Surgery on Metric Spaces: Cut‐Glue Methods

• Surgery on metric spaces is a family of operations—cut–glue, remove–repair, replace–compress—that modify metrics while preserving key invariants.
• Pseudo-isometric surgery replaces subspaces via controlled maps, ensuring local distance bounds and avoiding large-scale collapse.
• Techniques span modulus arguments, Urysohn width control, curvature-preserving gluing, and spectral comparisons on quantum graphs.

“Surgery on metric spaces” denotes a family of cut–glue, remove–repair, and replace–compress operations in which the primary object is a metric, a metric measure space, a weighted metric space, or a space of metrics itself. In current arXiv usage, the phrase covers at least five distinct but related programs: puncture repair and removability for quasiconformal mappings on metric measure spaces; replacement of a subspace $S\subset X$ by another metric space $T$ via a map $f:S\to T$; connected-sum-type surgery and its effect on large-scale invariants such as Urysohn width; surgery preserving curvature conditions on manifolds and on spaces of Riemannian metrics; and local graph modifications whose effect on Laplace spectra can be tracked exactly or monotonically (Warhurst, 2018, Clay et al., 31 Jul 2025, Berdnikov et al., 17 Feb 2026, Kordaß, 2018, Reiser, 2021, Reiser et al., 2024, Berkolaiko et al., 2018). This suggests an umbrella viewpoint in which surgery is not a single construction but a repertoire of operations whose admissibility is governed by modulus, curvature, coarse distortion, or spectral monotonicity.

1. Metric surgery as a family of constructions

One concrete model replaces a subspace $S$ of a metric space $X$ by another metric space $T$ using a map $f:S\to T$. The resulting surgered space is denoted $\widehat{X}_f$, and the induced map $F:X\to \widehat{X}_f$ is the main object of study in “pseudo-isometric surgery” (Clay et al., 31 Jul 2025). A second model removes a compact region $A$ from two complete manifolds and identifies the boundary copies, producing

$T$0

with the focus shifted from topology to the behavior of the $T$1-dimensional Urysohn width $T$2 under this operation (Berdnikov et al., 17 Feb 2026).

A third model is local and analytic rather than topological. In puncture repair, one removes a point $T$3 from a neighborhood $T$4, studies a quasiconformal map on $T$5, and proves that under modulus and Loewner hypotheses the image must again be an open set minus a single point. In that sense, the singularity is removable and the puncture can be “repaired” (Warhurst, 2018). A fourth model treats surgery on the underlying manifold as an operation on the topological spaces $T$6 of Riemannian metrics satisfying a curvature condition $T$7, and shows that under surgery-stability hypotheses the homotopy type of $T$8 is unchanged by surgeries of suitable codimension (Kordaß, 2018).

The phrase also appears in smooth metric measure geometry. There, weighted manifolds $T$9 are modified by connected sums and higher-dimensional surgeries while preserving positivity of the Bakry–Émery Ricci tensor. The analytic mechanism is a weighted Perelman gluing theorem plus neck metrics governed by ODE inequalities for doubly warped products (Reiser et al., 2024). In quantum graphs, finally, surgery is a systematic collection of local modifications—vertex gluing, edge unfolding, pendant insertion, and transplantation of volume—whose effect on the Laplacian spectrum is controlled by interlacing or Rayleigh quotient comparison (Berkolaiko et al., 2018).

2. Punctures, removability, and quasiconformal repair

In the metric measure setting, a metric measure space is $f:S\to T$0, where $f:S\to T$1 is a separable metric space and $f:S\to T$2 is a locally finite Borel regular measure. The central analytic invariant is the $f:S\to T$3-modulus of a curve family,

$f:S\to T$4

where the infimum is over all non-negative Borel functions $f:S\to T$5 satisfying $f:S\to T$6 for every rectifiable $f:S\to T$7. Warhurst’s puncture repair theorem assumes locally $f:S\to T$8-bounded geometry on the domain and target, path connectedness, and the $f:S\to T$9-Loewner property on the target. If $S$0, $S$1 is a neighborhood of $S$2, and

$S$3

is quasiconformal, with the additional hypothesis that for every $S$4 the set

$S$5

is a continuum, then

$S$6

for some open neighborhood $S$7 of a point $S$8 (Warhurst, 2018).

