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arxiv: 2604.15536 · v2 · submitted 2026-04-16 · 🧮 math.GT · math.SG

Building homology theories (ala Floer)

John B. Etnyre This is my paper

Pith reviewed 2026-05-10 09:10 UTC · model grok-4.3

classification 🧮 math.GT math.SG
keywords Floer homologyhomology theorieslow-dimensional topologysymplectic geometrycontact geometrychain complexes
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The pith

A common framework has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry.

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

The notes describe a shared framework for building homology theories that has appeared across low-dimensional topology and symplectic and contact geometry since the late 1980s. The framework adapts the algebraic structure of the chain groups to fit the concrete features of each setting, such as the type of manifolds or the geometric structures involved. A sympathetic reader would care because the approach explains why many distinct constructions, including various Floer homologies, follow similar patterns and why their differences arise naturally from the geometry or topology at hand.

Core claim

These notes indicate a common framework that has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry. In addition, the notes indicate how the specific nature of the situation being studied dictates the algebraic nature of the chain groups used to define the homology.

What carries the argument

The common framework for constructing homology theories, in which the algebraic structure of the chain groups is chosen according to the geometric or topological context.

If this is right

  • Homology theories in these fields can be understood as instances of one adaptable construction rather than separate inventions.
  • The algebraic choices for chain groups follow directly from the constraints of each specific situation.
  • Ideas and techniques can be transferred more readily between low-dimensional topology and symplectic or contact geometry.

Where Pith is reading between the lines

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

  • The framework may suggest how to build new homology theories by identifying analogous geometric situations not yet explored.
  • Viewing existing theories through this lens could reveal unexpected relations between invariants from different fields.
  • The approach highlights that the success of Floer-type theories depends on matching algebraic structure to geometry rather than on a single fixed algebra.

Load-bearing premise

The situations in low-dimensional topology and symplectic and contact geometry share enough structure for a single common framework to apply usefully across them.

What would settle it

A detailed comparison showing that the constructions in these different areas lack the shared structural steps and algebraic adaptations described in the framework would show the common approach does not hold.

Figures

Figures reproduced from arXiv: 2604.15536 by John B. Etnyre.

