Abstract
We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra.
As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2019R1A3B2067839
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2105451
Funding statement: G. Lim was partially supported by the National Research Foundation of Korea grant 2019R1A3B2067839. S. Weinberger was partially supported by the National Science Foundation grant DMS-2105451.
Acknowledgements
We would like to thank Jae Choon Cha and Fedya Manin for very stimulating conversations on a number of topics around the contents of this paper.
References
[1] P. Albin, E. Leichtnam, R. Mazzeo and P. Piazza, The signature package on Witt spaces, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 2, 241–310. 10.24033/asens.2165Suche in Google Scholar
[2] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’honneur de Henri Cartan, Astérisque 32–33, Société Mathématique de France, Paris (1976), 43–72. Suche in Google Scholar
[3] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. 10.2307/1970721Suche in Google Scholar
[4] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71–99. 10.1017/S0305004100052105Suche in Google Scholar
[5] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. 10.2307/1970717Suche in Google Scholar
[6] H. Bass, Algebraic 𝐾-theory, W. A. Benjamin, New York 1968. Suche in Google Scholar
[7] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980), no. 1, 1–47. 10.1016/0022-4049(80)90040-7Suche in Google Scholar
[8] W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242–245. 10.1090/S0002-9904-1967-11700-2Suche in Google Scholar
[9] S. E. Cappell and J. L. Shaneson, On four dimensional surgery and applications, Comment. Math. Helv. 46 (1971), 500–528. 10.1007/BF02566862Suche in Google Scholar
[10] J. C. Cha, A topological approach to Cheeger-Gromov universal bounds for von Neumann 𝜌-invariants, Comm. Pure Appl. Math. 69 (2016), no. 6, 1154–1209. 10.1002/cpa.21597Suche in Google Scholar
[11] J. C. Cha and G. Lim, Quantitative bordism over acyclic groups and Cheeger–Gromov 𝜌-invariants, in preparation. Suche in Google Scholar
[12] G. R. Chambers, D. Dotterrer, F. Manin and S. Weinberger, Quantitative null-cobordism, J. Amer. Math. Soc. 31 (2018), no. 4, 1165–1203. 10.1090/jams/903Suche in Google Scholar
[13] S. Chang, On conjectures of Mathai and Borel, Geom. Dedicata 106 (2004), 161–167. 10.1023/B:GEOM.0000033834.29385.66Suche in Google Scholar
[14] S. Chang and S. Weinberger, On invariants of Hirzebruch and Cheeger–Gromov, Geom. Topol. 7 (2003), 311–319. 10.2140/gt.2003.7.311Suche in Google Scholar
[15] S. Chang and S. Weinberger, A course on surgery theory, Ann. of Math. Stud. 211, Princeton University, Princeton 2021. 10.1515/9780691200354Suche in Google Scholar
[16]
J. Cheeger and M. Gromov,
Bounds on the von Neumann dimension of
[17] J. Cheeger and M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin (1985), 115–154. 10.1007/978-3-642-69828-6_9Suche in Google Scholar
[18] J. Cheeger, W. Müller and R. Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), no. 3, 405–454. 10.1007/BF01210729Suche in Google Scholar
[19]
T. D. Cochran, K. E. Orr and P. Teichner,
Knot concordance, Whitney towers and
[20] F. Costantino, Complexity of 4-manifolds, Exp. Math. 15 (2006), no. 2, 237–249. 10.1080/10586458.2006.10128960Suche in Google Scholar
[21] M. W. Davis and T. Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991), no. 2, 347–388. 10.4310/jdg/1214447212Suche in Google Scholar
[22] M. W. Davis, T. Januszkiewicz and S. Weinberger, Relative hyperbolization and aspherical bordisms: An addendum to “Hyperbolization of polyhedra”, J. Differential Geom. 58 (2001), no. 3, 535–541. 10.4310/jdg/1090348358Suche in Google Scholar
[23] M. Gromov, Volume and bounded cohomology, Publ. Math. Inst. Hautes Études Sci. 56 (1982), 5–99. Suche in Google Scholar
[24] J.-C. Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981), no. 1, 115–123. 10.1007/BF01218375Suche in Google Scholar
[25] F. Hirzebruch, Involutionen auf Mannigfaltigkeiten, Proceedings of the Conference on Transformation Groups, Springer, New York (1968), 148–166. 10.1007/978-3-642-46141-5_7Suche in Google Scholar