The mechanism is a modulus argument. A lemma shows that if $S$9 for $X$0, then for $X$1,

$X$2

Hence the modulus of curves escaping a shrinking ball tends to $X$3 as $X$4. Quasiconformality transfers this smallness to the image. The $X$5-Loewner condition on the target gives a positive lower bound for the modulus of curves joining two disjoint nondegenerate continua with controlled relative separation. The only way to avoid contradiction is for the “missing boundary” continuum in the image to degenerate to a single point. Under further topological hypotheses, one then extends $X$6 across the puncture.

The same paper places point-removal inside a broader removability theory for porous sets. In a $X$7-regular Loewner space, a compact set $X$8 is spherically $X$9-porous if for each $T$0 there is a sequence $T$1 with

$T$2

Balogh–Koskela’s theorem, as summarized there, states that in an unbounded $T$3-regular Loewner space every quasiconformal map

$T$4

that maps bounded sets to bounded sets extends to a quasisymmetric homeomorphism $T$5 when $T$6 is compact and spherically $T$7-porous. The paper also records refinements for general targets under properness, $T$8-regularity, $T$9-linear local connectivity, and a separation axiom $f:S\to T$0. A notable limitation is that spaces with boundary can behave differently if quasiconformal maps do not preserve boundary; the half-space translation example shows that naïve puncture repair can fail in that setting.

3. Replacement surgery and pseudo-isometric control

The most literal metric-space surgery in the corpus is the construction of $f:S\to T$1 from a metric space $f:S\to T$2, a subspace $f:S\to T$3, a target metric space $f:S\to T$4, and a map $f:S\to T$5. The paper first singles out pseudo-isometries: $f:S\to T$6 is a pseudo-isometry if there exist $f:S\to T$7 and $f:S\to T$8 such that

$f:S\to T$9

for all $\widehat{X}_f$0, and every $\widehat{X}_f$1 lies within distance $\widehat{X}_f$2 of some $\widehat{X}_f$3. The surgered topological space is

$\widehat{X}_f$4

A path-type pseudo-metric on $\widehat{X}_f$5 is then defined using admissible sequences that alternate between motion in $\widehat{X}_f$6 and motion in $\widehat{X}_f$7, with jumps through $\widehat{X}_f$8 assigned zero cost. After collapsing zero pseudo-distance classes, one obtains the metric space $\widehat{X}_f$9 and the natural map

$F:X\to \widehat{X}_f$0

(Clay et al., 31 Jul 2025).

The central theorem states that if $F:X\to \widehat{X}_f$1 is a pseudo-isometry, then $F:X\to \widehat{X}_f$2 is a pseudo-isometry as well. The proof depends on a lower bound for the length $F:X\to \widehat{X}_f$3 of an admissible sequence: $F:X\to \widehat{X}_f$4 This estimate blocks uncontrolled collapsing. A second lemma shows that if $F:X\to \widehat{X}_f$5, then

$F:X\to \widehat{X}_f$6

Thus the fibers of $F:X\to \widehat{X}_f$7 have uniformly bounded diameter. Since $F:X\to \widehat{X}_f$8 is also $F:X\to \widehat{X}_f$9-Lipschitz and coarsely surjective, it is a pseudo-isometry.

The paper also isolates the sharp obstruction to replacing “pseudo-isometry” by “quasi-isometry”. If one allows an additive constant in the upper bound,

$A$0

the additive error accumulates through arbitrarily long admissible sequences, and the argument fails. This is not merely technical. A concrete example takes $A$1, lets $A$2 be a union of intervals $A$3, and collapses each interval to its left endpoint. The local map $A$4 is a quasi-isometry, but the surgered space $A$5 is isometric to $A$6. The induced map $A$7 therefore cannot be a quasi-isometry. By contrast, the interval-collapsing construction on $A$8, where each $A$9 is collapsed to $T$00, yields a surgered space isometric to $T$01. The distinction is therefore between controlled replacement and large-scale collapse.

4. Urysohn width, connected sums, and universal covers

For a complete metric space $T$02, the $T$03-dimensional Urysohn width is

$T$04

It measures how efficiently $T$05 can be compressed into a $T$06-dimensional simplicial complex while keeping fiber diameters uniformly bounded. The surgery model in this setting removes the interior of an embedded compact $T$07-manifold $T$08 with boundary from each of two complete $T$09-manifolds and glues along $T$10 to form $T$11 (Berdnikov et al., 17 Feb 2026).

