Almost-Concordance in 3-Manifolds

• Almost-concordance is an equivalence relation on knots in closed oriented 3-manifolds (other than S³) defined by concordance in M×I with local knotting from S³.
• The framework adapts classical concordance theory to settings without a distinguished unknot, preserving π₁-sensitive invariants through techniques like Wall’s and covering link invariants.
• Recent advancements leverage Milnor invariants and algebraic methods in aspherical manifolds to establish infinitely many distinct almost-concordance classes.

Almost-concordance is the equivalence relation on knots in a closed oriented $3$-manifold $M\neq S^3$ obtained by taking concordance in $M\times I$ and then quotienting by local knotting with knots in $S^3$. Its central role is to adapt concordance theory to ambient manifolds that do not possess a distinguished unknot. In this setting one fixes a reference knot $K\subset M$, allows connected sum with a local knot $J\subset S^3$, and studies concordance classes modulo the resulting action of the knot concordance group of $S^3$. The resulting orbit set detects which concordance obstructions survive the insertion of local knotting; a key point is that local knotting does not change the $\mathbb Z[\pi_1(M)]$-homology type of the knot exterior, so many ambient-$\pi_1$-sensitive invariants remain available (Stees, 20 Aug 2025).

1. Definition and formal setup

Let $Y$ be a closed, connected, oriented $M\neq S^3$0-manifold, and let a knot mean an isotopy class of smooth embeddings

$M\neq S^3$1

If $M\neq S^3$2, then $M\neq S^3$3 and $M\neq S^3$4 are smoothly concordant if there is a smooth proper embedding

$M\neq S^3$5

with

$M\neq S^3$6

Concordance implies free homotopy, so the theory is organized inside a fixed free homotopy class $M\neq S^3$7 (Yildiz, 2017).

Almost-concordance weakens ordinary concordance by allowing local knotting. If $M\neq S^3$8 is a reference knot, then a knot $M\neq S^3$9 is obtained from $M\times I$0 by local knotting when

$M\times I$1

for some knot $M\times I$2. The concordance group $M\times I$3 of knots in $M\times I$4 acts on concordance classes of knots in $M\times I$5 by this operation, and the orbit set

$M\times I$6

is the set of almost-concordance classes. Equivalently, two knots are almost-concordant if they differ by a concordance in $M\times I$7 up to insertion of some local knot from $M\times I$8 (Stees, 20 Aug 2025).

The relation can be formulated in either the smooth or topological category. For $M\times I$9 or $S^3$0, one writes $S^3$1 for the set of $S^3$2-concordance classes in the free homotopy class $S^3$3, and the knot concordance group $S^3$4 acts by local knotting. Smooth almost concordance is coarser than smooth concordance, and topological almost concordance is coarser than topological concordance (Nagel et al., 2017).

A useful reformulation is that almost-concordance is essentially the same as PL-concordance in $S^3$5: if two knots are PL-concordant, one may assume there is a single cone singularity, and conversely local knotting can be pushed into a $S^3$6-ball and coned off to produce a PL-concordance. This identifies almost-concordance with a controlled singular version of concordance (Yildiz, 2017).

2. Conjectural structure and the lightbulb-trivial exception

The principal organizing conjecture is the Almost-Concordance Conjecture of Levine, Celoria, and Friedl–Nagel–Orson–Powell. For a free homotopy class $S^3$7, it asserts

$S^3$8

Equivalently, the only case in which a free homotopy class should contain finitely many almost-concordance classes is the “lightbulb-trivial” case, namely when the class admits an embedded dual sphere (Stees, 20 Aug 2025).

This conjectural dichotomy expresses a sharp contrast between ambient fundamental-group phenomena and the light bulb principle. The presence of an embedded dual $S^3$9-sphere is expected to collapse almost-concordance, while its absence should force infinite complexity. The conjecture is therefore not merely a counting statement; it isolates a geometric obstruction—the dual sphere—as the unique source of finiteness.

Early results already displayed both sides of this picture. For any closed $K\subset M$0-manifold $K\subset M$1 and any nontrivial $K\subset M$2, there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot (Yildiz, 2017). At the opposite extreme, if $K\subset M$3 is the free homotopy class of $K\subset M$4 in $K\subset M$5, then

$K\subset M$6

so every knot in that class is smoothly concordant to the standard core. This is the prototypical lightbulb-trivial phenomenon (Yildiz, 2017).

A parallel smooth/topological dichotomy was also established. In every closed oriented $K\subset M$7-manifold $K\subset M$8, the trivial free homotopy class contains an infinite family of null-homotopic knots that are all topologically concordant to each other, but pairwise distinct in smooth almost concordance. In every lens space and for every free homotopy class, there is a pair of topologically concordant but not smoothly almost-concordant knots, while every free homotopy class in every lens space contains infinitely many topological almost concordance classes (Nagel et al., 2017).

