compressing disk, subject to the usual exclusion of inessential spherical components. A compressing disk has boundary an essential closed curve on the topological surface and interior disjoint from it.

A hierarchy is a finite sequence of cuts along properly embedded two-sided incompressible surfaces, without spherical components, ending in disjoint three-balls. Disks are allowed. For an ambient manifold which is not orientable the two-sided cutting surfaces need not themselves be orientable. The existence of a hierarchy does not imply that every choice of incompressible cuts terminates.

A map between compact connected orientable irreducible 3-manifolds with nonempty boundaries, injective on the boundary and on the fundamental group, is homotopic to a homeomorphism relative to the boundary. The boundary determines degree $\pm 1$, and the cover corresponding to the image of the fundamental group consequently has one sheet. Straighten the preimage of an incompressible hierarchy surface, cut both manifolds, and repeat. At the terminal balls, extend the prescribed boundary homeomorphisms by coning. The homotopies and homeomorphisms glue across the straightened surface collars.

In an orientable irreducible three-manifold, an incompressible cut is fundamental group-injective by the Loop theorem. Lift a three-manifold hierarchy to the universal cover. The terminal ball pieces are contractible and their gluing surfaces have contractible universal covers. The incidence graph is a tree, by the van Kampen theorem normal form for successive cuts. Finite subtrees are contractible by successively attaching a terminal piece along a contractible intersection. Every sphere map has compact image contained in a finite subtree, so all higher homotopy groups of the universal cover vanish.

Let $P$ be a once-punctured torus and let $c$ be a nonseparating essential curve. The annulus $c\times [0,1]$ is an incompressible surface in $P\times [0,1]$. Cutting gives a pair of pants times an interval. Both products are genus-two handlebodies, since a compact connected surface with boundary times an interval is a handlebody of genus $1-\chi (P)$.

A compression disk for a two-sided embedded surface $F$ in a three-manifold is an embedded disk meeting $F$ exactly in its boundary, with boundary essential in $F$. Replacing an annular neighborhood of this boundary by two parallel copies of the disk reduces the surface's genus or separates it into simpler pieces.

A properly embedded topological surface is boundary-parallel if it cobounds a product region with a topological surface in the ambient boundary, with the remaining boundary of that product contained in the ambient boundary. An annulus parallel into one boundary component has zero net oriented boundary class on that component.

In an orientable irreducible Seifert fibered space, an essential two-sided incompressible surface that is also a boundary-incompressible surface can be isotoped to a horizontal surface in a Seifert fibered space or a vertical surface in a Seifert fibered space.

A properly embedded topological surface is boundary-incompressible when no disk in the ambient three-manifold joins an essential arc on the topological surface to an arc on the ambient boundary, with the disk's interior disjoint from both.

A Heegaard splitting divides a closed three-manifold into two handlebodies with common boundary. For a three-manifold with boundary, one uses compression bodies, which can retain negative boundary components.

Two nonseparating compressing disks on opposite sides with the same boundary form an embedded sphere. Cutting each handlebody along its disk leaves it connected, so the sphere is nonseparating in the glued three-manifold. Capping the two resulting sphere boundaries gives a splitting of one lower genus of the remaining summand; the removed summand is $S^1\times S^2$.

The least genus of a Heegaard surface among all Heegaard splittings of a closed orientable three-manifold. Having a splitting of genus $g$ only proves $g_H(M)\le g$; stabilization supplies larger-genus splittings of the same manifold.

Heegaard Floer homology assigns groups $H{F}^{\circ }(Y,\mathfrak{s})$ to a closed oriented three-manifold with a Spin-c structure. A pointed Heegaard diagram of genus $g$ gives product Lagrangian tori ${\mathbb{T}}_{\alpha },{\mathbb{T}}_{\beta }\subset {\mathrm{Sym}}^g(\Sigma )$; the chain complex is generated by their intersection points and its differential counts Whitney disk classes represented by holomorphic disks. The minus, infinity, plus and hat versions use different powers or quotients of the variable $U$. The Heegaard Floer cobordism maps and the Mixed Heegaard Floer invariant relate these groups to smooth four-dimensional topology.