The first general comparison is

$T$12

together with the reverse bound

$T$13

Under additional topological hypotheses, the factor $T$14 can be removed. For $T$15, this happens when $T$16 is simply connected, or more generally when $T$17 in the relevant complement, or when $T$18. For $T$19, it happens under orientation and connected-boundary hypotheses. In those cases one has

$T$20

and, under stronger assumptions or when $T$21,

$T$22

The same paper extends the theory to universal covers. If $T$23 and $T$24 are simply connected, then $T$25 decomposes as a tree-like connected sum of copies of $T$26 and $T$27 glued along copies of $T$28. Boundary-distance distortion is encoded by constants $T$29 and $T$30. Under these hypotheses,

$T$31

and, when $T$32 have no boundary,

$T$33

Conversely,

$T$34

for $T$35, for $T$36 in the closed case, and when $T$37 is a ball.

The paper also proves that these constants are essentially sharp. A cone example shows that the coefficient $T$38 in the general theorem cannot be uniformly lowered. Further examples show that the constants $T$39 and $T$40 for universal covers cannot be bounded by a universal constant independent of the group or the geometry. The broad conclusion is quantitative rather than invariant-theoretic: Urysohn width is quasi-stable under controlled surgery, but the control depends essentially on the topology of the interface and on ambient metric distortion.

5. Curvature-preserving surgery on manifolds and on spaces of metrics

A different branch of the subject treats surgery as an operation on manifolds together with a prescribed curvature condition. In Kordaß’s framework, a curvature condition in dimension $T$41 is an open, $T$42-invariant subset $T$43 of algebraic curvature operators satisfying the Bianchi identity. The key hypothesis is surgery stability in the sense of an inner cone condition with respect to the model operator $T$44. If $T$45 is also deformable, the parametrized Gromov–Lawson construction produces, for a compact family of metrics $T$46, a deformation through $T$47-metrics to a torpedo-standard family near the surgery sphere. The resulting inclusion

$T$48

is a weak homotopy equivalence, and surgery of codimension at least $T$49 induces a homotopy equivalence

$T$50

This generalizes the Chernysh–Walsh theorem for positive scalar curvature from $T$51 to any deformable surgery-stable curvature condition. The paper also derives bordism invariance statements and the application

$T$52

(Kordaß, 2018).

For positive Ricci curvature, generalized surgery proceeds by replacing the classical product handle with a sphere bundle over a manifold carrying a core metric. A core metric on $T$53 is a Ricci-positive metric admitting an embedded $T$54 whose boundary sphere is round of radius $T$55 and has positive-definite inward second fundamental form. Starting from a Ricci-positive manifold $T$56 containing an isometric copy of $T$57, and from a linear $T$58-bundle $T$59 over a base $T$60 with a core metric, one removes the surgery region from $T$61 and glues in $T$62. The neck carries a doubly warped product metric

$T$63

with Ricci positivity enforced by explicit inequalities for $T$64, $T$65, and $T$66. Perelman’s Ricci-positive gluing theorem then yields the generalized surgery theorem for $T$67, together with plumbing results and new core metrics on certain $T$68-bundles (Reiser, 2021).

In smooth metric measure geometry, the basic object is a weighted manifold $T$69 with measure $T$70 and $T$71-Bakry–Émery tensor

$T$72

Reiser and Tripaldi prove a weighted Perelman gluing theorem: if weighted manifolds with $T$73 have isometric boundary components with matching weights and satisfy

$T$74

then the glued manifold admits a smooth weighted metric with $T$75. This is used to construct weighted connected sums and to prove a higher-surgery theorem for $T$76 under local assumptions near a round, totally geodesic central sphere with constant weight and vanishing normal derivative. The same paper applies the theory to show that every closed, simply connected spin $T$77-manifold admits a weighted metric with $T$78, and notes that no example is known of a closed manifold that admits $T$79 but no $T$80 (Reiser et al., 2024).

6. Surgery processes in path spaces, outer space, and shape space

In Culler–Vogtmann outer space $T$81, the sphere model identifies points with weighted simple sphere systems in

$T$82

Given two sphere systems $T$83 and $T$84, Hatcher–Vogtmann’s construction uses normal form, innermost-disk surgery, doubling, simultaneous surgeries, and undoubling to produce a canonical combing path from $T$85 to $T$86. The path is built from repeated double surgery steps. Intersection numbers between sphere systems coincide with Guirardel’s intersection number of the corresponding trees, and on the $T$87-thick part one has

$T$88

where $T$89 is the asymmetric Lipschitz metric on $T$90. If the combing path stays in $T$91, then the number of intersection circles grows exponentially with the number of surgery steps, which yields the definitive estimate

$T$92

Accordingly, sphere-surgery combing paths are quasi-geodesics in the thick part (Horbez, 2012).