3. Invariants and obstruction mechanisms

Almost-concordance is designed so that local knotting by $K\subset M$9-knots is declared inessential. The central technical problem is therefore to isolate invariants that are unaffected by this operation but still distinguish ambient knot types.

A classical example is Wall’s self-intersection invariant, in Schneiderman’s formulation. For a null-homotopic knot $J\subset S^3$0, one chooses a singular disk $J\subset S^3$1 with $J\subset S^3$2, and defines

$J\subset S^3$3

After quotienting by the usual indeterminacies, one obtains

$J\subset S^3$4

and a well-defined surjective map

$J\subset S^3$5

A crucial lemma is that if $J\subset S^3$6 is null-homotopic and $J\subset S^3$7, then

$J\subset S^3$8

so $J\subset S^3$9 is an almost-concordance invariant on the free homotopy class of the unknot (Yildiz, 2017).

Another major family of techniques is based on covering links. A knot $S^3$0 is lifted to a cover $S^3$1, and one analyzes $S^3$2. Under suitable hypotheses, almost-concordance descends to concordance of the lifted link: if $S^3$3 and $S^3$4 are almost concordant and all components of $S^3$5 and $S^3$6 are unknotted, then

$S^3$7

This allows the use of classical smooth concordance invariants in $S^3$8, particularly Ozsváth–Szabó’s $S^3$9, the $\mathbb Z[\pi_1(M)]$0-invariant, and Levine–Tristram signatures (Nagel et al., 2017).

The 2025 framework extends Milnor’s link invariants to knots and links in arbitrary closed orientable $\mathbb Z[\pi_1(M)]$1-manifolds and then specializes them to almost-concordance. For a link $\mathbb Z[\pi_1(M)]$2, one constructs spaces $\mathbb Z[\pi_1(M)]$3 over $\mathbb Z[\pi_1(M)]$4 and defines a lower central homotopy invariant $\mathbb Z[\pi_1(M)]$5, a lower central homology invariant

$\mathbb Z[\pi_1(M)]$6

and a Milnor-type invariant $\mathbb Z[\pi_1(M)]$7. For knots $\mathbb Z[\pi_1(M)]$8 almost-concordant to a fixed $\mathbb Z[\pi_1(M)]$9,

$\pi_1$0

These are therefore genuine almost-concordance obstructions rather than merely concordance obstructions (Stees, 20 Aug 2025).

4. Aspherical $\pi_1$1-manifolds and the 2025 breakthrough

The main recent advance concerns homotopically essential knots in aspherical $\pi_1$2-manifolds. If $\pi_1$3 is such a knot and $\pi_1$4 denotes its exterior, one sets

$\pi_1$5

The key structural input is that in an aspherical $\pi_1$6-manifold this kernel is very large and has a lower central series with quotients behaving like those of a free group on countably many generators. The relevant series is

$\pi_1$7

and the quotients $\pi_1$8, together with peripheral data, are concordance invariants and in fact almost-concordance invariants (Stees, 20 Aug 2025).

The principal geometric construction is a $\pi_1$9-ambient connected sum with a weakly Brunnian link. A weakly Brunnian link $Y$0 is one with a distinguished component $Y$1 such that the remaining components form an unlink. Embedding a suitable handlebody in $Y$2, one uses $Y$3 to modify $Y$4 into a new knot $Y$5. If $Y$6 has vanishing Milnor invariants of lengths $Y$7, then $Y$8 admits an $Y$9-basing relative to $M\neq S^3$00, so $M\neq S^3$01 is defined. The comparison theorem identifies the change in $M\neq S^3$02 with Orr’s invariant of $M\neq S^3$03: $M\neq S^3$04 This creates an explicit bridge from classical link theory in $M\neq S^3$05 to almost-concordance in non-simply-connected manifolds (Stees, 20 Aug 2025).

The first main structural theorem states that for any homotopically essential knot $M\neq S^3$06 in any aspherical $M\neq S^3$07-manifold and any $M\neq S^3$08, there is a family

$M\neq S^3$09

all representing the same free homotopy class as $M\neq S^3$10, with the properties that $M\neq S^3$11 belongs to every family, distinct $M\neq S^3$12 at fixed $M\neq S^3$13 give pairwise non-almost-concordant knots, and families for different $M\neq S^3$14 intersect only in $M\neq S^3$15. The indexing set is

$M\neq S^3$16

Hence, if $M\neq S^3$17 is infinite, then the Almost-Concordance Conjecture holds for the class $M\neq S^3$18 (Stees, 20 Aug 2025).

