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Equivariant Instanton Homology EQUIVARIANT INSTANTON HOMOLOGY S. MICHAEL MILLER Abstract. We define four versions of equivariant instanton Floer homology (I`;I´;I8 and I) for a class of 3-manifolds and SOp3q-bundles over them including all rational homology spheres. These versions are analogous to the four flavors of monopoler and Heegaard Floer homology theories. This con- struction is functorial for a large class of 4-manifold cobordisms, and agrees with Donaldson's definition of equivariant instanton homology for integer ho- mology spheres. Furthermore, one of our invariants is isomorphic to Floer's instanton homology for admissible bundles, and we calculate I8 in all cases it is defined, away from characteristic 2. The appendix, possibly of independent interest, defines an algebraic con- struction of three equivariant homology theories for dg-modules over a dg- algebra, the equivariant homology H`pA; Mq, the coBorel homology H´pA; Mq, and the Tate homology H8pA; Mq. The constructions of the appendix are used to define our invariants. arXiv:1907.01091v1 [math.GT] 1 Jul 2019 1 2 S. MICHAEL MILLER Contents Introduction 3 Survey of the homology theories 8 Organization 11 1. Configuration spaces and their reducibles 13 1.1. Framed connections and the framed configuration space 14 1.2. The equivalent Up2q model 15 1.3. Reducibles on 3-manifolds 16 1.4. Configurations on the cylinder 20 1.5. Configurations on a cobordism 23 2. Analysis of configuration spaces 26 2.1. Tangent spaces 26 2.2. Hilbert manifold completions 27 2.3. The 4-dimensional case 29 3. Critical points and perturbations 33 3.1. Holonomy perturbations 33 3.2. Linear analysis 37 3.3. Reducible critical points 38 3.4. Transversality for critical points 40 3.5. Signature data of reducible critical points 42 4. Moduli spaces of instantons 46 4.1. Moduli spaces for cylinders and cobordisms 46 4.2. Linear analysis and index theory 52 4.3. Uhlenbeck compactness for framed instantons 56 4.4. Reducible instantons on the cylinder and cobordisms 65 4.5. Index calculations 72 4.6. Transversality for the cylinder and cobordisms 82 4.7. Gluing 87 4.8. Families of metrics and perturbations 102 5. Orientations 111 5.1. Orientability and determinant line bundles 111 5.2. A gluing operation for determinant lines 115 5.3. The gluing diffeomorphism is orientation-preserving 118 5.4. Canonical orientations for determinant lines 124 6. Floer homology 130 6.1. Geometric homology 130 6.2. Equivariant instanton homology 139 6.3. The index filtration 154 6.4. Four flavors of instanton homology 157 7. Examples, calculations, and comparisons 162 7.1. Equivariant instanton homology for admissible bundles 162 7.2. Comparison with Donaldson's theory 163 7.3. Instanton Tate homology 180 7.4. Examples of the I‚ and the index spectral sequence 183 7.5. Orientation reversal and equivariant cohomology 189 Appendix 192 A. Equivariant homology of dg-modules 192 A.1. Bar constructions 192 EQUIVARIANT INSTANTON HOMOLOGY 3 A.2. Invariance 195 A.3. The dualizing complex and Tate homology 200 A.4. Periodicity in Tate homology 204 A.5. Simplifying the twisted Borel homology 206 A.6. Spectral sequences 209 A.7. Group algebras 210 A.8. Periodic homological algebra 216 References 219 Introduction In [Flo88], Andreas Floer introduced the instanton homology groups IpY q, Z{8- graded abelian groups associated to integer homology 3-spheres. These form a sort of TQFT in which oriented cobordisms W : Y0 Ñ Y1 induce homomorphisms on the corresponding instanton homology groups. Since then, similar TQFT-style invariants have found themselves a powerful tool in 3- and 4-dimensional topology, especially the related monopole Floer homology of [KM07] and the Heegaard Floer homology of [OS04]. In ideal circumstances, the instanton homology groups are defined by a chain complex generated by irreducible flat SUp2q connections up to isomorphism (equiv- alently, representations π1pY q Ñ SUp2q whose image is non-abelian, modulo con- jugacy by elements of SUp2q); the component of the differential between two flat connections α´; α` is given by an algebraic count of solutions on the cylinder RˆY to the ASD equation ` FA “ 0; where A is a connection on the trivial SUp2q-bundle over the cylinder which is asymptotically equal to the α˘. We can think of this as the \Morse chain complex" of the Chern-Simons functional on the space of irreducible connections modulo ˚ ˚ gauge equivalence, BY “ AY {G, whose critical points are the flat connections and gradient flow equation is (formally) the ASD equation. While these equations depend on a choice of metric on the 3-manifold Y , the homology groups are an invariant of the Y itself. Floer's theory is constrained to homology 3-spheres because of the presence