Lecture notes on ECH 4: Three-dimensional Seiberg-Witten theory


Although the Seiberg-Witten equations that we reviewed previously are defined on a four-manifold, one can also use them to define invariants of three-manifolds. There are two basic ways one can do this.

First we need the following definition. Let Y be a closed oriented connected three-manifold. A spin-c structure on Y is a lift of the frame bundle to Spin^c(3)=Spin(3)\times_{{\mathbb Z}/2}S^1. Equivalently, a spin-c structure on Y is a rank 2 Hermitian vector bundle S\to Y with a Clifford action cl:TX\otimes S\to S satisfying the usual Clifford relation from before, plus an additional orientation condition that cl(e_1)cl(e_2)cl(e_3)=1 whenever (e_1,e_2,e_3) is an oriented orthonormal basis for T_xY. The set of isomorphism classes of spin-c structures on Y is an affine space over H^2(Y;{\mathbb Z}). A spin-c structure on Y is equivalent to an S^1-invariant spin-c structure on S^1\times Y. (The above facts are not immediately obvious, but it would be beyond the scope of this course to explain all of this.) The spin-c structure is called “torsion” if c_1(S) (which we also denote by c_1({\mathfrak s})) is a torsion class in H^2(Y;{\mathbb Z}).

Now the first thing one can do to define invariants of Y is to consider the Seiberg-Witten invariants of the four-manifold S^1\times Y. (Note that S^1\times Y has a canonical homology orientation, so there is no sign ambiguity in the definition.) If we restrict attention to S^1-invariant spin-c structures, then (for appropriate S^1-invariant perturbations) these invariants turn out to count S^1-invariant solutions to the Seiberg-Witten equations, which can alternately be described as solutions to a three-dimensional version of the Seiberg-Witten equations on Y. These invariants are the “Seiberg-Witten invariants” of Y. Since b_2^+(S^1\times Y)=b_1(Y), these invariants are well-defined when b_1(Y)>0, up to a choice of “chamber” when b_1(Y)=1. There is also a distinguished “zero” chamber to use when b_1(Y)=1 and {\mathfrak s} is not torsion.  In the previous installment, we computed this invariant when Y is a mapping torus (except that when b_1(Y)=1, we used the “symplectic” chamber, which sometimes disagrees with the “zero” chamber.)

In general, however, these invariants turn out to be not very interesting, because it was shown by Meng-Taubes and Turaev that they agree with a kind of Reidemeister torsion of Y.

The second, more interesting thing one can do is to “categorify” the previous invariant by defining a chain complex (over {\mathbb Z}) whose generators are {\mathbb R}-invariant solutions to the Seiberg-Witten equations on {\mathbb R}\times Y, and whose differential counts non-{\mathbb R}-invariant solutions to the Seiberg-Witten equations on {\mathbb R}\times Y which converge to two different {\mathbb R}-invariant solutions as the {\mathbb R}-coordinate converges to \pm\infty. If the spin-c structure {\mathfrak s} is non-torsion, then the homology of this chan complex is a well-defined invariant HM(Y,{\mathfrak s}), called “Seiberg-Witten Floer homology” or “monopole Floer homology”. This is a relatively {\mathbb Z}/d-graded {\mathbb Z}-module, where d denotes the divisibility of c_1({\mathfrak s}) in H^2(Y;{\mathbb Z}) mod torsion (which turns out to always be an even integer). This means that it splits into d summands, and there is a well-defined grading difference in {\mathbb Z}/d between any two of them, which is additive for the pairwise differences between any three summands. There is also a canonical {\mathbb Z}/2-grading, with respect to which the Euler characteristic of the Seiberg-Witten Floer homology HM(Y,{\mathfrak s}) is the Seiberg-Witten invariant SW(Y,{\mathfrak s}).

If {\mathfrak s} is torsion, then there is a difficulty in defining Seiberg-Witten Floer homology caused by reducible solutions to the Seiberg-Witten equations. There are two ways to resolve this difficulty, which lead to two versions of Seiberg-Witten Floer homology, which are denoted by \widehat{HM}(Y,{\mathfrak s}) and \check{HM}(Y,{\mathfrak s}). These are relatively {\mathbb Z}-graded; the former is zero in sufficiently positive grading, and the latter is zero in sufficiently negative grading. They fit into an exact triangle

\overline{HM}_*(Y,{\mathfrak s})\to \check{HM}_*(Y,{\mathfrak s})\to\widehat{HM}_*(Y,{\mathfrak s}) \to \overline{HM}_{*-1}(Y,{\mathfrak s}))\to\cdots

where \overline{HM}(Y,\frak{s}) is a third invariant which is computable in terms of the triple cup product on Y. In particular, Kronheimer-Mrowka showed that \overline{HM}(Y,\frak{s}) is nonzero (at least in half of the gradings) and \overline{HM}_*(Y,\frak{s})=\overline{HM}_{*+2}(Y,\frak{s}). In conjunction with the above exact triangle, this implies that \widehat{HM}_* and \check{HM}_* are likewise 2-periodic and nontrivial when the grading is sufficiently negative/positive. This fact is the key to the proof of the Weinstein conjecture and refinements as we will see later.

If {\mathfrak s} is not torsion, then both \widehat{HM}(Y,{\mathfrak s}) and \check{HM}(Y,{\mathfrak s}) are equal to the invariant HM(Y,{\mathfrak s}) discussed previously.

For any spin-c structure {\mathfrak s}, each version of Seiberg-Witten Floer homology is finitely generated in each grading (which I probably should have said before when I talked about Euler characteristic).

For the detailed construction of Seiberg-Witten Floer theory and the proofs of all the above facts and much more, please see the book by Kronheimer-Mrowka.

Towards ECH

The original motivation for defining ECH was to find an analogue of Taubes’s SW=Gr theorem for a three-manifold. That is, we would like to identify Seiberg-Witten Floer homology with an appropriate analogue of Taubes’s Gromov invariant for a three-manifold Y. The latter should be the homology of a chain complex which is generated by {\mathbb R}-invariant holomorphic curves in {\mathbb R}\times Y, and whose differential counts non-{\mathbb R}-invariant holomorphic curves in {\mathbb R}\times Y.

For this to make sense, {\mathbb R}\times Y needs to have a symplectic structure. This is the case for example when Y is the mapping torus of a symplectomorphism \phi (the symplectic form we previously defined on S^1\times Y also makes sense on {\mathbb R}\times Y). The analogue of Taubes’s Gromov invariant in this case is the “periodic Floer homology” of \phi (which is defined when all periodic orbits of \phi are nondegenerate); it is the homology of a chain complex which is generated by the unions of periodic orbits we discussed in the previous installment (i.e. no multiply covered hyperbolic orbits allowed), and its differential counts certain holomorphic curves in {\mathbb R}\times Y.

Which holomorphic curves exactly? This is a subtle issue which I will explain in the next installment. Except that I don’t want to discuss mapping tori since not every three-manifold is a mapping torus. Instead we will do an analogous construction for contact three-manifolds, which is more general since every oriented three-manifold admits a contact structure. Finding the appropriate definition of the chain complex is not obvious, but Taubes’s SW=Gr theorem and the computation for mapping tori give us a lot of hints about what to do. See the next installment for the answer.

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