[26] W. Jaco, H. Rubinstein and S. Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009), no. 1, 157–180. 10.1112/jtopol/jtp004Suche in Google Scholar
[27] M. Lackenby and J. S. Purcell, The triangulation complexity of fibred 3-manifolds, preprint (2019), https://arxiv.org/abs/1910.10914. Suche in Google Scholar
[28] E. Leichtnam and P. Piazza, On higher eta-invariants and metrics of positive scalar curvature, 𝐾-Theory 24 (2001), no. 4, 341–359. 10.1023/A:1014079307698Suche in Google Scholar
[29] G. Lim, Enhanced bounds for rho-invariants for both general and spherical 3-manifolds, J. Topol. Anal. (2022), 10.1142/S1793525322500029. 10.1142/S1793525322500029Suche in Google Scholar
[30] S. Mac Lane, Homology, Class. Math., Springer, Berlin 1995. Suche in Google Scholar
[31] F. Manin and S. Weinberger, Quantitative PL bordism, preprint (2023), https://arxiv.org/abs/2311.16389. Suche in Google Scholar
[32] S. V. Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990), no. 2, 101–130. 10.1007/BF00049576Suche in Google Scholar
[33] J. P. May, Simplicial objects in algebraic topology, Chic. Lectures in Math., University of Chicago, Chicago 1992. Suche in Google Scholar
[34] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. 10.1090/S0002-9904-1966-11484-2Suche in Google Scholar
[35] J. C. Moore, Semi-simplicial complexes and Postnikov systems, International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, México (1958), 232–247. Suche in Google Scholar
[36] P. Piazza and T. Schick, Rho-classes, index theory and Stolz’ positive scalar curvature sequence, J. Topol. 7 (2014), no. 4, 965–1004. 10.1112/jtopol/jtt048Suche in Google Scholar
[37] M. Ramachandran, von Neumann index theorems for manifolds with boundary, J. Differential Geom. 38 (1993), no. 2, 315–349. 10.4310/jdg/1214454297Suche in Google Scholar
[38] D. B. Ray and I. M. Singer, 𝑅-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210. 10.1016/0001-8708(71)90045-4Suche in Google Scholar
[39]
P. H. Siegel,
Witt spaces: A geometric cycle theory for
[40] C. Soulé, Perfect forms and the Vandiver conjecture, J. reine angew. Math. 517 (1999), 209–221. 10.1515/crll.1999.095Suche in Google Scholar
[41] C. T. C. Wall, Surgery on compact manifolds, 2nd ed., Math. Surveys Monogr. 69, American Mathematical Society, Providence 1999. 10.1090/surv/069Suche in Google Scholar
[42] S. Weinberger, Homotopy equivalent manifolds by pasting, Current trends in algebraic topology, CMS Conf. Proc. 2, American Mathematical Society, Providence (1982), 423–433. Suche in Google Scholar
[43] S. Weinberger and G. Yu, Finite part of operator 𝐾-theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds, Geom. Topol. 19 (2015), no. 5, 2767–2799. 10.2140/gt.2015.19.2767Suche in Google Scholar
[44] R. F. Williams, A useful functor and three famous examples in topology, Trans. Amer. Math. Soc. 106 (1963), 319–329. 10.1090/S0002-9947-1963-0146832-0Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Gluing Karcher–Scherk saddle towers I: Triply periodic minimal surfaces
- Nodal Enriques surfaces are Reye congruences
- Multi-localized time-symmetric initial data for the Einstein vacuum equations
- Matrix representations of arbitrary bounded operators on Hilbert spaces
- Hodge numbers of motives attached to Kloosterman and Airy moments
- Products of primes in arithmetic progressions
- A quantitative stability result for the sphere packing problem in dimensions 8 and 24
- Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds
Artikel in diesem Heft
- Frontmatter
- Gluing Karcher–Scherk saddle towers I: Triply periodic minimal surfaces
- Nodal Enriques surfaces are Reye congruences
- Multi-localized time-symmetric initial data for the Einstein vacuum equations
- Matrix representations of arbitrary bounded operators on Hilbert spaces
- Hodge numbers of motives attached to Kloosterman and Airy moments
- Products of primes in arithmetic progressions
- A quantitative stability result for the sphere packing problem in dimensions 8 and 24
- Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds