Embedding Seifert manifolds in 𝑆⁴

Donald, Andrew

doi:10.1090/S0002-9947-2014-06174-6

Embedding Seifert manifolds in $S^4$
HTML articles powered by AMS MathViewer

by Andrew Donald PDF

Trans. Amer. Math. Soc. 367 (2015), 559-595 Request permission

Abstract:

Using an obstruction based on Donaldson’s theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in $S^4$. We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in $S^4$ when either the base orbifold is non-orientable or the first Betti number is odd. In addition, we construct some new embeddings and use these, along with the $d$ and $\overline {\mu }$ invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.

References

Christian Bohr and Ronnie Lee, Homology cobordism and classical knot invariants, Comment. Math. Helv. 77 (2002), no. 2, 363–382. MR 1915046, DOI 10.1007/s00014-002-8344-0
R. Budney and B.A. Burton, Embeddings of 3-manifolds in ${S}^4$ from the point of view of the 11-tetrahedron census, arXiv:0810.2346v5 (2012).
J. S. Crisp and J. A. Hillman, Embedding Seifert fibred $3$-manifolds and $\textrm {Sol}^3$-manifolds in $4$-space, Proc. London Math. Soc. (3) 76 (1998), no. 3, 685–710. MR 1620508, DOI 10.1112/S0024611598000379
S. K. Donaldson, The orientation of Yang-Mills moduli spaces and $4$-manifold topology, J. Differential Geom. 26 (1987), no. 3, 397–428. MR 910015, DOI 10.4310/jdg/1214441485
Ronald Fintushel and Ronald Stern, Rational homology cobordisms of spherical space forms, Topology 26 (1987), no. 3, 385–393. MR 899056, DOI 10.1016/0040-9383(87)90008-5
Y. Fukumoto and M. Furuta, Homology 3-spheres bounding acyclic 4-manifolds, Math. Res. Lett. 7 (2000), no. 5-6, 757–766. MR 1809299, DOI 10.4310/MRL.2000.v7.n6.a8
M. Furuta, Monopole equation and the $\frac {11}8$-conjecture, Math. Res. Lett. 8 (2001), no. 3, 279–291. MR 1839478, DOI 10.4310/MRL.2001.v8.n3.a5
Patrick M. Gilmer and Charles Livingston, On embedding $3$-manifolds in $4$-space, Topology 22 (1983), no. 3, 241–252. MR 710099, DOI 10.1016/0040-9383(83)90011-3
Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
Joshua Greene and Stanislav Jabuka, The slice-ribbon conjecture for 3-stranded pretzel knots, Amer. J. Math. 133 (2011), no. 3, 555–580. MR 2808326, DOI 10.1353/ajm.2011.0022
J. A. Hillman, Embedding 3-manifolds with circle actions, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4287–4294. MR 2538589, DOI 10.1090/S0002-9939-09-09985-7
Fujitsugu Hosokawa, On trivial $2$-spheres in $4$-space, Quart. J. Math. Oxford Ser. (2) 19 (1968), 249–256. MR 232379, DOI 10.1093/qmath/19.1.249
Steve J. Kaplan, Constructing framed $4$-manifolds with given almost framed boundaries, Trans. Amer. Math. Soc. 254 (1979), 237–263. MR 539917, DOI 10.1090/S0002-9947-1979-0539917-X
Akio Kawauchi and Sadayoshi Kojima, Algebraic classification of linking pairings on $3$-manifolds, Math. Ann. 253 (1980), no. 1, 29–42. MR 594531, DOI 10.1007/BF01457818
Ana G. Lecuona, On the slice-ribbon conjecture for Montesinos knots, Trans. Amer. Math. Soc. 364 (2012), no. 1, 233–285. MR 2833583, DOI 10.1090/S0002-9947-2011-05385-7
Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007), 429–472. MR 2302495, DOI 10.2140/gt.2007.11.429
Paolo Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007), 2141–2164. MR 2366190, DOI 10.2140/agt.2007.7.2141
Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125–144. MR 585657
Walter D. Neumann and Frank Raymond, Seifert manifolds, plumbing, $\mu$-invariant and orientation reversing maps, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 163–196. MR 518415
Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, DOI 10.1016/S0001-8708(02)00030-0
Peter Ozsváth and Zoltán Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185–224. MR 1988284, DOI 10.2140/gt.2003.7.185
D. Rolfsen, Knots and links, AMS Chelsea, 2003.
Nikolai Saveliev, A surgery formula for the $\overline \mu$-invariant, Topology Appl. 106 (2000), no. 1, 91–102. MR 1769335, DOI 10.1016/S0166-8641(99)00075-9
Nikolai Saveliev, Fukumoto-Furuta invariants of plumbed homology 3-spheres, Pacific J. Math. 205 (2002), no. 2, 465–490. MR 1922741, DOI 10.2140/pjm.2002.205.465
Martin Scharlemann, Smooth spheres in $\textbf {R}^4$ with four critical points are standard, Invent. Math. 79 (1985), no. 1, 125–141. MR 774532, DOI 10.1007/BF01388659
V. G. Turaev, Classification of oriented Montesinos links via spin structures, Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 271–289. MR 970080, DOI 10.1007/BFb0082779
Masaaki Ue, The Neumann-Siebenmann invariant and Seifert surgery, Math. Z. 250 (2005), no. 2, 475–493. MR 2178795, DOI 10.1007/s00209-005-0764-2
E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8