Shadows of acyclic 4–manifolds with sphere boundary

Koda, Yuya¹ ; Naoe, Hironobu²

¹Department of Mathematics, Hiroshima University, Hiroshima, Japan
²Department of Mathematics, Chuo University, Tokyo, Japan

Algebraic and Geometric Topology, Tome 20 (2020) no. 7, pp. 3707-3731

Résumé

In terms of Turaev’s shadows, we provide a sufficient condition for a compact, smooth, acyclic 4–manifold with boundary the 3–sphere to be diffeomorphic to the standard 4–ball. As a consequence, we prove that if a compact, smooth, acyclic 4–manifold with boundary the 3–sphere has shadow-complexity at most 2, then it is diffeomorphic to the standard 4–ball.

Détail

DOI : 10.2140/agt.2020.20.3707

Classification : 57N13, 57M20, 57R55, 57R65
Keywords: $4$–manifold, shadow, differentiable structure, handlebody, polyhedron

Bibliographie

[1] U Burri, For a fixed Turaev shadow Jones–Vassiliev invariants depend polynomially on the gleams, Comment. Math. Helv. 72 (1997) 110 | DOI

[2] A Carrega, B Martelli, Shadows, ribbon surfaces, and quantum invariants, Quantum Topol. 8 (2017) 249 | DOI

[3] F Costantino, Shadows and branched shadows of 3 and 4–manifolds, PhD thesis, Scuola Normale Superiore (2004)

[4] F Costantino, A short introduction to shadows of 4–manifolds, Fund. Math. 188 (2005) 271 | DOI

[5] F Costantino, Complexity of 4–manifolds, Experiment. Math. 15 (2006) 237 | DOI

[6] F Costantino, Stein domains and branched shadows of 4–manifolds, Geom. Dedicata 121 (2006) 89 | DOI

[7] F Costantino, Branched shadows and complex structures on 4–manifolds, J. Knot Theory Ramifications 17 (2008) 1429 | DOI

[8] F Costantino, R Frigerio, B Martelli, C Petronio, Triangulations of 3–manifolds, hyperbolic relative handlebodies, and Dehn filling, Comment. Math. Helv. 82 (2007) 903 | DOI

[9] F Costantino, D Thurston, 3–manifolds efficiently bound 4–manifolds, J. Topol. 1 (2008) 703 | DOI

[10] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 | DOI

[11] D Gabai, Foliations and the topology of 3–manifolds, III, J. Differential Geom. 26 (1987) 479 | DOI

[12] H Ikeda, Acyclic fake surfaces, Topology 10 (1971) 9 | DOI

[13] M Ishikawa, Y Koda, Stable maps and branched shadows of 3–manifolds, Math. Ann. 367 (2017) 1819 | DOI

[14] M Ishikawa, H Naoe, Milnor fibration, A’Campo’s divide and Turaev’s shadow, from: "Singularities" (editors M Ishikawa, S Yokura), World Sci. (2020) 95

[15] Y Koda, B Martelli, H Naoe, Four-manifolds with shadow-complexity one, preprint (2018)

[16] B Martelli, Links, two-handles, and four-manifolds, Int. Math. Res. Not. 2005 (2005) 3595 | DOI

[17] B Martelli, Four-manifolds with shadow-complexity zero, Int. Math. Res. Not. 2011 (2011) 1268 | DOI

[18] S Matveev, Algorithmic topology and classification of 3–manifolds, 9, Springer (2003) | DOI

[19] H Naoe, Shadows of 4–manifolds with complexity zero and polyhedral collapsing, Proc. Amer. Math. Soc. 145 (2017) 4561 | DOI

[20] H Naoe, Mazur manifolds and corks with small shadow complexities, Osaka J. Math. 55 (2018) 479

[21] H Naoe, Corks with large shadow-complexity and exotic four-manifolds, Experiment. Math. (2019) | DOI

[22] G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)

[23] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)

[24] G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)

[25] A Shumakovitch, Shadow formula for the Vassiliev invariant of degree two, Topology 36 (1997) 449 | DOI

[26] D P Thurston, The algebra of knotted trivalent graphs and Turaev’s shadow world, from: "Invariants of knots and –manifolds" (editors T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts, V Turaev), Geom. Topol. Monogr. 4, Geom. Topol. Publ. (2002) 337 | DOI

[27] V G Turaev, Shadow links and face models of statistical mechanics, J. Differential Geom. 36 (1992) 35 | DOI

[28] V G Turaev, Quantum invariants of knots and 3–manifolds, 18, de Gruyter (1994) | DOI