Algebraic & Geometric Topology

On negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifolds

Yoshihiro Fukumoto

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Abstract

Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces L(a,b)#L(a,b) appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology 3–sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if 1m has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any L(m,1) must have a counterpart L(m,1) in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite 4–orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces L(m,1) and L(m,1) to admit a finite uniformization.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 4 (2019), 1837-1880.

Dates
Received: 11 December 2017
Revised: 23 July 2018
Accepted: 2 December 2018
First available in Project Euclid: 22 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1566439276

Digital Object Identifier
doi:10.2140/agt.2019.19.1837

Mathematical Reviews number (MathSciNet)
MR3995019

Zentralblatt MATH identifier
07121515

Subjects
Primary: 57R18: Topology and geometry of orbifolds 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 57M05: Fundamental group, presentations, free differential calculus 57R90: Other types of cobordism [See also 55N22]

Keywords
Donaldson theory orbifolds homology cobordism fundamental group

Citation

Fukumoto, Yoshihiro. On negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifolds. Algebr. Geom. Topol. 19 (2019), no. 4, 1837--1880. doi:10.2140/agt.2019.19.1837. https://projecteuclid.org/euclid.agt/1566439276


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References

  • R C Alperin, An elementary account of Selberg's lemma, Enseign. Math. 33 (1987) 269–273
  • N Anvari, Extending smooth cyclic group actions on the Poincaré homology sphere, Pacific J. Math. 282 (2016) 9–25
  • N Anvari, I Hambleton, Cyclic group actions on contractible $4$–manifolds, Geom. Topol. 20 (2016) 1127–1155
  • D R Auckly, Topological methods to compute Chern–Simons invariants, Math. Proc. Cambridge Philos. Soc. 115 (1994) 229–251
  • S Bundgaard, J Nielsen, On normal subgroups with finite index in $F$–groups, Mat. Tidsskr. B. 1951 (1951) 56–58
  • A J Casson, C M Gordon, Cobordism of classical knots, from “À la recherche de la topologie perdue” (L Guillou, A Marin, editors), Progr. Math. 62, Birkhäuser, Boston, MA (1986) 181–199
  • S K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
  • S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge Univ. Press (2002)
  • S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Clarendon, New York (1990)
  • R Fintushel, R J Stern, Pseudofree orbifolds, Ann. of Math. 122 (1985) 335–364
  • R Fintushel, R Stern, Rational homology cobordisms of spherical space forms, Topology 26 (1987) 385–393
  • R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. 61 (1990) 109–137
  • R H Fox, On Fenchel's conjecture about $F$–groups, Mat. Tidsskr. B. 1952 (1952) 61–65
  • M Furuta, Perturbation of moduli spaces of self-dual connections, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 275–297
  • M Furuta, Homology cobordism group of homology $3$–spheres, Invent. Math. 100 (1990) 339–355
  • M Furuta, On self-dual pseudo-connections on some orbifolds, Math. Z. 209 (1992) 319–337
  • P Gilmer, C Livingston, The Casson–Gordon invariant and link concordance, Topology 31 (1992) 475–492
  • M Hedden, P Kirk, Chern–Simons invariants, ${\rm SO}(3)$ instantons, and $\mathbb Z/2$ homology cobordism, from “Chern–Simons gauge theory: $20$ years after” (J E Andersen, H U Boden, A Hahn, B Himpel, editors), AMS/IP Stud. Adv. Math. 50, Amer. Math. Soc., Providence, RI (2011) 83–114
  • M Hedden, P Kirk, Instantons, concordance, and Whitehead doubling, J. Differential Geom. 91 (2012) 281–319
  • M Kato, On uniformizations of orbifolds, from “Homotopy theory and related topics” (H Toda, editor), Adv. Stud. Pure Math. 9, North-Holland, Amsterdam (1987) 149–172
  • S-G Kim, C Livingston, Nonsplittability of the rational homology cobordism group of $3$–manifolds, Pacific J. Math. 271 (2014) 183–211
  • P Kirk, E Klassen, Chern–Simons invariants of $3$–manifolds decomposed along tori and the circle bundle over the representation space of $T^2$, Comm. Math. Phys. 153 (1993) 521–557
  • P B Kronheimer, Four-manifold invariants from higher-rank bundles, J. Differential Geom. 70 (2005) 59–112
  • H B Lawson, Jr, The theory of gauge fields in four dimensions, CBMS Regional Conference Series in Mathematics 58, Amer. Math. Soc., Providence, RI (1985)
  • P Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141–2164
  • M Namba, Branched coverings and algebraic functions, Pitman Research Notes in Mathematics Series 161, Longman Scientific & Technical, Harlow (1987)
  • J Pinzón-Caicedo, Independence of satellites of torus knots in the smooth concordance group, Geom. Topol. 21 (2017) 3191–3211
  • D Ruberman, Rational homology cobordisms of rational space forms, Topology 27 (1988) 401–414
  • H Sasahira, Instanton Floer homology for lens spaces, Math. Z. 273 (2013) 237–281
  • I Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 359–363
  • A Selberg, On discontinuous groups in higher-dimensional symmetric spaces, from “Contributions to function theory”, Tata Institute of Fundamental Research, Bombay (1960) 147–164
  • M Stoffregen, Manolescu invariants of connected sums, Proc. Lond. Math. Soc. 115 (2017) 1072–1117
  • W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
  • K K Uhlenbeck, Removable singularities in Yang–Mills fields, Bull. Amer. Math. Soc. 1 (1979) 579–581
  • K K Uhlenbeck, Connections with $L\sp{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31–42