\(G\)-corks and Heegaard Floer homology. (English) Zbl 1508.57026
Summary: In [D. Auckly et al., Algebr. Geom. Topol. 17, No. 3, 1771–1783 (2017; Zbl 1382.57010)], Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup \(G\) of \(S O(4)\) there exists a contractible smooth 4-manifold with an effective \(G\)-action on its boundary so that the twists associated to the non-trivial elements of \(G\) don’t extend to diffeomorphisms of the entire manifold. We give a different proof of this phenomenon using the Heegaard Floer techniques in [S. Akbulut and Ç. Karakurt, in: Proceedings of the 18th Gökova geometry-topology conference, Gökova, Turkey, May 30–June 4, 2011. Somerville, MA: International Press; Gökova: Gökova Geometry-Topology Conferences. 42–52 (2012; Zbl 1360.58008)].
MSC:
| 57K40 | General topology of 4-manifolds |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 57R55 | Differentiable structures in differential topology |
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
| 57R58 | Floer homology |
References:
| [1] | Akbulut, S., A fake compact contractible \(4\)-manifold, J. Differ. Geom.33(2) (1991) 335-356. · Zbl 0839.57015 |
| [2] | Akbulut, S. and Karakurt, Ç., Action of the cork twist on Floer homology, in Proc. Gökova Geometry-Topology Conf. 2011 (International Press, Somerville, MA, 2012), pp. 42-52. · Zbl 1360.58008 |
| [3] | Akbulut, S. and Yasui, K., Corks, plugs and exotic structures, J. Gökova Geom. Topol. GGT2 (2008) 40-82. · Zbl 1214.57027 |
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| [5] | Auckly, D., Kim, H. J., Melvin, P. and Ruberman, D., Equivariant corks, Algebr. Geom. Topol.17(3) (2017) 1771-1783. · Zbl 1382.57010 |
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| [12] | K. Yasui, Corks, exotic 4-manifolds, and knot concordance, preprint (2017) arXiv:1505.02551v4. |
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