Relative Donaldson invariants for 4-manifolds with boundary

Intuition Beginner

Instanton Floer homology is the state space assigned to a three-manifold. A four-manifold whose boundary is that three-manifold should then produce a state in that space. This is the relative Donaldson invariant.

Think of a four-manifold with boundary as a machine with an output port. The output port is the boundary $Y$, and the possible output states live in $H{F}_{\ast }(Y)$. Counting instantons on the four-manifold produces one such state.

When two four-manifolds are glued along a common boundary, their relative states pair. The result is the ordinary Donaldson invariant of the closed four-manifold obtained after gluing.

Visual Beginner

The boundary is the interface. Relative invariants live on the interface; closed Donaldson invariants appear when two interfaces are paired.

Worked example Beginner

Suppose a four-manifold $X_1$ has boundary $Y$ and produces the Floer state $2a-b$ in a two-generator group with basis $a,b$. Suppose another four-manifold $X_2$ has the opposite boundary and produces the dual state $3a^{\ast }+4b^{\ast }$.

Gluing $X_1$ and $X_2$ along $Y$ pairs these states: $$ (3a^+4b^)(2a-b)=6-4=2. $$ The number $2$ is the invariant of the closed glued four-manifold in this simplified model.

What this tells us: relative Donaldson invariants turn cut four-manifolds into algebraic vectors whose pairing reconstructs closed invariants.

Check your understanding Beginner

Formal definition Intermediate+

Let $X$ be a compact oriented smooth four-manifold with boundary $\partial X=Y$, and attach a cylindrical end to form $$ X^+=X\cup_Y([0,\infty)\times Y). $$ Fix a principal $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ bundle $P\to X^{+}$ with suitable topological data. An ASD connection on $X^{+}$ has finite energy when it converges along the cylindrical end to a flat connection $\alpha$ on $Y$.

For each generator $\alpha$ of the instanton Floer chain complex $C{F}_{\ast }(Y)$ 03.07.23, let $$ \mathcal M_X(\alpha) $$ be the moduli space of finite-energy ASD connections on $X^{+}$ asymptotic to $\alpha$, modulo gauge transformations that converge to the identity on the end in the chosen framing convention. After perturbations and insertions from homology classes in $X$, zero-dimensional components of these moduli spaces can be signed and counted.

The relative Donaldson chain is $$ \Phi_X=\sum_{\alpha}#\mathcal M_X^0(\alpha),\alpha\in CF_*(Y), $$ with degree shifted by the expected dimension formula and by any inserted cohomology classes. Compactness for one-dimensional relative moduli spaces shows that $$ \partial\Phi_X=0, $$ so ${\Phi }_X$ determines a class $$ [\Phi_X]\in HF_*(Y). $$

If $X=X_1{\cup }_Y{X}_2$ is a closed four-manifold split along $Y$, then the gluing formula has the form $$ D_X=\langle [\Phi_{X_1}],[\Phi_{X_2}]\rangle, $$ where $D_X$ is the closed Donaldson invariant and the pairing is the natural Floer pairing between $H{F}_{\ast }(Y)$ and $H{F}_{\ast }(-Y)$.

Counterexamples to common slips

• A relative invariant is not generally an integer. It is a Floer class; it becomes a number only after pairing with a complementary relative class or applying a functional.
• Cylindrical ends are analytic devices. They encode boundary asymptotics without changing the compact four-manifold's topology.
• The gluing formula requires matched bundles, perturbations, orientations, and insertions. It is not merely a topological union of sets.

Key theorem with proof Intermediate+

Theorem (relative invariants are cycles and glue to closed invariants). For a compact oriented four-manifold $X$ with boundary $Y$, the relative Donaldson chain ${\Phi }_X$ is a cycle in $C{F}_{\ast }(Y)$. If a closed four-manifold decomposes as $X=X_1{\cup }_Y{X}_2$, then the closed Donaldson invariant is obtained by pairing the relative Floer classes of $X_1$ and $X_2$.

Proof. Consider the one-dimensional relative moduli spaces on $X^{+}$ with fixed insertions. Uhlenbeck compactness on a cylindrical end says their compactifications have boundary strata of two kinds relevant to the chain equation: an ASD connection on $X^{+}$ limiting to an intermediate flat connection $\beta$, followed by an instanton Floer trajectory from $\beta$ to $\alpha$ on $\mathbb{R}\times Y$; and possible higher-codimension degenerations excluded by dimension and generic choices in the zero- and one-dimensional setup.

The signed boundary count of a compact oriented one-manifold is zero. The boundary strata just described are exactly the terms in $\partial {\Phi }_X$, with signs determined by the orientation and gluing conventions from 03.07.22. Hence $\partial {\Phi }_X=0$, so the relative chain defines a Floer homology class.

For gluing, stretch the neck in a closed decomposition $X_1{\cup }_Y{X}_2$. Compactness says a sequence of ASD connections on the stretched closed four-manifold converges to a pair of relative ASD connections on $X_1^{+}$ and $X_2^{+}$ with matching asymptotic flat connection on $Y$. The gluing theorem reverses this degeneration, giving a local correspondence between closed ASD moduli points and paired relative moduli points. The orientation convention identifies the signed closed count with the Floer pairing of the two relative classes. Therefore the closed Donaldson invariant equals $⟨[{\Phi }_{X_1}],[{\Phi }_{X_2}]⟩$. $\square$

Bridge. Relative Donaldson invariants build toward 03.07.25 because surgery exact triangles can be viewed through cobordism maps between Floer groups, and they appear again in 03.07.26 as the gauge-theoretic side of a TQFT-like boundary-state picture. The foundational reason is that ASD moduli spaces on manifolds with cylindrical ends identify boundary asymptotics with Floer generators; this is exactly what turns a cut four-manifold into a vector in $H{F}_{\ast }(Y)$. Putting these together, relative invariants generalise closed Donaldson polynomials, and the bridge is dual to the chain-level invariant in 03.07.23.

Exercises Intermediate+

Advanced results Master

The relative invariant is a cobordism-state assignment. A cobordism $W:Y_0\to Y_1$ gives, after choosing bundle data and insertions, a map $$ HF_*(Y_0)\longrightarrow HF_*(Y_1) $$ by counting ASD connections on $W$ with two cylindrical ends. The special case $Y_0=\varnothing$ gives a vector in $H{F}_{\ast }(Y_1)$, while $Y_1=\varnothing$ gives a covector on $H{F}_{\ast }(Y_0)$.

Insertions are handled by the universal bundle over $X\times \mathcal{M}$. In the closed theory, Donaldson polynomial invariants evaluate cohomology classes derived from the universal bundle on ASD moduli spaces. In the relative theory, the same insertions cut down the relative moduli spaces before counting their ends in Floer chain groups.

The gluing formula is compatible with composition of cobordisms. If $W_1:Y_0\to Y_1$ and $W_2:Y_1\to Y_2$, the map for $W_2\circ W_1$ agrees with the composition of the maps for $W_1$ and $W_2$, after the standard transversality and orientation choices. This is the precise sense in which instanton Floer homology supplies a TQFT-like extension of Donaldson theory.

For a closed decomposition $X=X_1{\cup }_Y{X}_2$, the pairing can be interpreted as summing over a basis of Floer generators at the cut: $$ D_X=\sum_{\alpha,\beta} c_\alpha,Q_{\alpha\beta},d_\beta, $$ where ${\Phi }_{X_1}=\sum c_{\alpha }\alpha$, ${\Phi }_{X_2}=\sum d_{\beta }\beta$, and $Q_{\alpha \beta }$ is the Floer intersection pairing between $H{F}_{\ast }(Y)$ and $H{F}_{\ast }(-Y)$.

Synthesis. The foundational reason relative Donaldson invariants exist is that cylindrical-end compactness converts boundary behavior of ASD connections into Floer generators. This is exactly what makes the boundary three-manifold carry a state space. The central insight identifies neck-stretching limits with tensor contractions over $H{F}_{\ast }(Y)$, while gluing is dual to composing cobordism maps. Putting these together, relative Donaldson theory supplies the bridge from closed four-manifold polynomials to a functorial gauge-theoretic field theory.

Full proof set Master

Proposition 1 (a relative chain is a cycle). Under the compactness, transversality, and orientation assumptions above, $\partial {\Phi }_X=0$.

Proof. Let ${\overline{\mathcal{M}}}_X^1(\alpha )$ be a compactified one-dimensional relative moduli space. Its boundary points are configurations consisting of a relative ASD connection limiting to some $\beta$, followed by a zero-dimensional Floer trajectory from $\beta$ to $\alpha$. The signed boundary count of ${\overline{\mathcal{M}}}_X^1(\alpha )$ is zero. Summing these signed boundary identities over $\alpha$ gives exactly the coefficient equations for $\partial {\Phi }_X=0$. $\square$

Proposition 2 (homology-class invariance under changing representatives). If two relative chains differ by a Floer boundary, then they define the same class in $H{F}_{\ast }(Y)$.

Proof. Suppose ${\Phi }_X^{\prime }-{\Phi }_X=\partial K$. Since ${\Phi }_X$ and ${\Phi }_X^{\prime }$ are cycles, both lie in $\ker \partial$. Their difference lies in $\mathrm{im}\partial$. The quotient defining Floer homology identifies cycles that differ by a boundary, so $[{\Phi }_X^{\prime }]=[{\Phi }_X]$. $\square$

Proposition 3 (pairing formula in a basis). If $\{e_i\}$ is a basis for $H{F}_{\ast }(Y)$ and $\{e_i^{\vee }\}$ is the dual basis for $H{F}_{\ast }(-Y)$, then $$ \left\langle \sum_i a_ie_i,\sum_j b_je_j^\vee\right\rangle=\sum_i a_ib_i. $$

Proof. Bilinearity of the pairing gives $$ \left\langle \sum_i a_ie_i,\sum_j b_je_j^\vee\right\rangle =\sum_{i,j}a_ib_j\langle e_i,e_j^\vee\rangle. $$ Duality gives $⟨e_i,e_j^{\vee }⟩={\delta }_{ij}$. Therefore only the diagonal terms remain, and the result is ${\sum }_i{a}_i{b}_i$. $\square$

Connections Master

• Instanton Floer homology 03.07.23. Relative Donaldson invariants use $H{F}_{\ast }(Y)$ as the state space assigned to the boundary of a four-manifold.

• Uhlenbeck compactness and gluing 03.07.20, 03.07.21. Compactness proves relative chains are cycles, and gluing proves the neck-stretching formula for closed Donaldson invariants.

• Orientations 03.07.22. The integer-valued relative counts depend on determinant-line orientations compatible with both relative moduli spaces and Floer trajectory moduli.

• Donaldson-Floer surgery triangle 03.07.25. Surgery exact sequences are built from cobordism maps between Floer groups, the same relative-invariant machinery in a local surgery setting.

Historical & philosophical context Master

Donaldson's polynomial invariants for closed smooth four-manifolds were formulated by integrating cohomology classes over instanton moduli spaces ^{[Donaldson 1990]}. Floer's instanton homology supplied the boundary state spaces needed to cut those closed invariants along three-manifolds.

Donaldson's Floer tract presents the relative theory and gluing formula as the bridge from three-dimensional Floer groups back to four-dimensional Donaldson invariants ^{[Donaldson 2002]}. This viewpoint made instanton Floer homology a TQFT-like extension of Donaldson theory rather than only a standalone three-manifold invariant.

Bibliography Master

@article{Donaldson1990Polynomial,
  author = {Donaldson, Simon K.},
  title = {Polynomial invariants for smooth four-manifolds},
  journal = {Topology},
  volume = {29},
  pages = {257--315},
  year = {1990}
}

@book{Donaldson2002Relative,
  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{DonaldsonKronheimer1990Relative,
  author = {Donaldson, Simon K. and Kronheimer, Peter B.},
  title = {The Geometry of Four-Manifolds},
  publisher = {Oxford University Press},
  year = {1990}
}

@article{Floer1988Relative,
  author = {Floer, Andreas},
  title = {An instanton-invariant for 3-manifolds},
  journal = {Communications in Mathematical Physics},
  volume = {118},
  pages = {215--240},
  year = {1988}
}