Topological spines of 4–manifolds

Kim, Hee Jung¹ ; Ruberman, Daniel²

¹Department of Mathematics, Western Washington University, Bellingham, WA, United States
²Department of Mathematics, Brandeis University, Waltham, MA, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 7, pp. 3589-3606

Résumé

We show that infinitely many of the simply connected 4–manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.

Détail

DOI : 10.2140/agt.2020.20.3589

Classification : 57N13, 57Q40, 57R67
Keywords: spine, topological manifold, surgery, 4-manifold

Bibliographie

[1] S Akbulut, A note on homology surgery and the Casson–Gordon invariant, Math. Proc. Cambridge Philos. Soc. 85 (1979) 335 | DOI

[2] M Boileau, S Wang, Non-zero degree maps and surface bundles over S1, J. Differential Geom. 43 (1996) 789 | DOI

[3] S Boyer, Simply-connected 4–manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986) 331 | DOI

[4] W Browder, Surgery on simply-connected manifolds, 65, Springer (1972) | DOI

[5] S E Cappell, J L Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. 99 (1974) 277 | DOI

[6] S E Cappell, J Shaneson, Totally spineless manifolds, Illinois J. Math. 21 (1977) 231 | DOI

[7] A J Casson, C M Gordon, On slice knots in dimension three, from: "Algebraic and geometric topology, II" (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 39 | DOI

[8] A J Casson, C M Gordon, Cobordism of classical knots, from: "À la recherche de la topologie perdue" (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser (1986) 181

[9] J C Cha, K H Ko, Signature invariants of covering links, Trans. Amer. Math. Soc. 358 (2006) 3399 | DOI

[10] M Culler, N M Dunfield, M Goerner, J R Weeks, SnapPy, a computer program for studying the topology of 3–manifolds

[11] M H Freedman, F Quinn, Topology of 4–manifolds, 39, Princeton Univ. Press (1990)

[12] S Friedl, P Teichner, New topologically slice knots, Geom. Topol. 9 (2005) 2129 | DOI

[13] S Gadgil, Degree-one maps, surgery and four-manifolds, Bull. Lond. Math. Soc. 39 (2007) 419 | DOI

[14] T G Group, GAP: groups, algorithms, and programming, software (2004)

[15] D L Goldsmith, A linking invariant of classical link concordance, from: "Knot theory" (editor J C Hausmann), Lecture Notes in Math. 685, Springer (1978) 135 | DOI

[16] I Hambleton, L R Taylor, A guide to the calculation of the surgery obstruction groups for finite groups, from: "Surveys on surgery theory, I" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 225 | DOI

[17] K Hayden, L Piccirillo, The trace embedding lemma and spinelessness, preprint (2019)

[18] B Jahren, S Kwasik, Three-dimensional surgery theory, UNil-groups and the Borel conjecture, Topology 42 (2003) 1353 | DOI

[19] R C Kirby, L C Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969) 742 | DOI

[20] R C Kirby, L R Taylor, A survey of 4–manifolds through the eyes of surgery, from: "Surveys on surgery theory, II" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 387 | DOI

[21] A S Levine, T Lidman, Simply connected, spineless 4–manifolds, Forum Math. Sigma 7 (2019) | DOI

[22] J Levine, A characterization of knot polynomials, Topology 4 (1965) 135 | DOI

[23] J Levine, K E Orr, A survey of applications of surgery to knot and link theory, from: "Surveys on surgery theory, I" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 345 | DOI

[24] Y Matsumoto, A 4–manifold which admits no spine, Bull. Amer. Math. Soc. 81 (1975) 467 | DOI

[25] Y Matsumoto, Wild embeddings of piecewise linear manifolds in codimension two, from: "Geometric topology" (editor J C Cantrell), Academic (1979) 393 | DOI

[26] E K Pedersen, Spines of topological manifolds, Comment. Math. Helv. 50 (1975) 41 | DOI

[27] M Powell, Twisted Blanchfield pairings and symmetric chain complexes, Q. J. Math. 67 (2016) 715 | DOI

[28] J H Przytycki, A Yasuhara, Linking numbers in rational homology 3–spheres, cyclic branched covers and infinite cyclic covers, Trans. Amer. Math. Soc. 356 (2004) 3669 | DOI

[29] D Rolfsen, Knots and links, 7, Publish or Perish (1976)

[30] H Seifert, Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen, Abh. Math. Sem. Univ. Hamburg 11 (1935) 84 | DOI

[31] C T C Wall, Surgery on compact manifolds, 1, Academic (1970)

[32] C T C Wall, Classification of Hermitian forms, VI : Group rings, Ann. of Math. 103 (1976) 1 | DOI