Donaldson-Floer surgery exact triangle

Intuition Beginner

Surgery changes a three-manifold by cutting out a solid tube around a knot and gluing it back with a different slope. The surgery exact triangle says three closely related surgeries have Floer groups tied together in one repeating algebraic pattern.

If you know two corners of the triangle and the maps between them, the third is strongly constrained. This makes the triangle a computation tool: it lets one trade a difficult three-manifold for two related ones.

The word "exact" means that whatever dies under one map is exactly what came from the previous map. No information is lost without being accounted for at the preceding corner.

Visual Beginner

Each vertex is a three-manifold obtained by one surgery choice on the same knot. The arrows are cobordism maps, and exactness links the three Floer groups.

Worked example Beginner

Consider a simplified exact triangle $$ A\to B\to C\to A. $$ Suppose $A=0$ and $B\cong \mathbb{Z}$. Exactness at $B$ says the kernel of the map $B\to C$ equals the image of $A\to B$, which is zero. Therefore $B\to C$ is injective.

Exactness does not by itself determine all of $C$, but it forces a copy of $\mathbb{Z}$ from $B$ to survive inside $C$. With additional grading and rank information, exact triangles often determine unknown groups.

What this tells us: the surgery triangle is not just a relationship among names. It is a computational constraint among the Floer groups of related surgeries.

Check your understanding Beginner

Formal definition Intermediate+

Let $K\subset Y$ be a knot in an integral homology three-sphere. Let $$ Y_\infty,\quad Y_0,\quad Y_1 $$ denote the standard surgery triple determined by three slopes on $\partial (Y\setminus \nu K)$ with pairwise intersection number one; here $Y_{\infty }=Y$ is the original manifold in the usual notation. In the instanton setting one often uses admissible bundles or reduced variants when a surgery has positive first Betti number.

The Donaldson-Floer surgery exact triangle is a long exact sequence of instanton Floer groups $$ \cdots\longrightarrow HF_*(Y_\infty) \xrightarrow{F_\infty} HF_*(Y_0) \xrightarrow{F_0} HF_*(Y_1) \xrightarrow{F_1} HF_{*-1}(Y_\infty) \longrightarrow\cdots, $$ with grading shifts depending on the chosen orientation and bundle conventions.

The maps $F_{\infty },F_0,F_1$ are induced by two-handle cobordisms between the surgery manifolds, using the relative Donaldson/Floer cobordism maps from 03.07.24. Exactness means $$ \operatorname{im}F_\infty=\ker F_0,\qquad \operatorname{im}F_0=\ker F_1,\qquad \operatorname{im}F_1=\ker F_\infty $$ after the degree shift is accounted for.

Counterexamples to common slips

• The zero-surgery term may require an admissible bundle or variant of instanton Floer homology because $b_1(Y_0)$ is often positive.
• The triangle is not a short exact sequence. It is a long exact sequence with a degree shift after the third map.
• The maps are not arbitrary homomorphisms. They are cobordism maps defined by ASD moduli spaces on two-handle cobordisms.

Key theorem with proof Intermediate+

Theorem (surgery exact triangle). For a surgery triple $(Y_{\infty },Y_0,Y_1)$ associated to a knot in a homology three-sphere, the corresponding instanton Floer groups fit into a long exact triangle $$ \cdots\to HF_*(Y_\infty)\to HF_*(Y_0)\to HF_*(Y_1)\to HF_{*-1}(Y_\infty)\to\cdots. $$

Proof. The three surgery manifolds are related by two-handle cobordisms. Relative Donaldson theory assigns maps on instanton Floer homology to these cobordisms 03.07.24. Composing two consecutive cobordisms gives a four-dimensional family whose relevant ASD moduli spaces compactify with boundary strata corresponding to the two possible broken compositions.

At the chain level, the compactified one-dimensional parametrized moduli spaces show that consecutive maps compose to a chain homotopic zero map. Thus the three maps form a chain-level triangle. The deeper part of Floer's surgery theorem identifies the third complex with the mapping cone of the first map, up to the required grading shift. This mapping-cone identification is proved by analyzing the neck-stretched ASD equations over the surgery cobordisms and the local model near the torus boundary ^{[Floer surgery 1990]}.

For any chain map $f:A\to B$, the standard mapping-cone triangle $$ A\to B\to \operatorname{Cone}(f)\to A[-1] $$ induces a long exact sequence on homology. Applying this algebraic fact to the chain-level surgery triangle gives the displayed long exact sequence of instanton Floer homology groups. $\square$

Bridge. The surgery triangle builds toward 03.07.26 because it is one of the main structural tests any gauge-theoretic or symplectic model of Floer homology must reproduce, and it appears again in relative Donaldson theory as a composition statement for two-handle cobordism maps. The foundational reason is that changing a surgery slope changes the boundary condition in a controlled three-term family; this is exactly what identifies one Floer complex with a mapping cone of a cobordism map. Putting these together, the triangle generalises exact sequences from algebraic topology, and the bridge is dual to the cobordism-state formalism in 03.07.24.

Exercises Intermediate+

Advanced results Master

The surgery triangle is local near the knot complement. The three slopes on the boundary torus determine three fillings whose pairwise intersection numbers are one. The associated two-handle cobordisms form the geometric input for the three Floer maps. Gauge theory enters through ASD moduli spaces on these cobordisms with cylindrical ends.