For a closed smooth oriented four-manifold $M$ with $b_2^{+}(M)>1$, remove two balls to obtain $W:S^3\to S^3$, and choose an Admissible cut for a Heegaard Floer mixed map. Positivity and naturality show that the minus map of $W_1$ lands in $H{F}_{\mathrm{red}}^{-}$, while the plus map of $W_2$ descends from $H{F}_{\mathrm{red}}^{+}$. ThusThe coefficient of the bottom plus generator in this map, applied to $U^r{\Theta }^{-}\otimes \zeta$, defines ${\Phi }_{M,\mathfrak{s}}(U^r\otimes \zeta )\in \mathbb{Z}$, up to the orientation sign, with $\zeta \in {\Lambda }^{\ast }(H_1(M)/\mathrm{torsion})$. It vanishes unless $2r+\mathrm{deg}\zeta =(c_1(\mathfrak{s}{)}^2-2\chi (M)-3\sigma (M))/4$. The construction is independent of the admissible cut by the cobordism composition and associativity laws. Two positive pieces explain the condition $b_2^{+}>1$.

If ${\Phi }_{M,\mathfrak{s}}\ne 0$ and a connected closed oriented embedded surface $S$ has positive genus and $S^2\ge 0$, thenBlow up $S^2$ points on $S$ and choose the $\pm 1$ exceptional determinant classes which increase the Chern pairing on its proper transform by $S^2$. The Blowup formula for the mixed Heegaard Floer invariant preserves nonvanishing, while the proper transform has square zero. A cut disjoint from it allows the relevant cobordism map to pass through $H{F}^{\pm }(S\times S^1,\mathfrak{s}{|}_{S\times S^1})$. These groups vanish when the Chern pairing exceeds $2g-2$, so the mixed map would then vanish, a contradiction. Reverse the surface orientation for the absolute value. The positive-genus condition is essential: a small nullhomologous sphere has square zero and zero pairing and would violate the displayed inequality.

Let $\tilde{M}=M\#{\overline{\mathbb{CP}}}^{,2}$, and let $E$ be the exceptional sphere. Extend a Spin-c structure by $c_1(\tilde{\mathfrak{s}})=c_1(\mathfrak{s})\pm \mathrm{PD}[E]$. Then ${\Phi }_{\tilde{M},\tilde{\mathfrak{s}}}={\Phi }_{M,\mathfrak{s}}$. More generally, an evaluation $⟨c_1(\tilde{\mathfrak{s}}),E⟩=\pm (2\ell +1)$ introduces the factor $U^{\ell (\ell +1)/2}$. This follows by putting the blowup on one side of an admissible cut: the punctured exceptional piece is negative definite, its minus and plus tower maps are multiplication by this power of $U$, and the cobordism composition law gives the formula. In the $\pm 1$ case the exponent is zero, so nonvanishing is preserved.

For a knot $K$ in an integral homology three-sphere and $n>0$, fix a Spin-c structure ${\mathfrak{t}}_k$ on the $n$-surgery. Write ${\mathfrak{s}}_i$ for structures on zero-surgery with $⟨c_1({\mathfrak{s}}_i),[\hat{F}]⟩=2i$, using a capped Seifert surface. In compatible labels the completed-minus triangle isOne can equally use the plus theory without completing. The first map is the congruence-restricted zero-surgery trace map; the last is the sum of reverse, orientation-reversed $n$-surgery trace maps. The middle map uses a Heegaard triple with an auxiliary lens-space generator. For $n=1$ that lens space is $S^3$, giving the usual three two-ended cobordism maps. For $n>1$ it is important to retain the auxiliary boundary and its fixed Spin-c structure. The U-adic completion of Heegaard Floer homology makes the sums of maps well-defined.

A smooth oriented cobordism $W:Y_1\to Y_2$ with a Spin-c structure induces $U$-equivariant maps $F_{W,\mathfrak{s}}^{\circ }$ on the corresponding Heegaard Floer homology groups. A handle decomposition defines the maps using generator selection for one- and three-handles and holomorphic triangle counts for two-handles. Composition sums over extensions of the prescribed Spin-c structures. When the boundary determinant classes are torsion, the absolute grading shift isThe square is defined using the rational relative lift. If $b_2^{+}(W)>0$, the infinity map vanishes. Between rational homology spheres, when $b_1(W)=b_2^{+}(W)=0$, the infinity map is an isomorphism of Laurent towers.

