Geodesic
Rational homology 3–spheres and simply connected definite bounding
Sato, Kouki1 ; Taniguchi, Masaki1
1Graduate School of Mathematical Sciences, University of Tokyo, Meguro, Japan
Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 865-882
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For each rational homology 3–sphere Y which bounds simply connected definite 4–manifolds of both signs, we construct an infinite family of irreducible rational homology 3–spheres which are homology cobordant to Y but cannot bound any simply connected definite 4–manifold. As a corollary, for any coprime integers p and q, we obtain an infinite family of irreducible rational homology 3–spheres which are homology cobordant to the lens space L(p,q) but cannot be obtained by a knot surgery.

DOI : 10.2140/agt.2020.20.865
Classification : 57M25, 57M27
Keywords: homology $3$–sphere, $4$–manifold, gauge theory, Chern–Simons functional

Sato, Kouki  1   ; Taniguchi, Masaki  1

1 Graduate School of Mathematical Sciences, University of Tokyo, Meguro, Japan 
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Sato, Kouki; Taniguchi, Masaki. Rational homology 3–spheres and simply connected definite bounding. Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 865-882. doi: 10.2140/agt.2020.20.865

[1] D Auckly, An irreducible homology sphere which is not Dehn surgery on a knot, preprint (1993)

[2] S Boyer, D Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990) 181 | DOI

[3] D H Choe, K Park, On intersection forms of definite 4-manifolds bounded by a rational homology 3–sphere, Topology Appl. 238 (2018) 59 | DOI

[4] T D Cochran, R E Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3–spheres and property P, Topology 27 (1988) 495 | DOI

[5] T D Cochran, S Harvey, P Horn, Filtering smooth concordance classes of topologically slice knots, Geom. Topol. 17 (2013) 2103 | DOI

[6] S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279 | DOI

[7] R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. 61 (1990) 109 | DOI

[8] M Furuta, Homology cobordism group of homology 3–spheres, Invent. Math. 100 (1990) 339 | DOI

[9] P M Gilmer, C Livingston, On embedding 3–manifolds in 4–space, Topology 22 (1983) 241 | DOI

[10] C M Gordon, J Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. 20 (1989) 83 | DOI

[11] R C Lyndon, P E Schupp, Combinatorial group theory, 89, Springer (1977) | DOI

[12] J Milnor, On manifolds homeomorphic to the 7–sphere, Ann. of Math. 64 (1956) 399 | DOI

[13] B Owens, S Strle, Dehn surgeries and negative-definite four-manifolds, Selecta Math. 18 (2012) 839 | DOI

[14] C H Taubes, Gauge theory on asymptotically periodic 4–manifolds, J. Differential Geom. 25 (1987) 363 | DOI

[15] Y Tsutsumi, Universal bounds for genus one Seifert surfaces for hyperbolic knots and surgeries with non-trivial JSJT-decompositions, Interdiscip. Inform. Sci. 9 (2003) 53 | DOI

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