The zero-surgery corner is the main technical wrinkle in the homology-sphere setting. Since zero-surgery usually has $b_1=1$, the simplest instanton Floer group for homology spheres is not directly applicable without modification. Donaldson-Floer theory handles this by using admissible bundles or closely related variants so that reducible flat connections do not spoil the exact sequence.

At chain level, the theorem is a mapping-cone statement. One constructs chain maps between two surgery complexes and proves the third complex is chain homotopy equivalent to the cone. The proof uses parametrized moduli spaces over a family of metrics and perturbations adapted to the surgery cobordisms. Boundary strata of the compactified parametrized spaces give the chain homotopies needed for exactness.

The triangle is a computational engine. Starting from known groups, such as the vanishing of the reduced instanton Floer homology of $S^3$ in the standard convention, and using exactness plus grading, one can compute or constrain Floer groups of surgeries on knots. Fintushel-Stern computations for Seifert fibered homology spheres are among the classical applications.

Synthesis. The foundational reason the surgery triangle exists is that three Dehn fillings of one knot complement form a controlled local modification family. This is exactly what lets cobordism maps fit into a mapping-cone triangle. The central insight identifies four-dimensional two-handle cobordisms with algebraic maps between Floer complexes, while exactness is dual to compactified parametrized moduli-space boundaries. Putting these together, surgery turns instanton Floer homology into a computable invariant rather than only a definition.

Full proof set Master

Proposition 1 (mapping-cone long exact sequence). A chain map $f:A\to B$ gives a long exact sequence $$ \cdots\to H_k(A)\to H_k(B)\to H_k(\operatorname{Cone}(f))\to H_{k-1}(A)\to\cdots. $$

Proof. The mapping cone fits into a short exact sequence of chain complexes $$ 0\to B\to \operatorname{Cone}(f)\to A[-1]\to0. $$ The snake-lemma or connecting-homomorphism construction for a short exact sequence of chain complexes gives the associated long exact sequence in homology. Reindexing $H_k(A[-1])=H_{k-1}(A)$ gives the displayed sequence. $\square$

Proposition 2 (exactness gives rank constraints over a field). In an exact sequence of finite-dimensional vector spaces $$ A\xrightarrow{f}B\xrightarrow{g}C, $$ one has $\dim B=\dim \mathrm{im}f+\dim \mathrm{im}g$.

Proof. Exactness at $B$ gives $\ker g=\mathrm{im}f$. Rank-nullity for $g:B\to C$ gives $$ \dim B=\dim\ker g+\dim\operatorname{im}g. $$ Substituting $\ker g=\mathrm{im}f$ gives the formula. $\square$

Proposition 3 (consecutive maps vanish on homology). In any exact triangle, the composition of two consecutive maps is zero.

Proof. For consecutive maps $A\stackrel{f}{\to }B\stackrel{g}{\to }C$, exactness at $B$ says $\mathrm{im}f=\ker g$. Hence for any $a\in A$, $f(a)\in \mathrm{im}f=\ker g$, so $g(f(a))=0$. Thus $g\circ f=0$. The same argument applies at every corner of the triangle. $\square$

Connections Master

• Instanton Floer homology 03.07.23. The exact triangle is a structural theorem about the groups $H{F}_{\ast }(Y)$ and their behavior under Dehn surgery.

• Relative Donaldson invariants 03.07.24. The maps in the triangle are cobordism maps, built using the same relative ASD counting machinery.

• Atiyah-Floer conjecture 03.07.26. Any proposed symplectic model of instanton Floer homology must reproduce surgery exact triangles or explain their symplectic counterpart.

• Floer homology in symplectic geometry 05.08.02. Mapping cones and exact triangles are shared algebraic patterns across Floer theories, even though the analytic moduli spaces differ.

Historical & philosophical context Master

Floer announced the instanton surgery exact sequence as part of his program relating instanton homology, knots, and Dehn surgery ^{[Floer surgery 1990]}. The sequence quickly became one of the main computational tools in the subject.

Braam and Donaldson gave a detailed account of Floer's surgery work in the Floer Memorial Volume, placing the exact triangle inside the four-dimensional cobordism-map framework ^{[Braam-Donaldson 1995]}. Donaldson's later tract incorporated the triangle as part of the structural package surrounding instanton Floer homology ^{[Donaldson 2002]}.

Bibliography Master

@article{Floer1990Surgery,
  author = {Floer, Andreas},
  title = {Instanton homology, surgery, and knots},
  journal = {Bulletin of the American Mathematical Society},
  volume = {22},
  pages = {363--370},
  year = {1990}
}

@incollection{BraamDonaldson1995,
  author = {Braam, Peter J. and Donaldson, Simon K.},
  title = {Floer's work on instanton homology, knots and surgery},
  booktitle = {The Floer Memorial Volume},
  publisher = {Birkhauser},
  pages = {195--256},
  year = {1995}
}

@book{Donaldson2002Surgery,
  author = {Donaldson, Simon K.},
  title = {Floer Homology Groups in Yang-Mills Theory},
  series = {Cambridge Tracts in Mathematics},
  volume = {147},
  publisher = {Cambridge University Press},
  year = {2002}
}

@article{FintushelStern1990Surgery,
  author = {Fintushel, Ronald and Stern, Ronald J.},
  title = {Instanton homology of Seifert fibred homology three spheres},
  journal = {Proceedings of the London Mathematical Society},
  volume = {61},
  pages = {109--137},
  year = {1990}
}