Dehn–Goodman–Fried Surgery

• Dehn–Goodman–Fried surgery is a set of operations that generalize classical Dehn surgery on 3-manifolds to create dynamically rich Anosov, pseudo-Anosov, and expansive flows.
• It combines cut-and-paste topology with contact and foliation techniques to control hyperbolic structures, holonomy, and foliation branching.
• The approach has been applied to uniquely construct Anosov flows on hyperbolic manifolds and classify nonalgebraic flow behaviors by tuning surgery coefficients.

Dehn–Goodman–Fried surgery refers to a suite of operations that generalize the classical notion of Dehn surgery on 3-manifolds to the setting of Anosov flows and more generally to pseudo-Anosov and expansive flows. These procedures, introduced by Goodman and Fried in the 1980s and further systematically developed through modern contact and foliation-theoretic methods, provide powerful tools for constructing and classifying dynamically rich flows on 3-manifolds—including new Anosov flows on hyperbolic manifolds beyond algebraic suspensions. The process combines cut-and-paste topology with intricate dynamical control, ensuring preservation (or controlled modification) of hyperbolic structure, holonomy, foliation branching, and related dynamical invariants.

1. Foundations: Anosov Flows, Foliations, and Surgeries

An Anosov flow $\varphi^t$ on a closed, orientable 3-manifold $M$ admits a continuous invariant splitting $TM = E^s \oplus E^0 \oplus E^u$, with $E^0$ the flow direction and $E^s$, $E^u$ contracted exponentially in positive and negative time, respectively. The weak foliations $W^s$, $W^u$ integrate $E^s \oplus E^0$ and $E^u \oplus E^0$, and orientability is often assumed in applications of surgery.

Dehn–Goodman–Fried surgery modifies the flow by performing a Dehn-type operation along a periodic orbit or, in a more general vein, along a transverse annulus with carefully controlled framing. The basic recipe involves removing a solid torus neighborhood $M$0 of the orbit, then gluing back a torus $M$1 using a map specified by a slope $M$2 in $M$3, where $M$4 is meridional and $M$5 is the longitude along the flow. Classically, the gluing matrix for a $M$6 surgery is:

$M$7

corresponding to a $M$8-fold Dehn twist along the longitude direction (Asaoka, 2021).

The operation generalizes to surgeries along Legendrian–transverse knots distinguished by contact/bi-contact structures, by blowing up the knot to an invariant torus and regluing using twisted coordinates, yielding modified global flows (Salmoiraghi, 2021).

2. Topological and Dynamical Construction

(a) Local Model and Framing

A transverse annulus $M$9 or a tubular neighborhood $TM = E^s \oplus E^0 \oplus E^u$0 is selected around a periodic orbit (or more generally a suitable arc/circle with required transverse properties). On the boundary torus $TM = E^s \oplus E^0 \oplus E^u$1, coordinates $TM = E^s \oplus E^0 \oplus E^u$2 are chosen so that $TM = E^s \oplus E^0 \oplus E^u$3 is the meridian and $TM = E^s \oplus E^0 \oplus E^u$4 the flow-parallel longitude. The Dehn filling slope is specified by $TM = E^s \oplus E^0 \oplus E^u$5, realizing the surgery as a cut-and-paste with map:

$TM = E^s \oplus E^0 \oplus E^u$6

This construction is extendable to more general surgery data via matrices in $TM = E^s \oplus E^0 \oplus E^u$7 (Asaoka, 2021, Bonatti et al., 2020).

(b) Cut-and-Reglue and Extension of Dynamics

After cutting out the local model, a solid torus is reattached so that flow lines on the boundary match up to the Dehn twist. The flow is extended in this region by a model dynamical suspension (or via a suitable contact/bi-contact normal form), ensuring $TM = E^s \oplus E^0 \oplus E^u$8 (or smoother) structure and hyperbolicity. The return map's holonomy is twisted by the amount determined by the surgery coefficient, resulting in a new global Anosov (or pseudo-Anosov) flow (Asaoka, 2021, Tsang, 2024).

3. Bi-Contact Geometry and Reeb Dynamics

The link between contact geometry and Dehn–Goodman–Fried surgery crystallizes in the bi-contact framework. A bi-contact structure $TM = E^s \oplus E^0 \oplus E^u$9 comprises two transverse contact plane fields, each determined by 1-forms $E^0$0 with opposite coorientations and (dynamically consistent) contact conditions:

$E^0$1

Mitsumatsu's criterion states that a flow is projectively Anosov iff its flow direction lies in the intersection $E^0$2 and the normal dynamics split hyperbolically (Salmoiraghi, 2021). Under volume-preserving hypotheses and correct Reeb vector field alignment (dynamically positive Reeb field), deforming via $E^0$3-twist yields new contact forms $E^0$4, which, if remaining contact, guarantee the surgically modified line field is Anosov. This “Anosovity via Reeb dynamics” mechanism provides necessary and sufficient criteria for the contact-geometric version of the surgery to produce a bona fide Anosov (contact Anosov) flow (Salmoiraghi, 2021).

Legendrian–transverse knots for the bi-contact structure play a distinguished role: the surgery operation can be realized along these, and explicit normal forms are available near geodesic flow orbits (Salmoiraghi, 2021).

