Casson's invariant and the Euler characteristic of instanton Floer homology

Intuition Beginner

A three-manifold that looks like a sphere to homology can still carry hidden structure inside its fundamental group. Casson's invariant measures that structure with a single whole number. It counts the ways the loops of the space can be represented by rotations in three-dimensional space, the rotation group that physicists call $\mathrm{SU}(2)$.

The count is signed. Each such representation is tagged with a plus or minus according to how two pieces of the manifold meet, and the signed total is Casson's invariant $\lambda (Y)$.

The deep surprise is that this plain count is the shadow of a far richer object. Instanton Floer homology assigns whole graded groups to the same space. Casson's invariant is recovered by taking the alternating sum of the sizes of those groups, the Euler characteristic. The number came first; the groups that explain it came later.

Visual Beginner

On the left, two representation surfaces cross. Each crossing dot carries a sign, and the signed total is $\lambda (Y)$. On the right, the same number appears as the alternating sum of the graded Floer groups. One count, two faces.

Worked example Beginner

Picture the simplest version of the signed count. Suppose the two representation surfaces meet at exactly three points. Two of the crossings are positive and one is negative.

The signed total is $2-1=1$. So this model space would have Casson invariant equal to one.

Now look at the Floer side. Imagine the graded groups hold one generator in an even degree and none in odd degrees. The Euler characteristic is the even count minus the odd count, which is $1-0=1$. Doubling rules of the theory aside, the point to absorb is that both the crossing count and the alternating sum of the groups land on the same number. The signed geometry and the graded algebra agree.

Check your understanding Beginner

Formal definition Intermediate+

Let $Y$ be an oriented integral homology three-sphere, so $H_{\ast }(Y;\mathbb{Z})\cong H_{\ast }(S^3;\mathbb{Z})$. Fix a Heegaard splitting $$ Y=H_1\cup_\Sigma H_2, $$ where $\Sigma$ is a closed surface of genus $g$ and $H_1,H_2$ are handlebodies. For a compact group $G=\mathrm{SU}(2)$, write $$ R(\Sigma)=\operatorname{Hom}(\pi_1\Sigma,G)/G,\qquad R(H_i)=\operatorname{Hom}(\pi_1 H_i,G)/G $$ for the conjugation quotients of the representation spaces. Restriction along ${\pi }_1\Sigma \to {\pi }_1{H}_i$ embeds $R(H_i)$ into $R(\Sigma )$. By van Kampen, a representation of ${\pi }_{1}Y$ is a representation of ${\pi }_1\Sigma$ that extends over both handlebodies, so $$ R(Y)=R(H_1)\cap R(H_2)\subset R(\Sigma). $$ On the open dense locus $R^{\ast }(\Sigma )$ of irreducible representations, $R^{\ast }(H_1)$ and $R^{\ast }(H_2)$ are smooth half-dimensional submanifolds of the symplectic Goldman manifold $R^{\ast }(\Sigma )$ 03.07.17.

Definition (Casson's invariant). After a small perturbation making the intersection transverse and avoiding the reducible stratum, set $$ \lambda(Y)=\tfrac12,(-1)^{g},\bigl\langle R^(H_1),,R^(H_2)\bigr\rangle_{R^*(\Sigma)}, $$ the signed algebraic intersection number, where each transverse intersection point carries the sign of its local orientation comparison. The half and the genus sign are normalizing conventions; the resulting integer is independent of the splitting and the perturbation.

Equivalently, the irreducible flat $\mathrm{SU}(2)$ connections on $Y$ are exactly the intersection points, since a flat connection is a conjugacy class of holonomy representation ${\pi }_{1}Y\to \mathrm{SU}(2)$ 03.07.17. So $\lambda (Y)$ is a signed count of irreducible flat connections.

Counterexamples to common slips

• The reducible (abelian) representations are not counted. On a homology sphere there is only the central one, isolated and removed by the construction.
• The intersection must be taken inside the irreducible locus. The full representation variety is singular along reducibles, and the naive count there is ill-posed.
• The factor of one half is part of the definition, not an afterthought; it is what makes the surgery formula and the relation to Floer homology come out with integer values.

Key theorem with proof Intermediate+

Theorem (Taubes 1990). For an oriented integral homology three-sphere $Y$, $$ \chi\bigl(HF_*(Y)\bigr)=2,\lambda(Y), $$ where $H{F}_{\ast }(Y)$ is instanton Floer homology 03.07.23 and $\chi$ is the Euler characteristic of the mod-eight graded group.

Proof. The chain group $C{F}_{\ast }(Y)={⨁}_{\alpha \in {\mathcal{R}}^{\ast }(Y)}\mathbb{Z}⟨\alpha ⟩$ is generated by the irreducible perturbed-flat connections 03.07.23, each placed in its mod-eight spectral-flow degree. The Euler characteristic of a finitely generated chain complex equals the Euler characteristic of its homology, so $$ \chi(HF_*(Y))=\sum_{\alpha\in\mathcal R^(Y)}(-1)^{\deg\alpha}. $$ Taubes establishes that the spectral-flow parity $(-1{)}^{\mathrm{deg}\alpha }$ of a flat connection equals the local intersection sign of the corresponding point of $R^(H_1)\cap R^(H_2)$.Thisistheanalyticcore:thesignattachedtoaflatconnectionbythedeterminantlineofitsdeformationoperatormatchestheorientationcomparisonofthetwoLagrangian-typerepresentationsubmanifoldsatthatpoint.Summingthesignsoverallgeneratorsgivesthesignedintersectionnumber,whichbythedefinitionaboveis$2\lambda(Y)$oncethegenussignandthefactorofonehalfareunwound.Therefore$\chi(HF_(Y))=2\lambda(Y)$.

Bridge. This identity builds toward the relative-invariant TQFT of four-manifolds with boundary and appears again in the surgery exact triangle 03.07.25, where the alternating Euler characteristic is additive. The foundational reason the two sides agree is that both are signed counts of the same flat connections; the central insight is that instanton Floer homology categorifies Casson's invariant, so $\lambda (Y)$ is exactly the decategorified shadow of $H{F}_{\ast }(Y)$. Putting these together, the Euler characteristic generalises the classical count: it survives when individual generators are created or cancelled in pairs, which is what makes the Floer groups a strict refinement rather than a restatement.

Exercises Intermediate+

Advanced results Master

Casson defined $\lambda$ in 1985 by the Heegaard intersection above and proved its key structural properties by surgery. The surgery formula $$ \lambda(Y_{1/(n+1)})-\lambda(Y_{1/n})=\tfrac12,\Delta_K''(1) $$ expresses the change of $\lambda$ under $1/n$ Dehn surgery on a knot $K$ in a homology sphere through the second derivative of the symmetrized Alexander polynomial at one. Together with $\lambda (S^3)=0$ and $\lambda (-Y)=-\lambda (Y)$, the surgery formula determines $\lambda$ on every homology sphere, since any such manifold is obtained from $S^3$ by a sequence of $1/n$ surgeries on knots. The reduction of $\lambda$ modulo two recovers the Rokhlin invariant $\mu (Y)\in \mathbb{Z}/2$, the signature mod sixteen of any spin four-manifold bounding $Y$ divided appropriately 03.06.19; this congruence was Casson's route to detecting homology spheres with $\mu \ne 0$ that bound no acyclic smooth four-manifold.

Walker extended the invariant to rational homology spheres. The Casson-Walker invariant ${\lambda }_W(Y)$ adds correction terms accounting for the finite group $H_1(Y;\mathbb{Z})$ and the reducible representations that now appear in positive-dimensional families; Lescop later gave a single closed formula valid for all closed oriented three-manifolds. These extensions keep the surgery-additive character while widening the domain past the integral-homology-sphere case.

The gauge-theoretic reformulation reframes the entire construction. Taubes replaced the Heegaard intersection by the count of irreducible flat $\mathrm{SU}(2)$ connections on $Y$, weighted by spectral-flow signs, and proved the two counts agree. Floer's instanton homology then organizes these same connections into a graded complex whose differential records ASD cylinders 03.07.23. The Euler characteristic collapses the grading back to the signed count, giving $\chi (H{F}_{\ast }(Y))=2\lambda (Y)$. This is the precise sense in which instanton Floer homology was built as a categorification: Casson's number is the decategorified invariant, and the theory that lifts it was constructed to explain why that number is a smooth invariant at all.

These threads converge on the homology-cobordism group ${\Theta }_{\mathbb{Z}}^3$ of integral homology spheres modulo bounding acyclic four-manifolds. Casson's invariant and its Floer refinements give homomorphisms and filtrations on ${\Theta }_{\mathbb{Z}}^3$; Frøyshov and later Pin(2)-equivariant Floer invariants proved ${\Theta }_{\mathbb{Z}}^3$ has free summands and resolved the triangulation question for high-dimensional manifolds, with Casson's $\lambda$ standing as the first numerical probe of this group.

Synthesis. The foundational reason Casson's invariant and instanton Floer homology fit together is that both are signed counts of the same flat connections, read at two levels of structure. This is exactly the decategorification pattern: $\lambda (Y)$ is the Euler characteristic, and $H{F}_{\ast }(Y)$ is the chain-level object that generalises it. The central insight is that the surgery formula, the orientation-reversal antisymmetry, and the Rokhlin congruence all survive categorification, so each property of $\lambda$ is dual to a structural feature of the Floer groups. Putting these together, the bridge is complete: a single signed count of representations, born from a Heegaard splitting, becomes the shadow of a homology theory that probes the homology-cobordism group, and the factor of two records the spectral-flow normalization linking the two descriptions.

Full proof set Master

Proposition 1 (orientation reversal). For an oriented integral homology three-sphere $Y$, $\lambda (-Y)=-\lambda (Y)$.