Figure 1
Figure 1. Figure 1: The gradient flow of f(x1, . . . , xn) = −x 2 1 − · · · − x 2 k + x 2 k+1 + · · · x 2 n . Example 2.11. Consider the unit sphere S 2 in R 3 with the metric induced from the Euclidean metric on R 3 . If the function f : S 2 → R is given by projection to the last coordinate, then we see that there are exactly two critical points N = (0, 0, 1) and S = (0, 0, −1). It is clear that N has index 2 and S has index… view at source ↗
Figure 2
Figure 2. Figure 2: The gradient flow of the height function on the sphere. Exercise 2.12. Show that a flow line of the gradient in the previous example through the point x = (cos θ,sin θ, 0) is given by γ(t) =  cos θ cosh t , sin θ cosh t ,tanh t  . Hint: Show that in stereographic coordinates given by ϕ(u, v) = 1 1 + u 2 + v 2 (2u, 2v, u2 + v 2 − 1) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The gradient flow of projection the the z-axis for the skew torus in R 3 . d c c b b a a a a [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The torus represented as the square with opposite edges identified by translation. We see the gradient flow of the function f. The points in the purple curves are in one-to-one correspondence with flow lines from d to a. Morse Theorem 1 (Transversality) For a generic choice of metric g on M and Morse function f : M → R the space Mp q is a manifold of dimension index p − index q − 1 [PITH_FULL_IMAGE:figure… view at source ↗
Figure 5
Figure 5. Figure 5: The symmetric torus in R 3 . metric (here the metric is induced from R 3 ) in the sense of Morse Theorem 1, then Mc b would have dimension −1 and hence be empty. Thus it is clear that Morse Theorem 1 is not true for all Morse functions and metrics. We now move to the second main theorem in the case of Morse homology. Morse Theorem 2 (Orientability) If M is oriented and we choose orientations on all the neg… view at source ↗
Figure 6
Figure 6. Figure 6: A sequence of flow lines converging to a broken flow line. borhood of the union of the images of γe1, and γe2, and we see that for large enough i, the images of the γi are contained in the neighborhood. A similar figure will show that for large i the images of γe1, and γe2 will be in a small neighborhood of the images of the γi . Corollary 2.19. If index p = index q + 1, then Mp q is a compact 0-manifold. … view at source ↗
Figure 7
Figure 7. Figure 7: The compactification of the gradient flow lines in the lower left corner are the broken flow lines γ1 ∪ δ4 and γ3 ∪ δ2. More generally we see that Md b = {γ1, γ2}, Md c = {γ3, γ4}, Mb a = {δ3, δ4}, and Mc a = {δ1, δ2}, and ∂Md a = (Md b × Mb a ) ∪ (Md c × Mc a ). So we saw that Md a consists of 4 open intervals and thus its closure consists of 4 closed intervals with boundary being 8 broken flow lines. Thi… view at source ↗
Figure 8
Figure 8. Figure 8: The compactification of the gradient flow lines in the lower left corner are the broken flow lines γ1 ∪ δ4 and γ3 ∪ δ2. We can now define the i th Morse homology of M to be Hi(M) = ker (∂ : Ci → Ci−1) image (∂ : Ci+1 → Ci) . Exercise 2.27. Show that Hi(M) agrees with the i th homology of M as is usually defined in algebraic topology. Hint: This might be more challenging than some previous exercises. Maybe … view at source ↗
Figure 9
Figure 9. Figure 9: A component of Mfp q . Now if p ∈ Crit(f0) and q ∈ Crit(f1) then we set Mcp q = {(γ, s) : γ ∈ Mfp q (s)}, where Mfp q (s) is Mfp q for the path Γs = (fs,t, gs,t). We have the “same” Morse Theorems 1 through 4 for Mcp q except that dimMcp q = index p − index q + 1 and the “obvious” changes to the compactness and gluing theorems. Exercise 2.32. Figure out what the “obvious” changes are. Now we can define K :… view at source ↗
Figure 10
Figure 10. Figure 10: The blue is a path γs in P p q through γ (shown in orange). The green curves are the paths s 7→ γs(t0) for some fixed t0 ∈ R and the black vectors give the vector field v along γ. for each γ(t) the vector v(t) ∈ Tγ(t)M. Summing up, given a path in P p q through γ its “derivative” at γ(t) is a vector field v(t) along γ. We also know that since γs ∈ Pp q we must have that v(t) → 0 as t → ±∞ [PITH_FULL_IMAG… view at source ↗
Figure 11
Figure 11. Figure 11: Converging to a broken flow line. In the top left, we see the sequence of gradient flow lines in Mp q near p. They intersect ∂B1 in points xi that converge to x∞. The flow line γe1 is the flow line through x∞. On the bottom left we see the critical point r2 to which γe1 flows. We also see the ball B2 around r2 used to find the next flow line in the broken flow line. On the right we see the complete broken… view at source ↗
Figure 12
Figure 12. Figure 12: On the left we see a broken flow line, and on the right we see an approximate flow line near the broken flow line. and the stable manifold of p to be Sp = {x ∈ M such that limt→∞ γx(t) = p}. These are also sometimes referred to as the descending manifold and ascending manifold, respectively. One may prove, [3], that Up and Sp are both open disks of dimension k and n − k, respectively, where k = index p. W… view at source ↗
Figure 13
Figure 13. Figure 13: The properly embedded curves γi on Σ. each γi to a point, which we denote by zi . See [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The cusped surface Σ and the images b zi of the γi . of Σ to be b s(Σ) = b {z1, . . . , zn}. Now, given an oriented Riemannian surface (Σ, j) we get a metric on Σ by setting h(v, w) = Ω(v, jw) where Ω is any araa form on Σ giving the orientation on Σ. Given an almost-complex structure J compatible with a symplectic form ω on a manifold X can now define the action of a map u: Σ → X to be A(u) = Z Σ ∥du∥ 2Ω… view at source ↗
Figure 15
Figure 15. Figure 15: On the left are the images of the uk, shown in red, and the u l ∞, shown in blue. On the right we see the same picture where the spheres have not been punctured (though they are still, necessarily, half-dimensional, so the figure can be drawn). To define the boundary map, we will consider “pseudo-holomorphic strips”. That is we will consider the surface R × [0, 1] with coordinate (s, t) and the complex st… view at source ↗
Figure 16
Figure 16. Figure 16: Two Lagrangian subspaces of C. The map I is indicated by the arrow. The path γΛ0,Λ1 rotates Λ1 anticlockwise to Λ0. Λ0 Λ1 x y [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The gray is the image of a pseudo-holomorphic curve in C with bound￾ary on Λ0 and Λ1. We drew red and blue arrows whose span gives the tangent space to the Λi . The green arrows give the interpolation between the end of the red and blue arrows given by the paths γγ1(∞),γ0(∞) and γγ0(−∞),γ1(−∞) . (Note that the first path rotates anticlockwise a bit more than π/2 while the second path rotates clockwise a b… view at source ↗
Figure 18
Figure 18. Figure 18: Cusped curves to which a pseudo-holomorphic strip can converge. In the first example, a sphere “bubbles off” in the limit. In the second example, a disk “bubbles off” in the limit. In the last case, the pseudo-holomorphic strip limits to a “broken strip”. Finally, we have a gluing theorem. We state a vague version of the theorem as the full theorem can be a bit technical. Floer Theorem 4 (Gluing) Any cusp… view at source ↗
Figure 19
Figure 19. Figure 19: The proof that ∂ 2 = 0. p+ x p− [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: How ∂ 2 = 0 can go wrong if one has undesirable limits. The limit on the right is supposed to represent a disk bubling off. Given Lagrangian submanifolds L0 and L1 satisfying (∗) and for which we do not have limits (1) and (2) in [PITH_FULL_IMAGE:figures/full_fig_p037_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The standard contact structure on R 3 . A submanifold L of a (2n + 1)-dimensional contact manifold (M, ξ) is called Legendrian if it has dimension n and the tangent space of L is contained in ξ: TxL ⊂ ξx for all x ∈ L. Legendrian submanifolds are essential objects in contact geometry, see for example, [16, 17]. Suppose that L is a Legendrian submanifold of (X × R, dt − λ) and let π : (X × R) → X be the pr… view at source ↗
Figure 22
Figure 22. Figure 22: Part of the projection π(L) is shown on the left. The middle diagram shows the image of u while the right-hand diagram shows the image of u ′ . formed when u and u ′ are glued. We notice that this family will now break into pseudo-holomorphic [PITH_FULL_IMAGE:figures/full_fig_p042_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: On the left, we see the image of a pseudo-holomorphic strip obtained by gluing u and u ′ . Notice the branched point on the boudnary near the crossing y. In the middle, we see the middle of the family of pseudo-holomorphic strips where the “branched point” is “at infinity” or “at z”. On the right, we see a pseudo￾holomorphic strip with the branched point approaching w. disks as shown in [PITH_FULL_IMAGE:… view at source ↗
Figure 24
Figure 24. Figure 24: The two pseudo-holomorphic disks in the boundary of the family in [PITH_FULL_IMAGE:figures/full_fig_p043_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: A pseudo-holomorphic disk converging to a broken pseudo-holomorphic disk. Remark 4.3. We note that, for simplicity, this theorem is not quite accurate in some cases, as we defined Mx0 x1,...,xk to be pseudo-holomorphic disks moded out by pseudo-holomorphic reparam￾eterization. When k is large, there are no such reparameterizations and the theorem is correct as stated, but for small k, the statement should… view at source ↗
read the original abstract

