Shadows of 4-manifolds with complexity zero and polyhedral collapsing

Naoe, Hironobu

doi:10.1090/proc/13595

Shadows of 4-manifolds with complexity zero and polyhedral collapsing
HTML articles powered by AMS MathViewer

by Hironobu Naoe PDF

Proc. Amer. Math. Soc. 145 (2017), 4561-4572 Request permission

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