The Hausmann-Weinberger 4–manifold invariant of abelian groups
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- by Paul Kirk and Charles Livingston PDF
- Proc. Amer. Math. Soc. 133 (2005), 1537-1546 Request permission
Abstract:
The Hausmann-Weinberger invariant of a group $G$ is the minimal Euler characteristic of a closed orientable 4–manifold $M$ with fundamental group $G$. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.References
- Beno Eckmann, $4$-manifolds, group invariants, and $l_2$-Betti numbers, Enseign. Math. (2) 43 (1997), no. 3-4, 271–279. MR 1489886
- Beno Eckmann, Introduction to $l_2$-methods in topology: reduced $l_2$-homology, harmonic chains, $l_2$-Betti numbers, Israel J. Math. 117 (2000), 183–219. Notes prepared by Guido Mislin. MR 1760592, DOI 10.1007/BF02773570
- Jean-Claude Hausmann and Shmuel Weinberger, Caractéristiques d’Euler et groupes fondamentaux des variétés de dimension $4$, Comment. Math. Helv. 60 (1985), no. 1, 139–144 (French). MR 787667, DOI 10.1007/BF02567405
- Jonathan A. Hillman, An homology 4-sphere group with negative deficiency, Enseign. Math. (2) 48 (2002), no. 3-4, 259–262. MR 1955602
- F. E. A. Johnson and D. Kotschick, On the signature and Euler characteristic of certain four-manifolds, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 3, 431–437. MR 1235990, DOI 10.1017/S0305004100071711
- Rob Kirby (ed.), Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR 1470751, DOI 10.1090/amsip/002.2/02
- D. Kotschick, Four-manifold invariants of finitely presentable groups, Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 89–99. MR 1312175
- C. Livingston, Four-manifolds of large negative deficiency, preprint (2003), arXiv:math/0302026. To appear, Math. Proc. Camb. Phil. Soc.
- Wolfgang Lück, $L^2$-Betti numbers of mapping tori and groups, Topology 33 (1994), no. 2, 203–214. MR 1273782, DOI 10.1016/0040-9383(94)90011-6
Additional Information
- Paul Kirk
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 266369
- Email: pkirk@indiana.edu
- Charles Livingston
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 193092
- Email: livingst@indiana.edu
- Received by editor(s): October 6, 2003
- Received by editor(s) in revised form: December 31, 2003
- Published electronically: October 18, 2004
- Additional Notes: The first named author gratefully acknowledges the support of the National Science Foundation under grant no. DMS-0202148.
- Communicated by: Ronald A. Fintushel
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 1537-1546
- MSC (2000): Primary 57M05, 57M07
- DOI: https://doi.org/10.1090/S0002-9939-04-07652-X
- MathSciNet review: 2111955