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Knot Homology Groups from Instantons

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Knot Homology Groups from Instantons Knot homology groups from instantons P. B. Kronheimer and T. S. Mrowka Harvard University, Cambridge MA 02138 Massachusetts Institute of Technology, Cambridge MA 02139 1 Introduction 1.1 An observation of some coincidences For a knot or link K in S 3, the Khovanov homology Kh.K/ is a bigraded abelian group whose construction can be described in entirely combinatorial terms [15]. If we forget the bigrading, then as abelian groups we have, for example, Kh.unknot/ Z2 D and Kh.trefoil/ Z4 Z=2: D ˚ The second equality holds for both the right- and left-handed trefoils, though the bigrading would distinguish these two cases. The present paper was motivated in large part by the observation that the 4 group Z Z=2 arises in a different context. Pick a basepoint y0 in the com- ˚ plement of the knot or link, and consider the space of all homomorphisms 3 1 S K; y0 SU.2/ W n ! satisfying the additional constraint that  i 0à .m/ is conjugate to (1) 0 i for every m in the conjugacy class of a meridian of the link. (There is one such conjugacy class for each component of K, once the components are oriented. The orientation does not matter here, because the above element of SU.2/ is conjugate to its inverse.) Let us write 3 R.K/ Hom 1 S K; y0 ; SU.2/ n 2 for the set of these homomorphisms. Note that we are not defining R.K/ as a set of equivalence classes of such homomorphisms under the action of conju- gation by SU.2/. The observation, then, is the following: Observation 1.1. In the case that K is either the unknot or the trefoil, the Kho- vanov homology of K is isomorphic to the ordinary homology of R.K/, as an abelian group. That is, Kh.K/ H .R.K//: Š This observation extends to all the torus knots of type .2; p/. To understand this observation, we can begin with the case of the unknot, where the fundamental group of the complement is Z. After choosing a gen- erator, we have a correspondence between R.unknot/ and the conjugacy class of the distinguished element of SU.2/ in (1) above. This conjugacy class is a 2-sphere in SU.2/, so we can write R.unknot/ S 2: D For a non-trivial knot K, we always have homomorphisms which factor 3 through the abelianization H1.S K/ Z, and these are again parametrized n D by S 2. Every other homomorphism has stabilizer 1 SU.2/ under f˙ g the action by conjugation, so its equivalence class contributes a copy of SU.2/= 1 RP3 to R.K/. In the case of the trefoil, for example, there f˙ g D is exactly one such conjugacy class, and so R.trefoil/ S 2 RP3 D q I and the homology of this space is indeed Z4 Z=2, just like the Khovanov ˚ homology. This explains why the observation holds for the trefoil, and the case of the .2; p/ torus knots is much the same: for larger odd p, there are .p 1/=2 copies of RP3 in the R.K/. In unpublished work, the above observation has been shown to extend to all 2-bridge knots by Sam Lewallen [25]. The homology of the space R.K/, while it is certainly an invariant of the knot or link, should not be expected to behave well or share any of the more in- teresting properties of Khovanov homology; no should the coincidence noted above be expected to hold. A better way to proceed is instead to imitate the construction of Floer’s instanton homology for 3-manifolds, by constructing a framework in which R.K/ appears as the set of critical points of a Chern- Simons functional on a space of SU.2/ connections on the complement of the link. One should then construct the Morse homology of this Chern-Simons 3 invariant. In this way, one should associate a finitely-generated abelian group to K that would coincide with the ordinary homology of R.K/ in the very sim- plest cases. The main purpose of the present paper is to carry through this construction. The invariant that comes out of this construction is certainly not isomorphic to Khovanov homology for all knots; but it does share some of its formal properties. The definition that we propose is a variant of the orbifold Floer homology considered by Collin and Steer in [5]. In some generality, given a knot or link K in a 3-manifold Y , we will produce an “instanton Floer homology group” that is an invariant of .Y; K/. These groups will be functorial for oriented cobordisms of pairs. Rather than work only with SU.2/, we will work of much of this paper with a more general compact Lie group G, though in the end it is only for the case of SU.N / that we are able to construct these invariants. 