Instanton Floer homology $H{F}_{\ast }(Y)$

Intuition Beginner

Instanton Floer homology turns the Chern-Simons landscape of a three-manifold into algebra. The resting points are flat connections. The paths between them are instantons on a cylinder. Floer homology records which resting points remain after accounting for all the signed paths between them.

This is like Morse homology, where critical points of a height function generate a chain complex and downhill flow lines define the differential. The difference is that the landscape is infinite-dimensional and gauge symmetry must be divided out.

The output is a graded abelian group attached to the three-manifold. It is sensitive to the way the three-manifold can bound four-dimensional gauge fields.

Visual Beginner

Nodes are flat connections. Arrows are signed trajectories. Homology keeps the combinations of nodes that are cycles, modulo those that are already boundaries.

Worked example Beginner

Take a simplified chain complex with two generators, $a$ and $b$, and suppose there are no signed trajectories contributing from one to the other. Then the differential sends both generators to zero.

Every generator is a cycle, and none is a boundary. The resulting homology is a copy of the integers generated by $a$ plus a copy generated by $b$.

For the Poincare homology sphere in the standard instanton convention quoted in Donaldson-Floer theory, the basic computation has two generators in different mod-eight degrees and zero differential. What this tells us: instanton Floer homology can distinguish a three-manifold by the graded pattern of surviving flat connections.

Check your understanding Beginner

Formal definition Intermediate+

Let $Y$ be an oriented integral homology three-sphere. After choosing a Riemannian metric and a generic holonomy perturbation of the Chern-Simons functional, assume the irreducible critical points form a finite nondegenerate set $$ \mathcal R^*(Y). $$ These are perturbed flat $\mathrm{SU}(2)$ connections modulo gauge 03.07.17, 03.07.18.

The instanton Floer chain group is the free abelian group $$ CF_*(Y)=\bigoplus_{\alpha\in\mathcal R^*(Y)}\mathbb Z\langle \alpha\rangle, $$ graded relatively by $$ \mu(\alpha,\beta)\in\mathbb Z/8 $$ using spectral flow 03.07.19. For generators whose relative grading is one, let $$ \mathcal M^0(\alpha,\beta) $$ be the zero-dimensional moduli space of unparametrized ASD trajectories from $\alpha$ to $\beta$ on $\mathbb{R}\times Y$. Compactness 03.07.20, regularity, and orientations 03.07.22 give a finite signed count $$ #\mathcal M^0(\alpha,\beta)\in\mathbb Z. $$

The Floer differential is $$ \partial\alpha=\sum_{\beta:\ \mu(\alpha,\beta)=1} #\mathcal M^0(\alpha,\beta),\beta. $$ The instanton Floer homology of $Y$ is $$ HF_*(Y)=\ker\partial/\operatorname{im}\partial, $$ with grading in $\mathbb{Z}/8$.

Counterexamples to common slips

• The generators are not all flat connections. Reducibles and degeneracies require restrictions or perturbations.
• The count is signed over $\mathbb{Z}$. Over $\mathbb{Z}/2$, the same construction forgets orientation signs.
• The differential counts unparametrized trajectories of expected dimension zero, not every ASD solution on the cylinder.

Key theorem with proof Intermediate+

Theorem (the instanton Floer differential squares to zero). With generic perturbation and coherent orientations, the map $\partial :C{F}_{\ast }(Y)\to C{F}_{\ast -1}(Y)$ satisfies $$ \partial^2=0. $$

Proof. Fix generators $\alpha$ and $\beta$ with relative grading two. The coefficient of $\beta$ in ${\partial }^2\alpha$ is $$ \sum_\gamma #\mathcal M^0(\alpha,\gamma)\cdot #\mathcal M^0(\gamma,\beta), $$ where $\gamma$ ranges over generators with the intermediate grading.

Consider the one-dimensional moduli space ${\mathcal{M}}^1(\alpha ,\beta )$ of unparametrized trajectories from $\alpha$ to $\beta$. Uhlenbeck compactness 03.07.20 compactifies it by adding broken trajectories $\alpha \to \gamma \to \beta$. In this index-two situation, bubbling is excluded by the expected-dimension and energy setup after generic perturbation, so the compactification is a compact oriented one-manifold whose boundary consists of those broken pairs.

The gluing theorem 03.07.21 identifies each broken pair with an actual boundary end of $\overline{{\mathcal{M}}^1(\alpha ,\beta )}$. Coherent orientations 03.07.22 identify the boundary sign of that end with the product of the signs of the two zero-dimensional trajectories. The signed boundary count of any compact oriented one-manifold is zero. Therefore the displayed coefficient of $\beta$ in ${\partial }^2\alpha$ is zero. Since this holds for every $\beta$, ${\partial }^2=0$. $\square$

Bridge. Instanton Floer homology builds toward 03.07.24 because relative Donaldson invariants of four-manifolds with boundary take values in these groups, and it appears again in 03.07.25 through the surgery exact triangle. The foundational reason is that Chern-Simons Morse theory identifies flat connections with generators and ASD cylinders with differentials; this is exactly what compactness, gluing, and orientations make algebraically valid. Putting these together, $H{F}_{\ast }(Y)$ generalises finite-dimensional Morse homology, and the bridge is dual to symplectic Floer homology in 05.08.02.

Exercises Intermediate+

Advanced results Master

For an integral homology sphere, the standard $\mathrm{SU}(2)$ instanton Floer chain complex is finite after generic perturbation. The nondegeneracy condition means the Hessian of the perturbed Chern-Simons functional has no zero modes transverse to gauge at each critical point. Spectral flow gives a relative $\mathbb{Z}/8$ grading, and the ASD deformation operator on a trajectory from $\alpha$ to $\beta$ has index equal to this relative grading before quotienting by translation.