An admissible cut splits a four-dimensional cobordism as $W=W_1{\cup }_N{W}_2$, with $b_2^{+}(W_1),b_2^{+}(W_2)>0$, and with the Mayer-Vietoris connecting map $H^1(N;\mathbb{Z})\to H^2(W;\mathbb{Z})$ equal to zero. The last condition removes the ambiguity in gluing Spin-c structures on the two pieces. Positivity makes both infinity cobordism maps vanish, allowing passage through the Reduced Heegaard Floer homology of $N$. A useful cut is the boundary of a neighborhood of a positive-square embedded surface, chosen so that its complement still has a positive direction. The nonzero normal Euler number makes $H^1$ of the neighborhood surject onto $H^1$ of its boundary, proving the connecting-map condition.

Use the convention that $C{F}^{-}$ is generated by $[\mathbf{x},i]$ with $i<0$, and $U[\mathbf{x},i]=[\mathbf{x},i-1]$. The chain complexes areThe last complex is generated by $[\mathbf{x},0]$. Equivalently it is $C{F}^{-}/UC{F}^{-}$ with its absolute grading shifted up by two. The short exact sequence $0\to C{F}^{-}\to C{F}^{\infty }\to C{F}^{+}\to 0$ gives the fundamental long exact sequence. For a rational homology sphere in each Spin-c structure, $H{F}^{\infty }\cong \mathbb{Z}[U,U^{-1}]$. The plus tower is ${\mathcal{T}}_d^{+}=\mathbb{Z}[U,U^{-1}]/U\mathbb{Z}[U]$, with bottom degree $d$; its corresponding minus tower is a free $\mathbb{Z}[U]$ module with top degree $d-2$.

Let ${\mathfrak{s}}_k$ on $S^1\times S^2$ have $⟨c_1({\mathfrak{s}}_k),[S^2]⟩=2k\ne 0$. The ordinary groups areA sufficiently wound genus-one diagram gives a two-generator summand whose chain differential is multiplication by $1-U^{|k|}$: the two relevant disks have opposite coherent signs and basepoint multiplicities differing by $|k|$. The map is injective over $\mathbb{Z}[U]$, so its homology is the displayed quotient. Multiplication by $U$ is invertible in the quotient, showing that minus-to-infinity is an isomorphism and that the plus and hat groups vanish. In the U-adic completion of Heegaard Floer homology, $1-U^{|k|}$ is a unit, so both completed minus and infinity groups vanish. This example shows why genus bounds for the completed-minus or plus groups must not be transferred unchanged to the ordinary minus theory.

The reduced minus and plus groups areThe connecting homomorphism of the fundamental long exact sequence induces an isomorphism $\delta :H{F}_{\mathrm{red}}^{+}\to H{F}_{\mathrm{red}}^{-}$ of degree $-1$. This follows immediately from exactness: its image is the indicated kernel, and its kernel is precisely the image quotiented out on the plus side. These groups isolate the information lost by inverting $U$.

Completing the minus chain complex in powers of $U$ replaces $\mathbb{Z}[U]$ by the formal power series ring $\mathbb{Z}[[U]]$; its homology is written ${\mathbf{HF}}^{-}$. This permits infinite sums of Heegaard Floer cobordism maps whose exponents tend to infinity, as occur in surgery formulas. A series with constant coefficient $\pm 1$ is a unit: its inverse is found recursively, or by the geometric series for its positive-order part. Such a series need not have a polynomial inverse, which is why exact triangles involving sums over all Spin-c extensions require a completion in the minus theory. Individual homogeneous cobordism maps are still defined on the ordinary minus groups.

For an admissible pointed Heegaard diagram, $C{F}^{-}$ is a free $\mathbb{Z}[U]$ chain complex with one generator for each point of ${\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }$. Each generator is an unordered selection containing exactly one intersection on each alpha circle and exactly one on each beta circle. After choosing coherent orientations,Here $\hat{\mathcal{M}}$ is the moduli space modulo strip translation, $\mu$ is the Maslov index, and $n_z$ is multiplicity at the basepoint. Holomorphic representatives have nonnegative domain multiplicities. The differential lowers grading by one and $U$ lowers it by two. Boundary strata of one-dimensional moduli spaces give the cancellation proving ${\partial }^2=0$. Its homology is $H{F}^{-}$.