1.2 Summary of results The basic construction. Let Y be a closed oriented 3-manifold, let K Y be an oriented link, and let P Y be a principal U.2/-bundle. Let K1;:::;Kr be ! the components of K. We will say that .Y; K/ and P satisfies the non-integrality condition if none of the 2r rational cohomology classes 1 1 1 c1.P / P:D:ŒK1 P:D:ŒKr (2) 2 ˙ 4 ˙ ˙ 4 is an integer class. When the non-integrality condition holds, we will define a finitely-generated abelian group I .Y; K; P /. This group has a canonical Z=2 grading, and a relative grading byZ=4. In the case that K is empty, the group I .Y; P / coincides with the familiar variant of Floer’s instanton homology arising from a U.2/ bundle P Y with ! odd first Chern class [8]. We recall, in outline, how this group is constructed. One considers the space C.Y; P / of all connections in the SO.3/ bundle ad.P /. This affine space is acted on by the “determinant-1 gauge group”: the group G .Y; P / of automorphisms of P that have determinant 1 everywhere. Inside C.Y; P / one has the flat connections: these can be characterized as critical points of the Chern-Simons functional, CS C.Y; P / R: W ! The Chern-Simons functional descends to a circle-valued function on the quo- tient space B.Y; P / C.Y; P /=G .Y; P /: D 4 The image of the set of critical points in B.Y; P / is compact, and after per- turbing CS carefully by a term that is invariant under G .Y; P /, one obtains a function whose set of critical points has finite image in this quotient. If .1=2/c1.P / is not an integral class and the perturbation is small, then the crit- ical points in C.Y; P / are all irreducible connections. One can arrange also a Morse-type non-degeneracy condition: the Hessian if CS can be assumed to be non-degenerate in the directions normal to the gauge orbits. The group I .Y; P / is then constructed as the Morse homology of the circle-valued Morse function on B.Y; P /. In the case that K is non-empty, the construction of I .Y; K; P / mimics the standard construction very closely. The difference is that we start not with the space C of all smooth connections in ad.P /, but with a space C.Y; K; P / of connections in the restriction of ad.P / to Y K which have a singularity along n K. This space is acted on by a group G .Y; K; P / of determinant-one gauge transformations, and we have a quotient space B.Y; K; P / C.Y; K; P /=G .Y; K; P /: D In the case that c1.P / 0, the singularity is such that the flat connections in D the quotient space B.Y; K; P / correspond to conjugacy classes of homomor- phisms from the fundamental group of Y K to SU.2/ which have the behavior n (1) for meridians of the link. Thus, if we write C.Y; K; P / B.Y; K; P / for this set of critical points of the Chern-Simons functional, then we have C.Y; K; P / R.Y; K/= SU.2/ (3) D where R.Y; K/ is the set of homomorphisms 1.Y K/ SU.2/ satisfying W n ! (1) and SU.2/ is acting by conjugation. The non-integrality of the classes (2) is required in order to ensure that there will be no reducible flat connections. Application to classical knots. Because of the non-integrality requirement, the construction of I cannot be applied directly when the 3-manifold Y has first Betti number zero. In particular, we cannot apply this construction to “classi- cal knots” (knots in S 3). However, there is a simple device we can apply. Pick 3 a point y0 in Y K, and form the connected sum at y0 of Y and T , to obtain n 3 a new pair .Y #T ;K/. Let P0 be the trivial U.2/ bundle on Y , and let Q be the U.2/ bundle on T 3 S 1 T 2 whose first Chern class is Poincare´ dual to 1 D 3 S point . We can form a bundle P0#Q over Y #T . This bundle satisfies f g the non-integral condition, so we define 3 FI .Y; K/ I .Y #T ; K; P0#Q/: D 5 and call this the framed instanton homology of the pair .Y; K/.
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