Equivariant aspects of singular instanton Floer homology Aliakbar Daemi∗ Christopher Scaduto Abstract We associate several invariants to a knot in an integer homology 3-sphere using SUp2q singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern–Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Frøyshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowka’s singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the 4-sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples. ∗The work of AD was supported by NSF Grant DMS-1812033. 1 Contents 1 Introduction4 2 Background on singular SUp2q gauge theory 14 2.1 Singular SUp2q connections......................... 14 2.2 The Chern-Simons functional and flat connections.............. 15 2.3 The flip symmetry............................... 16 2.4 Perturbing the critical set........................... 17 2.5 Gradient trajectories and gradings...................... 18 2.6 Moduli spaces for cobordisms........................ 21 2.7 Reducible connections and negative definite pairs.............. 23 2.8 Counting critical points............................ 27 2.9 Orienting moduli spaces........................... 28 3 Instanton Floer homology groups for knots 31 3.1 An analogue of Floer’s instanton homology for knots............ 31 3.2 The operators δ1 and δ2 ............................ 33 3.3 Holonomy operators and v-maps....................... 35 3.3.1 The case of closed loops....................... 35 3.3.2 The case of a cylinder: the v-map.................. 37 3.3.3 Curves with boundary in cobordisms................ 39 3.4 A framed instanton homology for knots................... 40 4 The algebra of S-complexes 44 4.1 An equivalent formulation of S-complexes.................. 44 4.2 Equivariant homology theories associated to S-complexes......... 45 4.3 Small equivariant complexes and the h-invariant............... 47 4.4 Dual S-complexes.............................. 51 4.5 Tensor products of S-complexes....................... 53 4.6 Local equivalence groups........................... 56 4.7 Nested sequences of ideals.......................... 57 5 Equivariant invariants from singular instanton theory 59 5.1 Equivariant Floer homology groups..................... 59 5.2 Local equivalence and the concordance invariant h ............. 61 6 The connected sum theorem 63 6.1 Topology of the connected sum theorem................... 63 6.2 Moduli spaces on the cobordisms pW; Sq and pW 1;S1q ........... 65 6.3 Proof of the connected sum theorem..................... 66 6.3.1 Definition of the map λpW;Sq ..................... 66 6.3.2 Definition of the map λpW 1;S1q .................... 70 r r 2 6.3.3 Chain homotopies of compositions.................. 71 6.3.4 Naturality of the equivalences.................... 80 7 Local coefficient systems and filtrations 81 7.1 Local coefficients............................... 81 7.2 The Chern-Simons filtration......................... 84 7.3 Enriched S-complexes............................ 85 7.4 Local equivalence for enriched complexes.................. 87 R 7.5 The concordance invariant ΓpY;Kq ...................... 88 8 Connections to Kronheimer and Mrowka’s constructions 90 8.1 Instanton homology for admissible links................... 90 8.2 Computing I6pY; Kq and I#pY; Kq from the framed complex....... 93 8.3 Local coefficients and the concordance invariant s#pKq ........... 96 8.4 Instanton homology for strongly marked webs................ 97 8.5 Connect summing with a theta web...................... 99 8.6 Local coefficients from theta webs...................... 100 8.7 Relations to equivariant homology groups.................. 104 8.8 Remarks on some concordance invariants.................. 107 9 Computations 109 9.1 Passing to the branched cover......................... 109 9.2 Two-bridge knots............................... 111 9.2.1 The irreducible chain complex with local coefficients........ 115 9.2.2 Sasahira’s instanton homology for lens spaces............ 116 9.2.3 Invariants for the right-handed trefoil................ 117 9.3 The p3; 5q torus knot............................. 119 9.4 The p3; 4q torus knot............................. 121 9.5 The irreducible homology of torus knots................... 122 9.6 More vanishing results for hpKq ....................... 