Shadows of 4-manifolds with complexity zero and polyhedral collapsing
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Abstract:
Our purpose is to classify acyclic 4-manifolds having shadow complexity zero. In this paper, we focus on simple polyhedra and discuss this problem combinatorially. We consider a shadowed polyhedron $X$ and a simple polyhedron $X_0$ that is obtained by collapsing from $X$. Then we prove that there exists a canonical way to equip internal regions of $X_0$ with gleams so that two 4-manifolds reconstructed from $X_0$ and $X$ are diffeomorphic. We also show that any acyclic simple polyhedron whose singular set is a union of circles can collapse onto a disk. As a consequence of these results, we prove that any acyclic 4-manifold having shadow complexity zero with boundary is diffeomorphic to a $4$-ball.References
- A. Carrega and B. Martelli, Shadows, ribbon surfaces, and quantum invariants, to appear in Quantum Topology, arXiv:1404.5983.
- Francesco Costantino, Complexity of 4-manifolds, Experiment. Math. 15 (2006), no. 2, 237–249. MR 2253547
- Francesco Costantino, Stein domains and branched shadows of 4-manifolds, Geom. Dedicata 121 (2006), 89–111. MR 2276237, DOI 10.1007/s10711-006-9092-x
- Francesco Costantino, Branched shadows and complex structures on 4-manifolds, J. Knot Theory Ramifications 17 (2008), no. 11, 1429–1454. MR 2469211, DOI 10.1142/S0218216508006683
- Francesco Costantino and Dylan Thurston, 3-manifolds efficiently bound 4-manifolds, J. Topol. 1 (2008), no. 3, 703–745. MR 2417451, DOI 10.1112/jtopol/jtn017
- A. Hatcher, Pants decompositions of surfaces, arXiv:9906084.
- Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematics Studies, No. 80, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0415630
- Hiroshi Ikeda, Acyclic fake surfaces, Topology 10 (1971), 9–36. MR 326735, DOI 10.1016/0040-9383(71)90013-9
- M. Ishikawa and Y. Koda, Stable maps and branched shadows of 3-manifolds, to appear in Math. Ann., arXiv:1403.0596.
- Bruno Martelli, Four-manifolds with shadow-complexity zero, Int. Math. Res. Not. IMRN 6 (2011), 1268–1351. MR 2806506, DOI 10.1093/imrn/rnq110
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744
- Osamu Saeki, Topology of singular fibers of differentiable maps, Lecture Notes in Mathematics, vol. 1854, Springer-Verlag, Berlin, 2004. MR 2106689, DOI 10.1007/b100393
- V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
Additional Information
- Hironobu Naoe
- Affiliation: Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
- Email: hironobu.naoe.p5@dc.tohoku.ac.jp
- Received by editor(s): May 30, 2016
- Received by editor(s) in revised form: September 27, 2016, and November 16, 2016
- Published electronically: June 22, 2017
- Communicated by: Ken Ono
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 4561-4572
- MSC (2010): Primary 57N13, 57M20; Secondary 57R65
- DOI: https://doi.org/10.1090/proc/13595
- MathSciNet review: 3690638