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arxiv: 2606.01903 · v1 · pith:4MIRYHBUnew · submitted 2026-06-01 · 🧮 math.AT · math.GT

Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2

Daiki Irikura This is my paper

Pith reviewed 2026-06-28 11:56 UTC · model grok-4.3

classification 🧮 math.AT math.GT
keywords long embeddingsrational homotopy groupshairy graphsinfinite-dimensionalcodimension twocompactly supported embeddingsuni-trivalent graphs
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The pith

The rational homotopy groups of the space of long embeddings Emb_c(R^{n-2}, R^n) are infinite-dimensional in infinitely many degrees when n is odd and at least 5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies compactly supported embeddings between Euclidean spaces using a hairy graph model of their rational homotopy. It constructs explicit elements in the rational homotopy groups of these embedding spaces that correspond to certain uni-trivalent graphs. These elements are shown to be nontrivial. The resulting infinite-dimensionality holds specifically for the codimension-two case Emb_c(R^{n-2}, R^n) with n odd and n at least 5. This reveals that the topology of these embedding spaces is more intricate than finite-dimensional expectations would suggest.

Core claim

By utilizing hairy graphs, we construct elements in the homotopy groups π_•(Emb̄_c(R^j, R^n)) ⊗ Q corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of Emb_c(R^{n-2}, R^n) are infinite-dimensional in infinitely many degrees when n ≥ 5 is odd.

What carries the argument

Hairy graph model, in which uni-trivalent graphs generate elements shown to be nontrivial in the rational homotopy groups of the embedding space.

If this is right

  • The space Emb_c(R^{n-2}, R^n) has infinite-dimensional rational homotopy groups in infinitely many degrees for each odd n ≥ 5.
  • Nontrivial rational homotopy classes arise directly from uni-trivalent graphs via the hairy graph model.
  • The result applies specifically to compactly supported long embeddings of codimension two.
  • The construction works for the one-point compactification Emb̄_c and descends to the original embedding space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar graph constructions might detect infinite-dimensional rational homotopy in embedding spaces of other codimensions or dimensions not covered by the current proof.
  • The infinite-dimensionality could imply that the embedding spaces are not rationally formal or have rich rational homotopy Lie algebra structures.
  • One could test whether the same graphs remain nontrivial after applying forgetful maps to lower-codimension embedding spaces.

Load-bearing premise

The elements constructed from uni-trivalent graphs in the hairy graph model are nontrivial in the rational homotopy groups of the embedding space.

What would settle it

An explicit computation or relation in the hairy graph complex that forces one of the constructed graph elements to bound or become zero in the rational homotopy group of Emb_c(R^{n-2}, R^n) for some odd n ≥ 5.

Figures

Figures reproduced from arXiv: 2606.01903 by Daiki Irikura.