These notes are an expanded version of evening talks at the 2025 Georgia International Topology Conference, and an abbreviated version of talks at Georgia Tech, which were aimed at graduate students. The hope was to indicate a common framework that has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry. In addition to this, we also try to indicate how the specific nature of the situation being studied dictates the algebraic nature of the chain groups used to define the homology.

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

0 major / 2 minor

Summary. The manuscript consists of expository notes (expanded from evening talks at the 2025 Georgia International Topology Conference and abbreviated versions of Georgia Tech talks) aimed at graduate students. It outlines a common methodological framework employed since the late 1980s to construct homology theories in low-dimensional topology and in symplectic and contact geometry, while explaining how the geometric setting determines the algebraic structure of the associated chain groups.

Significance. If the descriptive account is accurate, the notes provide a useful pedagogical synthesis of a standard pattern in the field. By emphasizing the shared template across constructions such as Floer homology, Heegaard Floer homology, and contact homology, and by highlighting the geometry-to-algebra dependence, the work can help students recognize unifying principles without requiring them to extract the pattern from scattered research papers. Its value is therefore primarily educational rather than the introduction of new theorems or computations.

minor comments (2)
  1. As an expository piece, the notes would benefit from an explicit list of recommended further reading or primary references at the end to guide students to the original papers on each example.
  2. Consider adding a short concluding section that summarizes the key takeaways about how the algebraic choices (e.g., chain groups) are dictated by the geometry, to reinforce the central pedagogical point for the target audience.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. We are pleased that the expository intent and pedagogical synthesis of the common framework for Floer-type constructions have been recognized.

Circularity Check

0 steps flagged

Expository notes with no derivations, predictions or self-referential steps

full rationale

The manuscript consists of expository notes aimed at graduate students that describe an existing methodological template used since the late 1980s for constructing homology theories (Floer, Heegaard Floer, contact homology, etc.). No equations, fitted parameters, uniqueness theorems, ansatzes, or predictions are introduced; the text simply outlines how algebraic details vary by geometric setting. Because the central claim is descriptive rather than deductive, no load-bearing step reduces to its own inputs by construction, self-citation, or renaming. The paper is therefore self-contained against external benchmarks with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Expository notes with no new mathematical claims, parameters, or entities introduced.

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