Algebraic & Geometric Topology

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

Yoshihiro Fukumoto

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Donaldson theory orbifolds homology cobordism fundamental group


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.

Export citation


  • 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 {\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