Geometric Surgery Protocol

• Geometric surgery protocol is a set of explicit procedures for modifying geometric objects by removing, deforming, quotienting, or regluing controlled regions while ensuring global compatibility conditions.
• It spans diverse applications including hyperbolic and Seifert manifold constructions, discrete Ricci and mean curvature flows, and quantum code deformations, offering practical methods for topological and metric transformations.
• The protocol framework extends to four-dimensional surgery, diagrammatic and fold-map realizations, and spacetime surgery, linking geometric operations to algebraic invariants and quantum computational models.

“Geometric surgery protocol” denotes a family of explicit procedures for modifying a geometric object by removing, deforming, quotienting, or regluing a controlled local region while tracking the resulting global structure. In the cited literature, the phrase is used for Dehn-type and cone-manifold constructions on hyperbolic and Seifert manifolds, surgery-through-singularity in discrete Ricci flow and mean curvature flow, homotopy-theoretic and $4$-dimensional surgery schemes, diagrammatic conversions from Heegaard or trisection data, metric replacement operations, and fault-tolerant code deformations in CSS and surface-code settings (Molnár et al., 2020, Lozano et al., 2014, Alsing et al., 2017, Haslhofer, 2023, Crabb et al., 2016, Nikolic et al., 2023, Clay et al., 31 Jul 2025, Chamberland et al., 2021, Chamberland et al., 2022, Poirson et al., 2 May 2025). This suggests a common structural pattern: identify an interface, impose compatibility conditions, replace it by a model piece or quotient, and read off the new object through holonomy, bordism, curvature, or logical action.

1. Recurring formal pattern

Across these works, a geometric surgery protocol is not a single theorem but a recurrent method. The local datum may be a cusp neighborhood, a neck region, a regular-value neighborhood of a map, a handlebody around a bouquet of closed leaves, a metric subspace $S\subseteq X$, or a subcode in a CSS complex. The replacement datum may be a cap, a compact core, a quotient by a subspace, a gluing by a mapping class, or a merge/split map. The output is then characterized by induced geometric structures, new singular loci, bordism invariants, or logical operators (Falbel et al., 2024, Clay et al., 31 Jul 2025, Poirson et al., 2 May 2025).

Setting	Local datum	Output
Hyperbolic / flag geometry	cusp, bouquet, handlebody	compactified manifold or glued flag manifold
Geometric flows	neck, pinched edge, surgery scale $r_\sharp$	continued flow past singularity
Metric spaces	subspace $S\subseteq X$, map $f:S\to T$	metric quotient $\widehat{X}_f$
Surgery theory / QEC	kernel complex, subcode, annular boundary	obstruction class or logical gate

A plausible implication is that “geometric” here always indicates that the interface is modeled explicitly enough to support a compatibility theorem. In some papers that theorem is analytic, as in hybrid compactness for mean curvature flow; in others it is algebraic, as in long exact sequences for CSS surgery or induced quadratic structures from the geometric Hopf invariant (Haslhofer, 2023, Poirson et al., 2 May 2025, Crabb et al., 2016).

2. Hyperbolic, Seifert, and flag-structure constructions

For the Gieseking hyperbolic ideal simplex manifold $\mathcal S$, the protocol begins with a one-cusped non-orientable hyperbolic $3$-manifold realized by a single ideal tetrahedron with vertices $0,1,z,\infty$ and anti-holomorphic face pairings $z_1,z_2$. The deformation keeps the edge-cycle constraint $S\subseteq X$0, computes the cusp stabilizer on a horosphere, isolates the “critical surgery transform” $S\subseteq X$1, and forces this holonomy to be a finite-order rotation by imposing

$S\subseteq X$2

The ideal cusp is then replaced by a compact “solid Klein bottle” type piece built around a segment of the axis $S\subseteq X$3, yielding a compact polyhedral fundamental domain and a compact group presentation. The four root branches produce the families Gies.1–Gies.4. The corrected interpretation is that Gies.1 and Gies.2 are cone manifolds for $S\subseteq X$4, with cone angle $S\subseteq X$5, and orbifolds only at $S\subseteq X$6; Gies.3 and Gies.4 are orbifolds for all $S\subseteq X$7. This resolves the earlier mistaken claim that the small-volume first family consisted of orbifolds. The asymptotics separate sharply: in Gies.1 the volume drops from $S\subseteq X$8 at $S\subseteq X$9 to $r_\sharp$0 at $r_\sharp$1 and tends to $r_\sharp$2, whereas Gies.3 tends back to the regular ideal simplex with limiting volume $r_\sharp$3 (Molnár et al., 2020).