Over $\mathbb{Z}$, holomorphic disk moduli spaces in a Heegaard Floer chain complex must be oriented compatibly with gluing: the orientation of a concatenated disk class is the product determinant-line orientation of its two factors. Boundary orientations of compactified one-dimensional moduli spaces then make broken disks cancel, yielding ${\partial }^2=0$. Isolated regular disks contribute $+1$ or $-1$. Changing the chosen orientation at an intersection generator changes the corresponding row and column signs of the differential and yields an isomorphic integral chain complex. Cobordism triangle counts require compatible orientation choices as well; an overall orientation ambiguity accounts for the usual sign ambiguity of the closed invariant.

An embedded positive domain which is a bigon or rectangle, has convex alternating alpha and beta corners, and contains no other selected coordinate in its interior, has Maslov index one. A bigon has Euler measure $1/2$ and two corner averages $1/4$; a rectangle has Euler measure zero and four corner averages $1/4$. For the usual regular complex structure, its holomorphic disk count modulo translation is $\pm 1$. For a bigon the Riemann map to the strip is unique up to translation. For a rectangle, the tautological correspondence represents the disk in the symmetric product by a branched double cover of the strip onto the rectangular domain: its conformal modulus fixes the branch parameter, and the only remaining freedom is translation. The resulting isolated regular disk gives one point in the oriented moduli space. The differential coefficient is $\pm U^{n_z}$, so a basepoint-free domain contributes a unit.

For generators in the same Spin-c structure and a Whitney disk class $\psi$ joining them,The ambiguity is the divisibility of $c_1(\mathfrak{s})$ on $H_2(Y)$; a torsion determinant class gives an integer relative grading. Adding a sphere class changes $\mu$ by two and $n_z$ by one, so it leaves the difference unchanged. For a domain $D$, the index can be calculated as $\mu (D)=e(D)+n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)$, where $e$ is Euler measure and each $n_{\mathbf{x}}$ is the sum of averages of the four adjacent region multiplicities at the selected intersections.

For a torsion Spin-c structure, the relative grading of Heegaard Floer homology lifts canonically to a rational absolute grading, normalized by the bottom plus generator of $S^3$ having degree zero. Multiplication by $U$ has degree $-2$, and a Heegaard Floer cobordism map has degree $(c_1^2-2\chi -3\sigma )/4$. The correction term $d(Y,\mathfrak{s})$ is the lowest grading of the image of $H{F}^{\infty }$ in $H{F}^{+}$ for a rational homology sphere. A pure tower then has bottom plus degree $d$ and top minus degree $d-2$ in the $i<0$ minus convention. Keeping that shift explicit prevents errors of two in surgery and mixed-invariant calculations.

For intersection generators $\mathbf{x},\mathbf{y}$, a Whitney disk is a map of a disk into the symmetric product, with its two boundary semicircles mapped to the two product tori and its endpoints mapped to $\mathbf{x},\mathbf{y}$. Its homotopy class belongs to ${\pi }_2(\mathbf{x},\mathbf{y})$. Projecting it to the Heegaard surface gives an integral domain. With the surface oriented, the convention is ${\partial }_{\alpha }D=\mathbf{y}-\mathbf{x}$. The obstruction to a disk is the class $ϵ(\mathbf{x},\mathbf{y})\in H_1(Y)$ obtained by joining the points along alpha arcs and returning along beta arcs. In the Heegaard Floer setup, its vanishing is equivalent to equality of the generators' Spin-c structures; for the rational homology sphere diagrams this is equivalent to ${\pi }_2(\mathbf{x},\mathbf{y})\ne \varnothing$.

A compression body is obtained from a product $F\times I$ by attaching one-handles to $F\times \{1\}$, allowing three-ball components as well. Its negative boundary is the retained copy of $F$ and its positive boundary is the other boundary component. A handlebody has empty negative boundary.

The Heegaard surface is the common positive boundary of the two pieces in a Heegaard splitting.

A Heegaard diagram marks the boundaries of compressing disks from the two sides of a Heegaard surface. Attaching two-handles along the two collections, and capping resulting spherical boundary components, reconstructs the three-manifold.

A generalized Heegaard diagram permits compression bodies and a nonempty negative boundary. Equivalently, a suitable handlebody with two-handles attached along only part of a compressing system can retain torus boundary components. Disk crossings supply generators and attaching-curve words supply relators for its fundamental group.

Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.

A genus-$g$ handlebody is a three-ball with $g$ one-handles attached. Its boundary is a closed orientable topological surface of genus $g$.

Every connected closed orientable three-manifold with finite fundamental group admits spherical geometry. With cyclic fundamental group, it is a lens space.