{
  "id": "2210.14071",
  "version": "v1",
  "published": "2022-10-25T14:55:13.000Z",
  "updated": "2022-10-25T14:55:13.000Z",
  "title": "Instantons and rational homology spheres",
  "authors": [
    "Aliakbar Daemi",
    "Mike Miller Eismeier"
  ],
  "comment": "Preliminary version. Comments welcome!",
  "categories": [
    "math.GT"
  ],
  "abstract": "In previous work, the second author defined 'equivariant instanton homology groups' $I^\\bullet(Y,\\pi;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $\\pi$, and a PID $R$. These objects are modules over the cohomology ring $H^{-*}(BSO_3;R)$. We prove that the equivariant instanton homology groups $I^\\bullet(Y;R)$ are independent of the auxiliary data $\\pi$, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components. For any rational homology sphere $Y$, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres $I_*(Y, \\pi; R)$ which now depends on the auxiliary data $\\pi$, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise \"wall-crossing formula'' for $I_*(Y, \\pi; R)$ as the auxiliary data $\\pi$ moves between adjacent chambers. We use this to define an instanton invariant $\\lambda_I(Y) \\in \\Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant. Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism $W: Y \\to Y'$, which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of $Y$ and $Y'$. However, it does define a chain map between the instanton chain complex of $Y$ and a sort of suspension of the instanton chain complex of $Y'$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-25T14:55:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57R58"
    ],
    "keywords": [
      "rational homology sphere",
      "equivariant instanton homology groups",
      "instanton chain complex",
      "auxiliary data",
      "defined equivariant instanton homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}