For $r_\sharp$4-surgery on the left-handed trefoil knot $r_\sharp$5, the protocol is organized by the Seifert structure

$r_\sharp$6

with singular set the core of the surgery and cone angle $r_\sharp$7. The key parameters are

$r_\sharp$8

and the base-orbifold parameter

$r_\sharp$9

The sign of $S\subseteq X$0 determines the $S\subseteq X$1-geometry: $S\subseteq X$2 gives $S\subseteq X$3, $S\subseteq X$4 gives Nil, and $S\subseteq X$5 gives $S\subseteq X$6. Equivalently, for $S\subseteq X$7, the cone-manifold is spherical if

$S\subseteq X$8

Nil if

$S\subseteq X$9

and $f:S\to T$0 if

$f:S\to T$1

The central geometric transition is therefore

$f:S\to T$2

with Nil at the Euclidean threshold of the base orbifold (Lozano et al., 2014).

Three-dimensional flag structures provide a different higher-rank model. Here the geometry is $f:S\to T$3, where

$f:S\to T$4

equipped with the two circle foliations

$f:S\to T$5

A loxodromic element has attracting and repelling $f:S\to T$6-$f:S\to T$7 bouquets

$f:S\to T$8

The surgery removes genus-two handlebodies around bouquets in two flag manifolds and reglues the complements using an anti-flag involution

$f:S\to T$9

or a conjugate $\widehat{X}_f$0, arranged so that $\widehat{X}_f$1 exchanges the inner and outer boundaries of a nested handlebody pair. The main existence theorem states that if the chosen bouquets admit neighborhoods flag-isomorphic to open subsets of $\widehat{X}_f$2, then the surgery exists. A second theorem states that a flag surgery of Kleinian flag manifolds is again Kleinian. By contrast, a special construction on $\widehat{X}_f$3 yields surgeries whose developing map is surjective onto $\widehat{X}_f$4, and since the fundamental group is infinite such structures are not virtually Kleinian (Falbel et al., 2024).

3. Surgery in geometric flows

In discrete Ricci flow, surgery is implemented on an axially symmetric piecewise-linear dumbbell geometry. The edge-length evolution is

$\widehat{X}_f$5

with

$\widehat{X}_f$6

The model uses $\widehat{X}_f$7 icosahedral cross-sections, evolves by RK4 with $\widehat{X}_f$8, and remeshes every $\widehat{X}_f$9 time steps by cubic spline to keep the circumcenter inside each frustum block. The neckpinch occurs at $\mathcal S$0. The surgery protocol is explicit: remove the pinched axial edge $\mathcal S$1, thereby disconnecting the geometry into left and right lobes; cap the open ends by gluing an icosahedron of edge length $\mathcal S$2 on one side and $\mathcal S$3 on the other; remesh each capped lobe using a cubic spline; then continue the same discrete Ricci flow on both components. A more refined cap replacing the last three $\mathcal S$4-variables and two $\mathcal S$5-variables by spherical-cap values was also implemented, but the simpler protocol gave the same result. The post-surgery components each evolve toward a collapsing $\mathcal S$6-sphere geometry, giving the expected decomposition into two constant-curvature Thurston geometries (Alsing et al., 2017).

