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Peer-reviewedCC BYCompositio Mathematica

The Casson–Sullivan invariant for homeomorphisms of 4-manifolds

  • Daniel Galvin1
  1. 1University of Texas at Austin
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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 double struck upper Z divided by 2 Z / 2 ℤ/2 -coefficients. We prove that for all pairs of orientable, homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single upper S squared times upper S squared S 2 × S 2 S2× S2 . 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.

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Cite

BibTeXRIS
BibTeX
@article{galvin2026cassonsullivan,
  title = {{The Casson–Sullivan invariant for homeomorphisms of 4-manifolds}},
  author = {Daniel Galvin},
  journal = {Compositio Mathematica},
  year = {2026},
  month = oct,
  doi = {10.1017/s0010437x26103510},
  url = {https://doi.org/10.1017/s0010437x26103510},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The Casson–Sullivan invariant for homeomorphisms of 4-manifolds
AU  - Daniel Galvin
PY  - 2026
DA  - 2026/10/28
JO  - Compositio Mathematica
DO  - 10.1017/s0010437x26103510
UR  - https://doi.org/10.1017/s0010437x26103510
AB  - 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 double struck upper Z divided by 2 Z / 2 ℤ/2 -coefficients. We prove that for all pairs of orientable, homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single upper S squared times upper S squared S 2 × S 2 S^2× 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.
SN  - 0010-437X
SN  - 1570-5846
ER  -