A stability range for topological $4$-manifolds
HTML articles powered by AMS MathViewer
- by Ian Hambleton
- Trans. Amer. Math. Soc. 376 (2023), 8769-8793
- DOI: https://doi.org/10.1090/tran/9020
- Published electronically: August 29, 2023
- HTML | PDF | Request permission
Abstract:
We introduce a new stable range invariant for the classification of closed, oriented topological $4$-manifolds (up to $s$-cobordism), after stabilization by connected sum with a uniformly bounded number of copies of $S^2\times S^2$.References
- A. Bartels, F. T. Farrell, and W. Lück, The Farrell-Jones conjecture for cocompact lattices in virtually connected Lie groups, J. Amer. Math. Soc. 27 (2014), no. 2, 339–388. MR 3164984, DOI 10.1090/S0894-0347-2014-00782-7
- Arthur Bartels and Wolfgang Lück, The Borel conjecture for hyperbolic and $\textrm {CAT}(0)$-groups, Ann. of Math. (2) 175 (2012), no. 2, 631–689. MR 2993750, DOI 10.4007/annals.2012.175.2.5
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- Hyman Bass, Unitary algebraic $K$-theory, Algebraic $K$-theory, III: Hermitian $K$-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 343, Springer, Berlin-New York, 1973, pp. 57–265. MR 371994
- Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330, DOI 10.1007/s002220050168
- Mladen Bestvina, Koji Fujiwara, and Derrick Wigglesworth, The Farrell-Jones conjecture for hyperbolic-by-cyclic groups, Int. Math. Res. Not. IMRN 7 (2023), 5887–5904. MR 4565703, DOI 10.1093/imrn/rnac012
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
- Sylvain E. Cappell and Julius L. Shaneson, On four dimensional surgery and applications, Comment. Math. Helv. 46 (1971), 500–528. MR 301750, DOI 10.1007/BF02566862
- Diarmuid Crowley and Jörg Sixt, Stably diffeomorphic manifolds and $l_{2q+1}(\Bbb Z[\pi ])$, Forum Math. 23 (2011), no. 3, 483–538. MR 2805192, DOI 10.1515/FORM.2011.016
- F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic $K$-theory, J. Amer. Math. Soc. 6 (1993), no. 2, 249–297. MR 1179537, DOI 10.1090/S0894-0347-1993-1179537-0
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Ross Geoghegan and Michael L. Mihalik, Free abelian cohomology of groups and ends of universal covers, J. Pure Appl. Algebra 36 (1985), no. 2, 123–137. MR 787167, DOI 10.1016/0022-4049(85)90065-9
- Ian Hambleton and Alyson Hildum, Topological 4-manifolds with right-angled Artin fundamental groups, J. Topol. Anal. 11 (2019), no. 4, 777–821. MR 4040013, DOI 10.1142/s1793525319500328
- I. Hambleton and A. Hildum, Topological $4$-manifolds with right-angled Artin fundamental groups: corrigendum, (in preparation), 2021.
- Ian Hambleton and Matthias Kreck, On the classification of topological $4$-manifolds with finite fundamental group, Math. Ann. 280 (1988), no. 1, 85–104. MR 928299, DOI 10.1007/BF01474183
- Ian Hambleton and Matthias Kreck, Cancellation of hyperbolic forms and topological four-manifolds, J. Reine Angew. Math. 443 (1993), 21–47. MR 1241127
- Ian Hambleton and Matthias Kreck, Homotopy self-equivalences of 4-manifolds, Math. Z. 248 (2004), no. 1, 147–172. MR 2092726, DOI 10.1007/s00209-004-0657-9
- Ian Hambleton and Matthias Kreck, On the classification of topological 4-manifolds with finite fundamental group: corrigendum [ MR0928299], Math. Ann. 372 (2018), no. 1-2, 527–530. MR 3856820, DOI 10.1007/s00208-018-1656-1
- Ian Hambleton, Matthias Kreck, and Peter Teichner, Topological 4-manifolds with geometrically two-dimensional fundamental groups, J. Topol. Anal. 1 (2009), no. 2, 123–151. MR 2541758, DOI 10.1142/S1793525309000084
- I. Hambleton, R. J. Milgram, L. Taylor, and B. Williams, Surgery with finite fundamental group, Proc. London Math. Soc. (3) 56 (1988), no. 2, 349–379. MR 922660, DOI 10.1112/plms/s3-56.2.349
- Ian Hambleton and Erik K. Pedersen, Identifying assembly maps in $K$- and $L$-theory, Math. Ann. 328 (2004), no. 1-2, 27–57. MR 2030369, DOI 10.1007/s00208-003-0454-5
- Daniel Kasprowski, Markus Land, Mark Powell, and Peter Teichner, Stable classification of 4-manifolds with 3-manifold fundamental groups, J. Topol. 10 (2017), no. 3, 827–881. MR 3797598, DOI 10.1112/topo.12025
- D. Kasprowski, M. Powell, and P. Teichner, Algebraic criteria for stable diffeomorphism of spin 4-manifolds, arXiv:2006.06127 [math.GT], 2021.