123 3 1 Introduction Instanton Floer homology [Flo88] and Heegaard Floer homology [OS04b] provide two powerful invariants of 3-manifolds, each of which have knot-theoretic variations: singular instanton Floer homology [KM11b] and knot Floer homology [OS04a,Ras03]. These knot invariants share many formal properties: they are both functorial with respect to surface cobordisms, they each have skein exact triangles, and it is even conjectured that some versions of the theories agree with one another [KM10b, Conjecture 7.25]. Despite their similarities, each of the two theories have some advantage over the other. On the one hand, singular instanton Floer homology is more directly related to the fundamental group of the knot complement. For example, this Floer homology can be used to show that the knot group of any non-trivial knot admits a non-abelian representation into the Lie group SUp2q [KM04,KM10b]. On the other hand, knot Floer homology currently has a richer algebraic structure which can be used to obtain invariants of closed 3-manifolds obtained by surgery on a knot [OS08, OS11]. Moreover, knot Floer homology is more computable, and in fact has combinatorial descriptions [MOS09, OS19]. A natural question is whether there is a refinement of singular instanton Floer homology that helps bridge the gap between the two theories. An important step in this direction was recently taken by Kronheimer and Mrowka [KM19c]. The main goal of the present paper is to propose a different approach to this question. Like [KM11b, KM11a], we construct invariants of knots in integer homology spheres using singular instantons. However, in contrast to those constructions, we do not avoid reducibles, and instead exploit them to derive equivariant homological invariants. As we explain below, the relevant symmetry group in this setting is S1. The knot invariants in this paper recover various versions of singular instanton Floer homology in the literature, including all of the ones constructed in [KM11a, KM13, KM19c]. Moreover, some of the structures of our invariants do not seem to have any obvious analogues in the context of Heegaard Floer invariants. For instance, a filtration by the Chern-Simons functional and a Floer homology group categorifying the knot signature can be derived from the main construction of the present work. Motivation The basic idea behind the main construction of the present paper is to construct a config- uration space of singular connections with an S1-action. Let K be a knot in an integer homology sphere Y and fix a basepoint on K. Consider the space of connections on the trivial SUp2q-bundle E over Y which are singular along K and such that the holonomy along any meridian of K is asymptotic to a conjugate of i 0 P SUp2q (1.1) 0 ´i „ as the size of meridian goes to zero. (See Section2 for a more precise review of the definition of such singular connections.) A framed singular connection is a singular connection with a 4 trivialization of E at the basepoint of K such that the holonomy of the connection along a meridian of K at the basepoint is asymptotic to (1.1) (rather than just conjugate to it). The space of automorphisms of E acts freely on the space of framed singular connection and we denote the quotient by BpY; Kq. There is an S1-action on BpY; Kq given by changing the framing at the basepoint. An important featurer of this S1-action is that the stabilizersr of elements in BpY; Kq are not all the same. The element ´1 P S1 acts trivially on BpY; Kq. Thus the action factors through S1 – S1{t˘1u which acts freely on a singular framed connection in BpY;r Kq unless the underlying singular connection is S1-reducible, namely,r it respects a decomposition of E into a sum of two (necessarily dual) complex line bundles. Although framedr connections do not appear in the sequel, our constructions are motivated by the above S1-action and the interactions between framed singular connections with different stabilizers. An important source of inspiration for the authors was a similar story for non-singular connections which is developed in [Don02, Frø02, Mil19]. S-complexes associated to knots The fundamental object that we associate to a knot K Ă Y in an integer homology 3-sphere 2 is a chain complex pC˚pY; Kq; dq which is a module over the graded ring Zrχs{pχ q, where χ has degree 1. The ring Zrχs{pχ2q should be thought of as the homology ring of S1 where the ring structure is inducedr byr the multiplication map. In particular, one expects a similar structure arising from the singular chain complex of a topological space with an S1-action. In our set up, singular homology is replaced with Floer homology. In fact, the chain complex C˚pY; Kq we associate to a knot K decomposes as: r C˚pY; Kq “ C˚pY; Kq ` C˚pY; Kqr1s ` Z: (1.2) Here C˚pY; Kq is Z{4-graded,r C˚pY; Kqr1s is the same complex as C˚pY; Kq with the grading shifted up by 1, and Z is in grading 0. The action of χ on C˚pY; Kq maps the first factor by the identity to the second factor, and maps the remaining two factors to zero. We call a chain complex over the graded ring Zrχs{pχ2q of the form (r1.2) an S-complex.