Proof. Reversing the orientation of $Y$ reverses the induced orientation of the Heegaard surface $\Sigma$ and exchanges the roles of the two handlebody representation submanifolds within $R^{\ast }(\Sigma )$. The Goldman symplectic orientation of $R^{\ast }(\Sigma )$ changes sign, so each transverse intersection point of $R^{\ast }(H_1)\cap R^{\ast }(H_2)$ keeps its location but reverses its local sign. The signed intersection number therefore negates. Since the normalizing genus sign and factor are unchanged, $\lambda (-Y)=-\lambda (Y)$. $\square$

Proposition 2 (vanishing on the standard sphere). $\lambda (S^3)=0$.

Proof. The fundamental group ${\pi }_1(S^3)$ is the one-element group, so every representation ${\pi }_1(S^3)\to \mathrm{SU}(2)$ is the central one and there are no irreducible representations. The irreducible intersection locus is empty, the signed count is the empty sum, and $\lambda (S^3)=0$. $\square$

Proposition 3 (Euler characteristic is a parity-weighted generator count). If $H{F}_{\ast }(Y)$ is computed from a finite chain complex with $c_k$ generators in mod-eight degree $k$, then $\chi (H{F}_{\ast }(Y))={\sum }_{k=0}^7(-1{)}^k{c}_k$.

Proof. For a bounded complex of free abelian groups the Euler characteristic of the homology equals the alternating sum of the ranks of the chain groups, because each application of the differential lowers degree by one and the rank-nullity contributions telescope. The chain group in degree $k$ has rank $c_k$, the number of generators there. Hence $\chi (H{F}_{\ast }(Y))={\sum }_k(-1{)}^k{c}_k$, the parity-weighted count of flat connections. $\square$

Connections Master

• Instanton Floer homology 03.07.23. Casson's invariant is one half of the Euler characteristic of $H{F}_{\ast }(Y)$. The Floer groups are the categorification: their generators are the very flat connections that Casson counts, and their differential is the extra data that the plain count forgets.

• Chern-Simons functional 03.07.17. The irreducible flat connections counted by $\lambda (Y)$ are the irreducible critical points of the Chern-Simons functional on the gauge quotient. Casson's Heegaard intersection and the variational picture describe the same finite set of representations.

• Donaldson-Floer surgery exact triangle 03.07.25. The surgery formula for $\lambda$ is the Euler-characteristic shadow of the surgery exact triangle: the alternating sum is additive along the triangle, so the change of $\lambda$ under surgery mirrors the long exact sequence relating the Floer groups of a surgery triple.

• Signature and intersection form 03.06.19. The reduction of $\lambda$ modulo two is the Rokhlin invariant, computed from the signature of a spin four-manifold bounding $Y$. This congruence ties Casson's count of representations to the four-dimensional intersection-form data and to the obstruction theory behind exotic smooth structures.

Historical & philosophical context Master

Andrew Casson introduced $\lambda$ in 1985 lectures, defining it through the Heegaard-splitting intersection of $\mathrm{SU}(2)$ representation varieties and proving the surgery formula; the construction circulated through notes taken by Akbulut and McCarthy before appearing as a Princeton monograph ^{[Casson 1985]}. Casson's motivation was four-dimensional: he sought obstructions to homology spheres bounding acyclic smooth four-manifolds, recovering and refining the Rokhlin invariant in the process. Taubes then reproved and reframed the invariant as a signed count of flat connections, opening the gauge-theoretic door ^{[Taubes 1990]}. Floer's instanton homology, whose Euler characteristic returns $2\lambda$, made the invariant the visible part of a graded theory, and Walker's extension to rational homology spheres widened the construction past Casson's original integral case ^{[Walker 1992]}. Donaldson's Cambridge tract presents the count and the Euler-characteristic identity as the historical motivation for the whole Floer program ^{[Donaldson 2002]}. The episode is a clean instance of a recurring pattern in mathematics: a numerical invariant defined by a clever count is later understood as the decategorified trace of a richer structure, and the richer structure is precisely what explains why the number was an invariant.

Bibliography Master

@book{AkbulutMcCarthy1990Casson,
  author = {Akbulut, Selman and McCarthy, John D.},
  title = {Casson's Invariant for Oriented Homology 3-Spheres: An Exposition},
  series = {Mathematical Notes},
  volume = {36},
  publisher = {Princeton University Press},
  year = {1990}
}

@article{Taubes1990CassonGauge,
  author = {Taubes, Clifford Henry},
  title = {Casson's invariant and gauge theory},
  journal = {Journal of Differential Geometry},
  volume = {31},
  pages = {547--599},
  year = {1990}
}

@book{Walker1992Extension,
  author = {Walker, Kevin},
  title = {An Extension of Casson's Invariant},
  series = {Annals of Mathematics Studies},
  volume = {126},
  publisher = {Princeton University Press},
  year = {1992}
}

@book{Donaldson2002FloerHomology,
  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}
}

@book{Saveliev2002Invariants,
  author = {Saveliev, Nikolai},
  title = {Invariants for Homology 3-Spheres},
  series = {Encyclopaedia of Mathematical Sciences},
  volume = {140},
  publisher = {Springer},
  year = {2002}
}