A different, genuinely infinite-dimensional setting is the shape space of unparameterized immersed submanifolds. There the basic space is

$T$93

and the metric is induced from a Sobolev-type inner metric

$T$94

with $T$95 elliptic, positive, symmetric, and reparametrization invariant. The model choice

$T$96

produces inner Sobolev metrics of order $T$97. Harms shows that the $T$98-metric yields vanishing geodesic distance on shape space, but if the metric is at least as strong as $T$99, then the induced distance on the embedding shape space $f:S\to T$00 is non-vanishing. The proof uses an area-swept-out lower bound and Lipschitz control of $f:S\to T$01. The geodesic equation is well posed under the stated ellipticity and smoothness hypotheses, and the framework is presented as one in which “surgery-like” localized or topologically complex deformations have a well-defined metric cost (Harms, 2012).

These two theories use the word “surgery” differently. In outer space it is a combinatorial move generating canonical paths in a metric space. In shape space it is a way of interpreting highly localized deformations inside a Riemannian metric on an infinite-dimensional quotient. The common feature is that surgery is converted into path geometry: the operation is meaningful because length, progress, or energy can be quantified.

7. Spectral surgery on quantum graphs

A compact metric graph is a combinatorial graph whose edges are intervals $f:S\to T$02, equipped with the path metric and total length

$f:S\to T$03

The Laplacian acts as $f:S\to T$04 on edges, with natural, Dirichlet, or $f:S\to T$05-type conditions at vertices. For $f:S\to T$06-type conditions one requires continuity and

$f:S\to T$07

The associated quadratic form is

$f:S\to T$08

and the spectrum is discrete. The paper organizes “spectral surgery principles” into three families: operations changing vertex conditions, operations increasing the volume, and operations transferring the volume (Berkolaiko et al., 2018).

Vertex gluing and strengthening a $f:S\to T$09-coupling produce sharp interlacing. If $f:S\to T$10 is obtained from $f:S\to T$11 by gluing vertices or by increasing a $f:S\to T$12-strength, then for all $f:S\to T$13,

$f:S\to T$14

Attaching a natural pendant graph, lengthening an edge, or inserting a graph at a natural vertex yields the opposite monotonicity for nonnegative eigenvalues: $f:S\to T$15 The most distinctive new principles are transplantation and unfolding. In transplantation, one cuts a graph into subgraphs $f:S\to T$16 and $f:S\to T$17, removes $f:S\to T$18, and reattaches the same total length elsewhere as graphs $f:S\to T$19. If $f:S\to T$20 is a $f:S\to T$21-eigenfunction and

$f:S\to T$22

then

$f:S\to T$23

with strict inequality under a stronger separation condition. Unfolding multiple parallel edges into a single longer edge, symmetrising parallel edges, and unfolding several pendant edges into one longer pendant all decrease the first nontrivial eigenvalue $f:S\to T$24 under the hypotheses stated in the paper.

These surgery rules culminate in isoperimetric-type estimates for the spectral gap. If $f:S\to T$25 is the doubly connected part of a connected graph $f:S\to T$26, with length $f:S\to T$27 and total length $f:S\to T$28, then

$f:S\to T$29

where $f:S\to T$30 is the symmetric dumbbell with total length $f:S\to T$31 and doubly connected part of length $f:S\to T$32. A sharper estimate uses the largest doubly connected component and the corresponding tadpole model $f:S\to T$33. In this 1-dimensional setting, surgery is therefore a precise spectral calculus: local cut-and-paste operations are admissible exactly when the quadratic form or eigenfunction geometry forces monotonicity.

Surgery on metric spaces is thus not a single theorem but a collection of technically specific doctrines. In quasiconformal analysis it is removability detected by modulus and Loewner estimates; in coarse geometry it is replacement controlled by pseudo-isometry; in large-scale topology it is quantitative stability of Urysohn width under connected-sum-type gluings; in curvature theory it is the preservation of positive scalar, Ricci, or Bakry–Émery curvature under explicit neck constructions; in outer space and shape spaces it is a path-generating or energy-measuring mechanism; and in quantum graphs it is a sharp spectral comparison toolkit. The common thread is that surgery becomes mathematically meaningful only when an ambient invariant—modulus, width, homotopy type, curvature tensor, path length, or eigenvalue—survives the cut-and-glue process in a controlled way.