4. Impact on Foliations and Orbit Structure

Surgeries of Goodman–Fried type affect the (bi-)foliated structure of the orbit space. In the universal cover, the orbit space $E^0$5 becomes a bi-foliated plane with transverse foliations $E^0$6. Under surgeries:

• If all surgery coefficients are of the same sign, the resulting flow is $E^0$7-covered and twisted in that sign, as established by Fenley (Bonatti et al., 2020).
• Surgeries along simple closed geodesics in geodesic flows leave the conjugacy class of the bi-foliated plane unchanged, preserving the $E^0$8-covered property (Bonatti et al., 2020).

Not all modifications induce nontrivial changes in the foliation’s branching structure: finite families of periodic “pivot” orbits in non-$E^0$9-covered flows can be surgered without altering the nonseparated leaf structure (Bonatti et al., 2020). However, mixed-sign surgeries produce complex holonomy and may destroy completeness in certain quadrants; with correct choices, non-$E^s$0-covered Anosov flows are created (Bonatti et al., 2020).

5. Classification Results and Examples

Goodman’s original construction produced the first known hyperbolic 3-manifolds admitting Anosov flows, realized as integer-slope Dehn surgeries on the figure-eight knot complement (Yu, 2021). For the manifolds $E^s$1 obtained by Dehn surgery with slope $E^s$2 on the figure-eight knot, the following holds:

• If $E^s$3 is integer, $E^s$4 admits a unique (up to topological equivalence) Anosov flow, realizable via Dehn–Goodman–Fried surgery,
• If $E^s$5 is not integer, $E^s$6 does not admit any Anosov flow (Yu, 2021).

The extension to pseudo-Anosov (expansive) flows via Dehn–Fried surgery reveals that the resulting flow is expansive if and only if the post-surgery stable and unstable foliations have no 1-prong singularities. Thus, the surgery controls not only the topology of the ambient manifold but the fine combinatorics of the foliation singularities (Iakovoglou, 2024). Surgeries can be performed so as to obtain, for example, nontransitive pseudo-Anosov flows with prescribed prong structure.

In table form, this can be summarized:

Surgery Data	Outcome	Reference
Integer slope $E^s$7 on fig-8 knot	Unique Anosov flow on $E^s$8	(Yu, 2021)
Non-integer $E^s$9 on fig-8 knot	No Anosov flow on $E^u$0	(Yu, 2021)
Same-sign coefficients on periodic orbits	$E^u$1-covered Anosov flow	(Bonatti et al., 2020)
Mixed-sign surgeries	Non-$E^u$2-covered flow	(Bonatti et al., 2020)
Surgeries on pivots	Branching structure preserved	(Bonatti et al., 2020)
Surgery inducing 1-prong in foliation	Not pseudo-Anosov/expansive	(Iakovoglou, 2024)

6. Horizontal and Generalized Surgeries

Horizontal Goodman surgery expands the classical procedure to allow for cuts along more general “horizontal” annuli, not necessarily neighborhood of closed orbits. Regluing is performed via a Dehn twist supported away from the boundary, modifying the flow’s dynamics in a controlled way. The almost equivalence theorem asserts that the surgically modified pseudo-Anosov flow is orbit equivalent to the original, modulo finitely many closed orbits (Tsang, 2024), thus generalizing the drilling/twisting operations of Goodman and Fried. When applied to mapping tori or scalloped tori, this construction reproduces Dehn twists on closed curves or more intricate foliation modifications.

Structural stability theorems guarantee that small $E^u$3 perturbations of pseudo-Anosov flows (away from singularities) preserve orbit equivalence, ensuring robustness of the surgery’s dynamical properties (Tsang, 2024).

7. Applications and Further Structure

Applications of Dehn–Goodman–Fried surgery include:

• Construction of contact Anosov flows, especially on hyperbolic 3-manifolds (Salmoiraghi, 2021).
• Generation of skewed $E^u$4-covered flows, and, via the converse, the orbit-equivalence of all positive skewed $E^u$5-covered Anosov flows to positive contact Anosov flows arising from surgeries on geodesic flows (Salmoiraghi, 2021).
• Classification of all Anosov flows on certain manifolds via branched surface and essential lamination techniques (Yu, 2021).
• Construction of a large class of non-algebraic Anosov and pseudo-Anosov flows with prescribed foliation branching and prong data (Iakovoglou, 2024, Bonatti et al., 2020).

A plausible implication is that Dehn–Goodman–Fried surgery acts as a unifying operation for the production and comparison of Anosov and pseudo-Anosov flows on 3-manifolds, with deep ramifications for 3-manifold topology, contact topology, and dynamical systems.

• Surgery on Anosov flows using bi-contact geometry (Salmoiraghi, 2021).
• Goodman-Fried surgery, Birkhoff section, and R-covered Anosov flows (Asaoka, 2021).
• Anosov flows on $E^u$6-manifolds: the surgeries and the foliations (Bonatti et al., 2020).
• Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows (Tsang, 2024).
• Dehn-Fried surgeries on non-transitive expansive flows (Iakovoglou, 2024).
• Anosov flows on Dehn surgeries on the figure-eight knot (Yu, 2021).