Metric and perturbation choices do not change the resulting homology. Given two choices, choose a generic one-parameter family connecting them. Counting parametrized ASD trajectories over this family defines a continuation chain map. Compactness and gluing for the one-parameter moduli spaces prove it is a chain map. Reversing the path gives an inverse up to chain homotopy, so the homology group is an invariant of $Y$.

For the Poincare homology sphere $\Sigma (2,3,5)$, the standard computation gives two basic irreducible generators in degrees $1$ and $5$ under one common orientation convention, with vanishing differential. Thus $$ HF_*(\Sigma(2,3,5))\cong \mathbb Z $$ in each of those two degrees and zero in the other mod-eight degrees. Orientation reversal dualizes and shifts conventions; the graded pattern is the useful invariant rather than the names of the generators.

Instanton Floer homology is the boundary state space for Donaldson theory. If $X$ is a smooth four-manifold with boundary $Y$, suitable ASD moduli spaces on $X$ define relative invariants valued in $H{F}_{\ast }(Y)$. Gluing four-manifolds along $Y$ pairs the two relative classes and recovers closed Donaldson invariants. This is the TQFT-like structure developed next in 03.07.24.

Synthesis. The foundational reason instanton Floer homology is a three-manifold invariant is that all auxiliary choices enter through continuation maps that are chain homotopy equivalences. This is exactly the infinite-dimensional Morse-homology pattern, with Chern-Simons replacing the height function and ASD cylinders replacing gradient lines. The central insight identifies compactified trajectory boundaries with algebraic differential compositions, while coherent orientations are dual to integer coefficients. Putting these together, $H{F}_{\ast }(Y)$ supplies the bridge from gauge analysis on cylinders to topological invariants of three- and four-manifolds.

Full proof set Master

Proposition 1 (the differential has degree minus one). With the mod-eight grading convention above, $\partial$ lowers degree by one.

Proof. The summation in the definition of $\partial \alpha$ includes only those $\beta$ for which $\mu (\alpha ,\beta )=1$. This means the expected dimension of parametrized trajectories from $\alpha$ to $\beta$ is one, so the unparametrized moduli space counted by the differential is zero-dimensional. Equivalently, $\mathrm{deg}(\beta )=\mathrm{deg}(\alpha )-1$ modulo eight. Therefore every term in $\partial \alpha$ has degree one less than $\alpha$. $\square$

Proposition 2 (homology is defined). Since ${\partial }^2=0$, the quotient $\ker \partial /\mathrm{im}\partial$ is well-defined.

Proof. The identity ${\partial }^2=0$ implies that every boundary is a cycle: if $x=\partial y$, then $\partial x={\partial }^{2}y=0$. Thus $\mathrm{im}\partial \subseteq \ker \partial$. The quotient of cycles by boundaries is therefore a well-defined abelian group in each mod-eight degree. $\square$

Proposition 3 (zero differential example). If $CF$ is freely generated by two generators $a,b$ and $\partial a=\partial b=0$, then $HF(CF,\partial )\cong \mathbb{Z}⟨a⟩\oplus \mathbb{Z}⟨b⟩$ with the inherited gradings.

Proof. Since the differential is zero, every chain is a cycle, so $\ker \partial =CF$. The image of $\partial$ is zero. Therefore $$ HF=\ker\partial/\operatorname{im}\partial=CF/0\cong CF. $$ The two generators survive with their original gradings. $\square$

Connections Master

• Chern-Simons functional 03.07.17. Instanton Floer homology is the Morse homology of the Chern-Simons functional on the gauge quotient, after perturbing to obtain nondegenerate critical points.

• Spectral flow and Floer grading 03.07.19. Spectral flow supplies the mod-eight relative grading and the expected dimension formula for trajectory moduli spaces.

• Compactness, gluing, and orientations 03.07.20, 03.07.21, 03.07.22. These three units are exactly the analytic package proving finiteness of counts, compactified-boundary descriptions, and the signed cancellation behind ${\partial }^2=0$.

• Symplectic Floer homology 05.08.02. The instanton theory is the gauge-theoretic sibling of Hamiltonian/Lagrangian Floer homology: critical points and flow lines are replaced by flat connections and ASD cylinder connections.

• Relative Donaldson invariants 03.07.24. Four-manifolds with boundary define classes or maps involving $H{F}_{\ast }(Y)$, making instanton Floer homology the boundary state space for Donaldson theory.

Historical & philosophical context Master

Floer introduced instanton homology in 1988 as a gauge-theoretic invariant of homology three-spheres built from the Chern-Simons functional and ASD equations on cylinders ^{[Floer 1988]}. The construction transformed Donaldson's four-dimensional instanton theory into a three-dimensional homology theory.

Donaldson's Cambridge tract gave the systematic treatment of the analytic foundations: configuration spaces, spectral flow, compactness, gluing, orientations, and invariance ^{[Donaldson 2002]}. Fintushel and Stern computed important families of examples, including Seifert fibered homology spheres, showing that the groups carry concrete three-manifold information ^{[Fintushel-Stern 1990]}.

Bibliography Master

@article{Floer1988InstantonHomology,
  author = {Floer, Andreas},
  title = {An instanton-invariant for 3-manifolds},
  journal = {Communications in Mathematical Physics},
  volume = {118},
  pages = {215--240},
  year = {1988}
}

@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{DonaldsonKronheimer1990Floer,
  author = {Donaldson, Simon K. and Kronheimer, Peter B.},
  title = {The Geometry of Four-Manifolds},
  publisher = {Oxford University Press},
  year = {1990}
}

@article{FintushelStern1990Instanton,
  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}
}