The second main theorem gives a large algebraic regime in which these groups are as large as the Milnor package permits. Under the hypotheses that $M\neq S^3$19 is primitive and of infinite order in $M\neq S^3$20, that the centralizer of any nontrivial power of $M\neq S^3$21 in $M\neq S^3$22 is cyclic, and that the left cosets of $M\neq S^3$23 admit a $M\neq S^3$24-invariant total order, one has

$M\neq S^3$25

with $M\neq S^3$26 a free abelian group of rank at least

$M\neq S^3$27

where $M\neq S^3$28 is the number of linearly independent Milnor invariants of length $M\neq S^3$29 for $M\neq S^3$30-component links up to relabeling. Since $M\neq S^3$31 with $M\neq S^3$32, this yields arbitrarily large families of distinct almost-concordance classes. In particular, the paper proves that for every nontrivial free homotopy class $M\neq S^3$33 in every aspherical $M\neq S^3$34-manifold $M\neq S^3$35, there are infinitely many almost-concordance classes (Stees, 20 Aug 2025).

5. Representative manifolds and ambient $M\neq S^3$36-effects

The general theory becomes especially transparent in manifolds where the ambient fundamental group can be controlled explicitly. One important class is that of surface bundles over $M\neq S^3$37,

$M\neq S^3$38

If $M\neq S^3$39 is a section arising from a fixed point of the monodromy $M\neq S^3$40, and if $M\neq S^3$41 is pseudo-Anosov for $M\neq S^3$42 or Anosov for $M\neq S^3$43, with $M\neq S^3$44 having real positive eigenvalues on $M\neq S^3$45, then the Almost-Concordance Conjecture holds for the class $M\neq S^3$46. These cases are notable because $M\neq S^3$47 can be primitive and of infinite order, even normally generating $M\neq S^3$48, so standard covering-link techniques do not immediately apply (Stees, 20 Aug 2025).

The ambient $M\neq S^3$49-action can also suppress distinctions that would be visible in a purely classical Milnor-theoretic setting. In some torus bundles, the $M\neq S^3$50-action can identify or negate Milnor invariants, causing some potential generators to have finite order in the orbit quotient. This shows that almost-concordance is sensitive not only to the existence of Milnor-type obstructions but also to the full $M\neq S^3$51-module structure of the relevant homology groups (Stees, 20 Aug 2025).

Lens spaces supply a complementary family of examples. In every lens space and for every free homotopy class, there exist knots that are topologically concordant but not smoothly almost concordant. At the same time, every free homotopy class in every lens space contains infinitely many topological almost concordance classes. The proofs use covering-link computations together with $M\neq S^3$52, $M\neq S^3$53, and Levine–Tristram signatures, and they confirm the conjectured richness of almost-concordance in the lens-space setting (Nagel et al., 2017).

The class of $M\neq S^3$54 in $M\neq S^3$55 remains the standard exceptional case. Every knot in that class is smoothly concordant to the core $M\neq S^3$56, so almost-concordance collapses even before taking the quotient by local knotting. This is the model example of a class with an embedded dual sphere (Yildiz, 2017).

6. Smooth versus topological almost-concordance and related directions

Almost-concordance is sharply sensitive to the distinction between smooth and topological categories. Smooth almost concordance always implies topological almost concordance, but not conversely. The null-homotopic families constructed in every $M\neq S^3$57 show that one may have knots that are all topologically concordant yet pairwise distinct in smooth almost concordance. In lens spaces this disparity persists in every free homotopy class (Nagel et al., 2017).

The Mazur-manifold examples refine this disparity further. On the boundary of a Mazur manifold, one can construct infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot; none of these classes bounds a PL-disk in the Mazur manifold, but all of the constructed knots are topologically slice. The proof combines Wall’s invariant, the Stein-surface adjunction inequality, and the calculation that the relevant Alexander polynomials are trivial, so Freedman–Quinn theory applies (Yildiz, 2017).

From a structural viewpoint, these examples show that almost-concordance is neither a minor perturbation of classical concordance nor a purely local quotient. The theory retains global information from the ambient manifold, especially from $M\neq S^3$58, and can distinguish classes that remain invisible after ordinary local stabilization. The 2025 aspherical-manifold results make this precise by converting almost-concordance into a problem about how classical Milnor invariants survive after transport through the topology of the ambient manifold (Stees, 20 Aug 2025).

A related line of work concerns concordance invariants coming from instanton homology with local coefficients. These invariants include a $M\neq S^3$59-parameter family of homomorphisms

$M\neq S^3$60

with inequalities of the form

$M\neq S^3$61

and a limiting invariant $M\neq S^3$62 that bounds the $M\neq S^3$63-dimensional clasp number. That work is explicitly about classical knot concordance rather than almost-concordance in general $M\neq S^3$64-manifolds. A plausible implication is that such genus- and double-point-sensitive invariants may eventually contribute refined obstructions for weaker concordance-like relations, including almost-concordance (Kronheimer et al., 2019).

In its present form, almost-concordance has become a framework for studying knot concordance beyond $M\neq S^3$65 when no canonical unknot exists and when ambient fundamental-group effects are unavoidable. Its most developed form is now the aspherical case, where every nontrivial free homotopy class contains infinitely many almost-concordance classes and, under natural algebraic hypotheses, these classes are distinguished in a maximally rich Milnor-theoretic sense (Stees, 20 Aug 2025).