Figure 1
Figure 1. Figure 1: Depiction of a chord diagram [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of a labeled plain graph. ♢ Similarly, P GCn,j is defined as the quotient of P GC′ n,j by the subspace spanned by graphs containing double edges or self-loops. We first define a differential dP GC′ on P GC′ n,j by dP GC′ (Γ) = X e∈E(Γ) sign(e) (Γ/e), where the sum is taken over all edges e that are neither loop edges nor chord edges. Here, a chord edge means an edge connecting two black vertices… view at source ↗
Figure 3
Figure 3. Figure 3: Example of a ribbon presentation Notation 3.1.3 (Orientation of a crossing). Since D ∪ B is an oriented disk, each disk D is oriented. We orient the core of each band Bi so that it goes from a leaf to the base disk D0. A crossing of a band with a disk (respectively, a crossing of a line with a disk) is called positive if the core of the band gives the oriented normal vector of Di in R 3 . Otherwise, it is … view at source ↗
Figure 4
Figure 4. Figure 4: S1, S3, S4, S7-moves ♢ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Applying S1-moves as shown in the figure. □ By this lemma, in this section, we assume that every disk of a ribbon presentation intersects at most one band. Moreover, near each intersection, we use the local model B = {(x1, x2, 0) | |x1| ≤ 3, |x2| ≤ 1 2 }, D = {(0, x2, x3) | x 2 2 + x 2 3 ≤ 1}. Notation 3.1.6. We regard R n as R 3 × R n−j−2 × R j−1 . Set VP = [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Type I planetary-like system. Definition 3.2.2 (Type II planetary-like system). A type II planetary-like system SII is a subset D+ ∪ D− ∪ D′ + ∪ D′ − ⊂ D j , where D+, D−, D′ +, and D′ − are disks, each with radius ϵ 2 . The centers of these disks are given as follows: • D+: (0, 0, . . . , 0) • D−: (− 1 2 , 0, . . . , 0) • D′ +: (ϵ, 0, . . . , 0) • D′ −: ( 1 2 , 0, . . . , 0) ♢ A planetary-like system is d… view at source ↗
Figure 7
Figure 7. Figure 7: Type II planetary-like system. • e T −: The composition of the reflection across the hyperplane x1 = 0 with the standard embedding onto the disk D− ⊂ T . (2) (Composition) If S is a monostellar planetary-like system and S ′ is a system of type T ∈ {I,II}, their composition S ◦ S′ is defined as the union of the images: S ◦ S′ = e T +(S) ∪ e T −(S) ∪ (S ′ \ (D+ ∪ D−)). (Note that this definition implies that… view at source ↗
Figure 8
Figure 8. Figure 8: Example of compositions of planetary-like systems. The red circle is the boundary of D j . ♢ Next, we introduce a rotation of the disk D′ + in the Type II planetary-like system. Definition 3.2.6 (Rotated Type II planetary-like system). Let θ ∈ S j−1 . The rotated type II planetary-like system, denoted by SII(θ), is defined by replacing the disk D′ + in SII with a disk D′ +(θ) of the same radius centered at… view at source ↗
Figure 10
Figure 10. Figure 10: RII ◦ RI ◦ RI with labeled disks Construction 3.2.9. Let C be a chord diagram. We define cC : (S j−1 ) g−1 × (S n−j−2 ) k → Embc(R j , R n) as follows. We assign Type I or Type II to each vertex in V (C), excluding the vertices lying on the x-axis. Analogously to the base cases, we define a composed system of embedded disks and bands, denoted by (R)i, by composing Type I or Type II systems along the i-th … view at source ↗
Figure 11
Figure 11. Figure 11: The chord diagram C0 3.3. Infiniteness of the top term of HGC. We write ∗HGCn,j for the corre￾sponding chain complex [1]; HGCn,j = Hom(∗HGCn,j , Q) We write B = Htop( ∗HGCn,j ), which is the space of open Jacobi diagrams. Next, we recall sl2 weight system Wsl2 . Instead of the construction using Lie algebra tensors, Wsl2 can be defined recursively using the local relations [CV97] shown in [PITH_FULL_IMAG… view at source ↗
Figure 12
Figure 12. Figure 12: The sl2 relations and values. We introduce Gp,g, which is the key graph used to construct the nontrivial cycle. Definition 3.3.2. The hairy graph Gp,g is defined as follows. It consists of a circle and g−1 parallel edges inside the circle (forming a graph with the first Betti number g). Furthermore, 2p hairs are attached to the upper arc of the circle. ♢ Proposition 3.3.3. Let n be an odd integer with n ≥… view at source ↗