- D. Kasprowski, M. Powell, and P. Teichner, The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds, arXiv:2105.12153 [math.GT], 2021.
- Daniel Kasprowski, Mark Powell, and Peter Teichner, Four-manifolds up to connected sum with complex projective planes, Amer. J. Math. 144 (2022), no. 1, 75–118. MR 4367415, DOI 10.1353/ajm.2022.0001
- Daniel Kasprowski and Peter Teichner, $\Bbb {CP}^2$-stable classification of 4-manifolds with finite fundamental group, Pacific J. Math. 310 (2021), no. 2, 355–373. MR 4229242, DOI 10.2140/pjm.2021.310.355
- Qayum Khan, Cancellation for 4-manifolds with virtually abelian fundamental group, Topology Appl. 220 (2017), 14–30. MR 3619277, DOI 10.1016/j.topol.2017.01.013
- Robion C. Kirby and Laurence R. Taylor, A survey of 4-manifolds through the eyes of surgery, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 387–421. MR 1818779
- M. Kreck and J. A. Schafer, Classification and stable classification of manifolds: some examples, Comment. Math. Helv. 59 (1984), no. 1, 12–38. MR 743942, DOI 10.1007/BF02566336
- Matthias Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no. 3, 707–754. MR 1709301, DOI 10.2307/121071
- Erkki Laitinen, End homology and duality, Forum Math. 8 (1996), no. 1, 121–133. MR 1366538, DOI 10.1515/form.1996.8.121
- Wolfgang Lück, Assembly maps, Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, [2020] ©2020, pp. 851–890. MR 4198000
- Wolfgang Lück and Holger Reich, The Baum-Connes and the Farrell-Jones conjectures in $K$- and $L$-theory, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 703–842. MR 2181833, DOI 10.1007/978-3-540-27855-9_{1}5
- Michael L. Mihalik and Steven T. Tschantz, One relator groups are semistable at infinity, Topology 31 (1992), no. 4, 801–804. MR 1191381, DOI 10.1016/0040-9383(92)90010-F
- Michael L. Mihalik and Steven T. Tschantz, Semistability of amalgamated products, HNN-extensions, and all one-relator groups, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 131–135. MR 1107833, DOI 10.1090/S0273-0979-1992-00257-4
- Jesper Michael Møller, Self-homotopy equivalences of group cohomology spaces, J. Pure Appl. Algebra 73 (1991), no. 1, 23–37. MR 1121629, DOI 10.1016/0022-4049(91)90104-A
- Mehmetcik Pamuk, Homotopy self-equivalences of 4-manifolds with $\textrm {PD}_2$ fundamental group, Topology Appl. 156 (2009), no. 8, 1445–1458. MR 2512597, DOI 10.1016/j.topol.2008.12.006
- Mehmetcik Pamuk, Homotopy self-equivalences of 4-manifolds with free fundamental group, Canad. J. Math. 62 (2010), no. 6, 1387–1403. MR 2760664, DOI 10.4153/CJM-2010-061-4
- J. T. Stafford, Absolute stable rank and quadratic forms over noncommutative rings, $K$-Theory 4 (1990), no. 2, 121–130. MR 1081655, DOI 10.1007/BF00533152
- John Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541–543. MR 158917, DOI 10.2307/2373106
- R. Strebel, A remark on subgroups of infinite index in Poincaré duality groups, Comment. Math. Helv. 52 (1977), no. 3, 317–324. MR 457588, DOI 10.1007/BF02567371
- Laurence Taylor and Bruce Williams, Surgery spaces: formulae and structure, Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978) Lecture Notes in Math., vol. 741, Springer, Berlin, 1979, pp. 170–195. MR 557167
- L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81(123) (1970), 328–351 (Russian). MR 269722
- C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR 163323, DOI 10.1112/jlms/s1-39.1.131
- C. T. C. Wall, On simply-connected $4$-manifolds, J. London Math. Soc. 39 (1964), 141–149. MR 163324, DOI 10.1112/jlms/s1-39.1.141
- C. T. C. Wall, Surgery on compact manifolds, 2nd ed., Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI, 1999. Edited and with a foreword by A. A. Ranicki. MR 1687388, DOI 10.1090/surv/069
- J. H. C. Whitehead, A certain exact sequence, Ann. of Math. (2) 52 (1950), 51–110. MR 35997, DOI 10.2307/1969511
Bibliographic Information
- Ian Hambleton
- Affiliation: Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
- MR Author ID: 80380
- Email: hambleton@mcmaster.ca
- Received by editor(s): March 7, 2023
- Received by editor(s) in revised form: June 30, 2023
- Published electronically: August 29, 2023
- Additional Notes: The research was partially supported by NSERC Discovery Grant A4000.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 8769-8793
- MSC (2020): Primary 57K40, 57R67, 57N65
- DOI: https://doi.org/10.1090/tran/9020
- MathSciNet review: 4669310