{
  "id": "2405.07928",
  "version": "v1",
  "published": "2024-05-13T16:58:58.000Z",
  "updated": "2024-05-13T16:58:58.000Z",
  "title": "The Casson-Sullivan invariant for homeomorphisms of 4-manifolds",
  "authors": [
    "Daniel A. P. Galvin"
  ],
  "comment": "40 pages, 1 figure. Comments welcome!",
  "categories": [
    "math.GT"
  ],
  "abstract": "We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\\mathbb{Z}/2$-coefficients. We prove that for all orientable pairs of homeomorphic, smooth $4$-manifolds this invariant can be realised fully after stabilising with a single $S^2\\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected $4$-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on $4$-manifolds which are diffeomorphic but not stably pseudo-isotopic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T16:58:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K40",
      "57R10",
      "57K10",
      "57N37",
      "57R50",
      "57R65"
    ],
    "keywords": [
      "casson-sullivan invariant",
      "homeomorphism",
      "stably pseudo-isotopic",
      "finite cyclic fundamental group",
      "sufficient external stabilisations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}