Figure 13
Figure 13. Figure 13: The hairy graph Gp,g. For the inductive step, applying the loop reduction to the bottom loop yields: Wsl2 (Gp,g) = 4ℏWsl2 (Gp,g−1). For the base case g = 1, we consider the relation involving the hairs. We obtain the following relation: · · · = 2ℏ · · · − 2ℏ · · · The second term vanishes due to the symmetry relation. Thus, we have the recurrence relation Wsl2 (Gp,1) = (2ℏC) · Wsl2 (Gp−1,1). Finally, we c… view at source ↗
Figure 14
Figure 14. Figure 14: Orientation of Gp,g Notation 3.4.1. We denote the ribbon presentation associated with D(Gp,g) by P(Gp,g) and we denote the resulting cycle by cp,g. ♢ [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The resulting chord diagram D(Gp,g) 4. Non-triviality of the cycle In this section, our goal is to establish the nontriviality of the cycle cp,g in H2p+g−1(Embc(R j , R n)). This section essentially follows the approach of [Yos25a, Section 7]. First, we recall the basic properties of the modified configuration space integral in § 4.1. Next, in § 4.2, we verify that the counting formula [Yos25a, Theorem 7.… view at source ↗
Figure 16
Figure 16. Figure 16: Configuration where Y and Y ′ are not linked. Proof. As the radii shrink, the direction vector x − x ′ ∥x − x ′∥ ∈ S j−1 varies only within an arbitrarily small neighborhood of a point on S j−1 ; hence the contribution to the integral tends to zero. □ Lemma 4.1.10 (Ingoing lemma). Let Γ be as above. Let Cin be the set of config￾urations for which there exist elementary components Y, Y ′ , Y ′′ such that … view at source ↗
Figure 17
Figure 17. Figure 17: Examples of nested components Y, Y ′ , Y ′′ in the In￾going Lemma. ♢ Proof. Since ωj−1 ∧ ωj−1 = 0, the integral vanishes in this limit. □ 4.2. Counting formula. To prove the nontriviality of the cycle, it remains only to check that the counting formula still holds. We recall the graph-chord pairing. Definition 4.2.1 (Graph–chord pairing [Yos25a, Definition 3.7]). Let C be a chord diagram on s directed lin… view at source ↗
Figure 18
Figure 18. Figure 18: The Type I boxed ribbon presentation Notation 5.2.2. Let c ′ I denote the element c ′ PI ∈ πn−j−2 [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The perturbed Type I boxed ribbon presentation We define b1, b2 ∈ Hn−j−1(I n \ (ν(I j ) ∪ ν(I j ) ∪ ν(I n−j−1 ))) so that b1 (resp. b2) is a normal sphere of I j ⊂ I n associated with D (resp. D′ ). b ′ 1 ∈ Hj (I n \ (ν(I j ) ∪ ν(I j )∪ν(I n−j−1 ))) is taken so that b ′ 1 is a normal sphere of I n−j−1 ⊂ I n associated with L. 5.3. Type II boxed ribbon presentation. We introduce the Type II ribbon presenta… view at source ↗
Figure 20
Figure 20. Figure 20: The Type II boxed ribbon presentation Proof. If L is removed, pull back D+ and D−, then perform an S4-move. Similarly, if L ′ is removed, pull back D′ + and D′ −, then perform an S4-move. There is nothing to show in the last case. □ We define b1 ∈ Hn−j−1(I n \ (ν(I j ) ∪ ν(I n−j−1 ) ∪ ν(I n−j−1 ))) so that b1 is a normal sphere of I j ⊂ I n associated with D . b ′ 1 , b′ 2 ∈ Hj (I n \ (ν(I j ) ∪ ν(I n−j−1… view at source ↗
Figure 21
Figure 21. Figure 21: Y-graphs G for Γ = G1,3 ♢ Thicken Gk ⊂ R n and denote the resulting manifold by Vk [PITH_FULL_IMAGE:figures/full_fig_p033_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Figures of Y-graph Gi 5.4.3 (Identification of thickened Y-graphs with standard models). If Gk corre￾sponds to a Type I vertex, we identify Vk with I n \ (ν(I j ) ∪ ν(I j ) ∪ ν(I n−j−1 )) as described in § 5.2, in such a way that bi corresponds to the link associated with the i-th incoming half-edge, and b ′ 1 corresponds to the component of the Hopf link associated with the outgoing half-edge. Similarly,… view at source ↗
Figure 23
Figure 23. Figure 23: Compositions c ′ I ◦1,1 c ′ I Example 5.4.8. Here is an example of the local model for the composition of two Type I ribbon presentations. At the two crossings labeled ”2”, the bands are perturbed simultaneously. ♢ 5.4.9 (Reinterpretation of Construction 5.4.4 via embedded boxed ribbon presen￾tations). Let P0 be a ribbon presentation with 2p labeled free disks D1, . . . , D2p and 2p bands with base disks.… view at source ↗
read the original abstract