For mean curvature flow with surgery, the object is a $\mathcal S$7-flow of closed mean-convex surfaces in $\mathcal S$8, where

$\mathcal S$9

A strong $3$0-neck is a region that is parabolically $3$1-close to the shrinking round cylinder $3$2. At a surgery time, a minimal collection of such necks at curvature scale $3$3 is selected to separate the high-curvature set $3$4 from the thick region $3$5. The necks are replaced by standard caps inside $3$6, with scale-invariant curvature bounds

$3$7

and components with

$3$8

everywhere are discarded; from each pair of facing surgery caps, precisely one is discarded. The novel analytical point is the hybrid compactness theorem: when the surgery scale remains visible in a blowup limit, the sequence converges to an ancient Brakke $3$9-flow, with smooth convergence near surgery regions and weak Brakke convergence elsewhere. This replaces the older estimate-first strategy by a compactness-and-classification argument and still yields existence of surgery flows for suitable $0,1,z,\infty$0 and $0,1,z,\infty$1 (Haslhofer, 2023).

4. Non-simply-connected and four-dimensional surgery theory

In homotopy-theoretic surgery, the core protocol starts from a stable map

$0,1,z,\infty$2

and constructs the geometric Hopf invariant

$0,1,z,\infty$3

For a representative $0,1,z,\infty$4, the unstable version is a $0,1,z,\infty$5-equivariant relative difference

$0,1,z,\infty$6

In the surgery application, an $0,1,z,\infty$7-dimensional normal map $0,1,z,\infty$8 yields a $0,1,z,\infty$9-equivariant Umkehr map

$z_1,z_2$0

Its geometric Hopf invariant induces the quadratic structure $z_1,z_2$1 on the kernel complex $z_1,z_2$2, where

$z_1,z_2$3

The resulting quadratic Poincaré complex $z_1,z_2$4 is precisely Wall’s obstruction

$z_1,z_2$5

The protocol is thus geometry $z_1,z_2$6 stable Umkehr map $z_1,z_2$7 geometric Hopf invariant $z_1,z_2$8 quadratic chain structure $z_1,z_2$9 $S\subseteq X$00-theory obstruction (Crabb et al., 2016).

Freedman and Krushkal formulate a different $S\subseteq X$01-dimensional protocol: the reduction of arbitrary unobstructed topological surgery problems to $S\subseteq X$02-$S\subseteq X$03-null universal models. Their kernels have the form

$S\subseteq X$04

intermediate between classical universal models and fully $S\subseteq X$05-null kernels. The group-theoretic engine is the $S\subseteq X$06-Engel relation

$S\subseteq X$07

together with the fact that any $S\subseteq X$08-Engel group is nilpotent of class $S\subseteq X$09. Deep commutators representing surgery-kernel attaching curves are rewritten in the Milnor group as products of special Engel-type commutators, realized geometrically by elementary Engel links. Handle slides then convert the kernel into an $S\subseteq X$10-trivial link with capped-grope duals, yielding the universality theorem for $S\subseteq X$11-$S\subseteq X$12-null models. The same technology shows that the weak $S\subseteq X$13-Null Disk Lemma is sufficient for good groups (Freedman et al., 2017).

Politarczyk studies a class of closed oriented $S\subseteq X$14-manifolds obtained from manifolds with free fundamental group by surgery on loops representing relators of a presentation. In the even case the normal $S\subseteq X$15-type is $S\subseteq X$16, with

$S\subseteq X$17

and the resulting manifolds $S\subseteq X$18 are produced by $S\subseteq X$19-realizing surgery on loops of type I. In the odd case one has type II surgery. The main stable classification statement is that two closed connected orientable smooth $S\subseteq X$20-manifolds obtained by $S\subseteq X$21-realizing surgery on loops with the same normal $S\subseteq X$22-type are stably diffeomorphic if and only if their signatures agree. In the odd formulation there is also the criterion

$S\subseteq X$23

A second strand of the paper realizes Tietze moves by Kirby calculus: some moves give diffeomorphic thickenings, while adding or removing a trivial relator yields

$S\subseteq X$24

This ties algebraic presentation changes directly to stable diffeomorphism (Politarczyk, 2013).

