A note on exotic families of 4-manifolds

Kato, Tsuyoshi; Konno, Hokuto; Nakamura, Nobuhiro

doi:10.1090/proc/16356

A note on exotic families of 4-manifolds
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by Tsuyoshi Kato, Hokuto Konno and Nobuhiro Nakamura;

Proc. Amer. Math. Soc. 151 (2023), 2695-2705

DOI: https://doi.org/10.1090/proc/16356

Published electronically: March 14, 2023

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Abstract:

We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.

References

Dave Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz, Isotopy of surfaces in 4-manifolds after a single stabilization, Adv. Math. 341 (2019), 609–615. MR 3873547, DOI 10.1016/j.aim.2018.10.040
Steven Boyer, Simply-connected $4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no. 1, 331–357. MR 857447, DOI 10.1090/S0002-9947-1986-0857447-6
Martin Čadek and Jiří Vanžura, On the classification of oriented vector bundles over $5$-complexes, Czechoslovak Math. J. 43(118) (1993), no. 4, 753–764. MR 1258434, DOI 10.21136/CMJ.1993.128427
Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
Jeffrey Giansiracusa, The stable mapping class group of simply connected 4-manifolds, J. Reine Angew. Math. 617 (2008), 215–235. MR 2400996, DOI 10.1515/CRELLE.2008.031
Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura, Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds, Compos. Math. 157 (2021), no. 4, 770–808. MR 4247572, DOI 10.1112/s0010437x2000771x
M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663. MR 561244
Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 645390, DOI 10.1515/9781400881505
J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999. MR 1702278
S. P. Novikov, On manifolds with free abelian fundamental group and their application, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 207–246 (Russian). MR 196765
Peter Ozsváth and Zoltán Szabó, The symplectic Thom conjecture, Ann. of Math. (2) 151 (2000), no. 1, 93–124. MR 1745017, DOI 10.2307/121113
Peter Ozsváth and Zoltán Szabó, Higher type adjunction inequalities in Seiberg-Witten theory, J. Differential Geom. 55 (2000), no. 3, 385–440. MR 1863729
B. Perron, Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique, Topology 25 (1986), no. 4, 381–397 (French). MR 862426, DOI 10.1016/0040-9383(86)90018-2
Frank Quinn, Isotopy of $4$-manifolds, J. Differential Geom. 24 (1986), no. 3, 343–372. MR 868975
Frank Quinn, A controlled-topology proof of the product structure theorem, Geom. Dedicata 148 (2010), 303–308. MR 2721629, DOI 10.1007/s10711-009-9406-x
Andrew Ranicki, Algebraic and geometric surgery, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2061749, DOI 10.1093/acprof:oso/9780198509240.001.0001
Daniel Ruberman, An obstruction to smooth isotopy in dimension $4$, Math. Res. Lett. 5 (1998), no. 6, 743–758. MR 1671187, DOI 10.4310/MRL.1998.v5.n6.a5
Daniel Ruberman, Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants, Geom. Topol. 5 (2001), 895–924. MR 1874146, DOI 10.2140/gt.2001.5.895
Yuli Rudyak, Piecewise linear structures on topological manifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. MR 3467983, DOI 10.1142/9887