## Algebraic & Geometric Topology

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

Yoshihiro Fukumoto

#### 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 $1∕m$ 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
Revised: 23 July 2018
Accepted: 2 December 2018
First available in Project Euclid: 22 August 2019

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

#### 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

#### 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