5. Diagrammatic and fold-map realizations

A planar Heegaard diagram $S\subseteq X$25 can be converted into a framed surgery link in $S\subseteq X$26 by first choosing an auxiliary system $S\subseteq X$27 so that $S\subseteq X$28 is the standard Heegaard diagram of $S\subseteq X$29. One then finds a product of right Dehn twists

$S\subseteq X$30

sending each $S\subseteq X$31 to a curve isotopic to $S\subseteq X$32. The surgery link is obtained by pushing the twisting curves $S\subseteq X$33 into the handlebody $S\subseteq X$34, ordered by depth according to the composition. The framing of a component corresponding to a twisting curve $S\subseteq X$35 is

$S\subseteq X$36

where

$S\subseteq X$37

is the self-linking relative to $S\subseteq X$38. This gives an explicit conversion from Heegaard data to surgery data (Nikolic et al., 2023).

For $S\subseteq X$39-knots in $S\subseteq X$40-manifolds, a doubly pointed trisection diagram

$S\subseteq X$41

encodes a $S\subseteq X$42-bridge trisection. Puncturing at $S\subseteq X$43 yields an annular arced relative trisection diagram for the exterior

$S\subseteq X$44

Surgery is then performed by gluing in standard filling pieces. If $S\subseteq X$45, sphere surgery is

$S\subseteq X$46

while Gluck surgery is

$S\subseteq X$47

If $S\subseteq X$48, one obtains $S\subseteq X$49-blowdown by gluing $S\subseteq X$50; if $S\subseteq X$51, one obtains $S\subseteq X$52-rational blowdown via

$S\subseteq X$53

The significance is that these cut-and-paste operations become local replacement rules on trisection diagrams rather than only manifold-level constructions (Gay et al., 2018).

For stable fold maps $S\subseteq X$54, Kitazawa introduces ATSS operations, a refined bubbling surgery based on a compatible normal system

$S\subseteq X$55

of immersed sphere-bundle neighborhoods in a regular-value region. One removes

$S\subseteq X$56

and replaces the trivial bundle projection there by local fold-map models built from a Morse function with one singular point. The new singular value set is

$S\subseteq X$57

and all new singular points have index $S\subseteq X$58. The paper isolates the case $S\subseteq X$59, where the number of connected components of the singular set increases by two. The novelty is that the generating spheres may be immersed with normal double crossings; the resulting Reeb-space cohomology ring then exhibits cup products not available in the earlier disjoint-embedding cases (Kitazawa, 2020).

6. Metric and quantum-code surgery

In metric geometry, the protocol begins with a metric space $S\subseteq X$60, a subspace $S\subseteq X$61, a target metric space $S\subseteq X$62, and a map

$S\subseteq X$63

One forms the quotient

$S\subseteq X$64

defines admissible alternating $S\subseteq X$65- and $S\subseteq X$66-sequences, assigns to such a sequence

$S\subseteq X$67

the length

$S\subseteq X$68

then sets

$S\subseteq X$69

and finally collapses zero pseudo-distance to obtain the metric space $S\subseteq X$70 with natural map

$S\subseteq X$71

If $S\subseteq X$72 is a pseudo-isometry, then $S\subseteq X$73 is a pseudo-isometry. The proof depends critically on the lack of an additive constant in the upper bound

$S\subseteq X$74

The quasi-isometry analogue fails: in the paper’s example, $S\subseteq X$75 is surgered to $S\subseteq X$76, so the output is not quasi-isometric to the original space (Clay et al., 31 Jul 2025).

In topological quantum coding, lattice surgery appears in two contrasting forms. The twist-free protocol for surface codes replaces direct $S\subseteq X$77-type surgery by two ordinary $S\subseteq X$78-type logical measurements mediated by a logical ancilla $S\subseteq X$79: first measure $S\subseteq X$80, then $S\subseteq X$81, then combine the outcomes and, if needed, apply a Pauli-frame correction after measuring $S\subseteq X$82 in the $S\subseteq X$83 basis. This avoids bulk twist defects and weight-five twist stabilizers, at the cost of roughly a $S\subseteq X$84 runtime overhead for $S\subseteq X$85-containing operations. The same paper introduces temporally encoded lattice surgery, where a redundant family of commuting Pauli measurements is chosen according to a classical $S\subseteq X$86 code with generator matrix $S\subseteq X$87; inconsistencies in $S\subseteq X$88 detect timelike measurement faults, and in realistic regimes this reduces runtime to about $S\subseteq X$89 of conventional sequential Pauli-based computation for $S\subseteq X$90 (Chamberland et al., 2021).