of reducible connections. While the instanton chain complex above is still a chain com- plex for rational homology spheres, the proof that its homology is independent of the choice of metric fails in the presence of Up1q-reducible connections (correspond- ing to representations π1pY q Ñ SUp2q with image lying inside a circle subgroup). One would need to take these reducible connections into account in the definition of the chain complex, but this cannot be done naively: while the instanton chain complex is a Morse complex for the space of irreducible connections, which is an infinite-dimensional manifold, the gauge group G does not act freely on the en- tire space of connections, and so the configuration space of all connections modulo gauge, BY , is not a manifold. Austin and Braam in [AB96] resolve this difficulty for a class of 3-manifolds (including all rational homology spheres) by defining an invariant called the equi- G variant instanton homology of Y , a Z{8-graded R-vector space I˚ pY q with an action 4 S. MICHAEL MILLER of RrUs “ H˚pBSOp3q; Rq, to be a form of SOp3q-equivariant Morse theory on an infinite-dimensional SOp3q-manifold BY with B SO 3 B : Y { r p q “ Y The manifold B might be called the configuration space of framed connections on Y r (the trivial SOp3q-bundle over) Y . Their invariant is defined using the equivariant de Rham complexr as a model form the equivariant (co)homology of a smooth G- manifold, and thus inherently uses real coefficients. Floer also defined, in [Flo91], instanton homology groups for SOp3q-bundles E 2 over 3-manifolds Y satisfying the admissibility criterion that w2pEq P H pY ; Z{2q 2 lifts to a non-torsion class in H pY ; Zq; in particular, b1pY q ¡ 0. In this case, there are no reducible connections, and the homology of the Floer complex is a well-defined invariant of the pair pY; Eq. This case is important for his work on surgery triangles in instanton homology. Using this, Kronheimer and Mrowka introduce framed instanton homology groups I#pY; Eq for an arbitrary SOp3q-bundle over a 3-manifold in [KM11b] by studying the instanton homology of pY #T 3;E#Qq for a certain admissible bundle Q over T 3. This is meant to be a version of the (non-equivariant) Morse homology of the space of framed connections BE. In this paper, we jointly generalize Floer's work on admissible bundles and Austin-Braam's work for rationalr homology spheres; to speak of both in the same breath, we say that an SOp3q-bundle E over a 3-manifold Y is weakly admissible if 2 either w2pEq has no lifts to a torsion class in H pY ; Zq, or if b1pY q “ 0. We take an alternate approach to Kronheimer and Mrowka's to the framed instanton homology groups: instead of taking a connected sum with T 3, we work on the space of framed connections BE itself. We do this with a sort of Morse-Bott complex for a smooth G-manifold equipped with an equivariant Morse function; our definition is partly inspired by ther Morse-Bott complex introduced for monopole Floer homology in [Lin18]. This uses Lipyanskiy's notion of the geometric chain gm complex C˚ pX; Rq of a smooth manifold X, introduced in [Lip14], whose homology gives the usual singular homology of X. While there are technical obstructions to carrying this out for all 3-manifolds, this has the advantage of providing more structure: for pY; Eq a weakly admissible bundle and R a commutative ground ring, we can define a Z{8-graded chain complex of R-modules, CIpY; E; π; Rq, which carries the action of the differential graded algebra C˚pSOp3q; Rq. (This is what we find to be the cleanest notion of a chain complex with anĂ action of the Lie group SOp3q.) This chain complex depends on further data π, including a metric on the 3-manifold itself and a perturbation of the functional defining the Morse complex, but this turns out to be mostly inessential: associated to a perturbation π on a pair pY; Eq is an element of a finite set σpY; Eq of signature data; this will be defined in section 3.5. For concreteness, we remark that if Y is a rational homology sphere whose universal abelian cover Y~ has H1pY~ ; Cq “ 0, then σpY; Eq consists of a single element for all E. The set σpY; Eq corresponds precisely to the \natural classes of perturbations" stated in the main theorem of [AB96]. In fact, the TQFT structure of the usual instanton Floer homology groups can be lifted to the level of the homology groups of CI. This is the main theorem of this paper. Ă EQUIVARIANT INSTANTON HOMOLOGY 5 Up2q;w Theorem 1. There is a category Cob3;b of based 3-manifolds pY; E; σ; bq equipped with weakly admissible Up2q-bundles and signature data, whose morphisms are cer- tain `weakly admissible' oriented cobordisms pW; Eq equipped with a path between the basepoints on the ends.
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