In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $\pi_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of $\mathrm{Emb}_c(\mathbb{R}^{n-2}, \mathbb{R}^n)$ are infinite-dimensional in infinitely many degrees when $n \ge 5$ is odd.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies the rational homotopy groups of the space of compactly supported long embeddings Emb_c(R^j, R^n). It uses the hairy graph model to construct classes in π_•(Emb̄_c(R^{n-2}, R^n)) ⊗ Q corresponding to uni-trivalent graphs, proves these classes are nontrivial, and concludes that the groups are infinite-dimensional in infinitely many degrees for odd n ≥ 5.

Significance. If the nontriviality of the constructed classes holds, the result would extend known computations of rational homotopy for embedding spaces to the codimension-2 case, providing explicit infinite-dimensionality statements. The explicit graph-theoretic construction is a strength when the detection map is verified.

major comments (1)
  1. [§4 (nontriviality argument)] The detection step mapping uni-trivalent graph classes in the hairy graph complex to nonzero elements in the rational homotopy groups (the load-bearing point for the infinite-dimensionality claim) requires explicit verification that no additional differentials or relations arise in codimension 2 for odd n ≥ 5; the abstract and construction alone do not confirm the pairing or chain map remains injective on these classes.
minor comments (2)
  1. [Introduction] Clarify the precise relationship between Emb_c and Emb̄_c in the introduction, as the bar notation is used without immediate definition.
  2. [§2] Ensure all graph complexes and their differentials are defined before the construction of the uni-trivalent classes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and the detailed comment on the nontriviality argument. We respond point-by-point below.

read point-by-point responses

  1. Referee: [§4 (nontriviality argument)] The detection step mapping uni-trivalent graph classes in the hairy graph complex to nonzero elements in the rational homotopy groups (the load-bearing point for the infinite-dimensionality claim) requires explicit verification that no additional differentials or relations arise in codimension 2 for odd n ≥ 5; the abstract and construction alone do not confirm the pairing or chain map remains injective on these classes.

    Authors: In Section 4 we construct an explicit chain map from the hairy graph complex to a model for the rational homotopy groups of the embedding space and prove that this map sends the indicated uni-trivalent graph classes to nonzero elements. The proof proceeds by exhibiting a dual pairing whose value on these classes is nonzero, together with a direct computation showing that the differential in the target model produces no additional boundaries in the relevant degrees when the codimension is 2 and n is odd and at least 5. These verifications appear in the statements and proofs of Theorems 4.1 and 4.5 (and the lemmas immediately preceding them), which treat the codimension-2 case separately from the higher-codimension results recalled from earlier literature. The abstract merely summarizes the outcome; the required injectivity statement is established in the body of the paper. revision: no

Circularity Check

0 steps flagged

No circularity detected; derivation relies on external hairy graph model and independent nontriviality proof

full rationale

The abstract and provided context describe a construction of homotopy classes from uni-trivalent graphs in the hairy graph model, followed by a separate proof that the images under the map to π_•(Emb̄_c(R^{n-2},R^n)) ⊗ Q are nonzero. This detection step is presented as an independent argument rather than a self-definition, fitted parameter, or self-citation chain that reduces the target infinite-dimensionality result to its own inputs by construction. No equations or steps in the given material exhibit the enumerated circularity patterns; the central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the hairy graph model providing a faithful representation of the rational homotopy groups of the embedding space, which is invoked to construct and detect the nontrivial elements; this is treated as a domain assumption from prior work in the field.

axioms (1)
  • domain assumption The hairy graph model accurately captures the rational homotopy of the embedding space Emb_c(R^j, R^n).
    The paper relies on this model to associate graphs to homotopy elements and prove nontriviality.

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