The twist-based alternative instead keeps direct $S\subseteq X$91-measurement capability. Its routing region includes domain-wall stabilizers, elongated checks, and weight-five twist-defect stabilizers measured with two ancillas prepared in a GHZ state. The protocol preserves a depth-four syndrome-extraction cadence but assumes enhanced degree-eight connectivity for some qubits. Circuit-level simulations with a biased depolarizing noise model show a slight decrease in the threshold for timelike logical failures relative to twist-free $S\subseteq X$92-type surgery, but comfortably below threshold—explicitly, for CNOT infidelities below about $S\subseteq X$93—the degradation is mild and preferable to an alternative twist-free scheme with about a $S\subseteq X$94 runtime penalty (Chamberland et al., 2022).

The CSS-surgery framework abstracts these ideas to arbitrary CSS codes. A CSS code is represented by

$S\subseteq X$95

with logical spaces

$S\subseteq X$96

The key notion is a subcode $S\subseteq X$97, and a quotient $S\subseteq X$98-merge is the quotient complex

$S\subseteq X$99

The induced logical action is governed by the long exact sequence of

$r_\sharp$00

For CNOT, one chooses an ancilla code $r_\sharp$01 and two subcodes: $r_\sharp$02 The first quotient identifies the control $r_\sharp$03-logical with the ancilla $r_\sharp$04-logical, the second identifies the target $r_\sharp$05-logical with the ancilla $r_\sharp$06-logical, and a final ancilla $r_\sharp$07-measurement completes a logical CNOT between any two logical qubits of any CSS code (Poirson et al., 2 May 2025).

7. Spacetime surgery and gauge-theory engineering

In $r_\sharp$08D and $r_\sharp$09D topological orders, spacetime surgery turns linked worldline and worldsheet processes into algebraic constraints on fusion and braiding data. One writes a closed spacetime manifold as

$r_\sharp$10

cuts along $r_\sharp$11, chooses a basis of boundary states created by operator insertions, and reglues by a mapping class group element $r_\sharp$12. The basic identity is

$r_\sharp$13

In $r_\sharp$14D, this reproduces the Verlinde formula from Hopf-link amplitudes

$r_\sharp$15

In $r_\sharp$16D it yields analogues involving particle-string Aharonov–Bohm phases

$r_\sharp$17

three-loop braiding amplitudes

$r_\sharp$18

and $r_\sharp$19 modular data on $r_\sharp$20. The protocol is “surgery” in the strong sense that the topology of spacetime and the linking of operator insertions are changed simultaneously (Wang et al., 2016).

A different physical use of surgery appears in $r_\sharp$21 gauge theories. A closed oriented $r_\sharp$22-manifold is presented by Dehn surgery on a framed link

$r_\sharp$23

Each surgery circle $r_\sharp$24 gives a $r_\sharp$25 gauge node with framing $r_\sharp$26, and pairwise linkings $r_\sharp$27 give mixed Chern–Simons couplings. Matter is added by non-compact Ooguri–Vafa Lagrangian defects $r_\sharp$28 intersecting $r_\sharp$29 along unknotted matter circles $r_\sharp$30. The corresponding chiral multiplet charges are read off from the winding/linking numbers

$r_\sharp$31

Kirby moves then become field-theoretic dualities: $r_\sharp$32-Kirby moves are interpreted as integrating gauge nodes in or out, and Rolfsen twists provide geometric realizations of $r_\sharp$33-moves and gauged mirror dualities (Cheng, 2023).

A unifying feature of these last examples is that surgery no longer means only cutting and gluing manifolds. It also means reorganizing operator sectors, defect data, or duality frames. This suggests that, in contemporary usage, a geometric surgery protocol is best understood as an explicit interface calculus: one specifies local replacement data, proves compatibility on the boundary or overlap, and derives the transformed global object from that local prescription.