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Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

Published online by Cambridge University Press:  20 January 2009

F. Rudolf Beyl ,
M. Paul Latiolais  and
Nancy Waller
F. Rudolf Beyl
Affiliation:
Department of Mathematical Sciences, Portland State University, Portland, OR 97207-0751, U.S.A.
M. Paul Latiolais
Affiliation:
Department of Mathematical Sciences, Portland State University, Portland, OR 97207-0751, U.S.A.
Nancy Waller
Affiliation:
Department of Mathematical Sciences, Portland State University, Portland, OR 97207-0751, U.S.A.
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Abstract

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We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental group G and minimal Euler characteristic 1. If the group ring ℤG satisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. If K1(ℤG) is represented by units and K is homotopy equivalent to a spine X, then K and X are simple homotopy equivalent. We exhibit several infinite families of non-abelian groups G for which these conditions apply.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 1 , February 1997 , pp. 69 - 84
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Browning, W., Finite CW-complexes of cohomological dimension 2 with finite abelian π1, ETH Zürich (unpublished), 05 1979.Google Scholar
2. Cartan, H. and Eilenberg, S., Homological Algebra (Princeton University Press, Princeton, 1956).Google Scholar
3. Cohen, M., A Course in Simple-Homotopy Theory (Graduate Texts in Math. 10, Springer-Verlag, New York, 1973).CrossRefGoogle Scholar
4. Curtis, C. W. and Reiner, I., Methods of Representation Theory with Applications to Finite Groups and Orders (2 volumes) (John Wiley & Sons, New York, 1981 and 1987).Google Scholar
5. Dyer, M., Homotopy classification of (π, m)-complexes, J Pure Appl. Algebra 7 (1976), 249–282.CrossRefGoogle Scholar
6. Dyer, M. and Sieradski, A., Trees of homotopy types of two-dimensional complexes, Comment. Math. Helv. 48 (1973), 31–44.CrossRefGoogle Scholar
7. Gutierrez, M. and Latiolais, M. P., Partial homotopy type of finite two-complexes, Math. Z. 207 (1991), 359–378.CrossRefGoogle Scholar
8. Hambleton, I. and Kreck, M., Cancellation of lattices and finite two-complexes, J. Reine Angew. Math. 442 (1993), 91–109.Google Scholar
9. Hasse, H., Number Theory (Grundlehren der Math. Wiss. 229, Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
10. Hog-Angeloni, C., Metzler, W. and Sieradski, A. (editors), Two-dimensional Homotopy and Combinatorial Group Theory (London Mathematical Soc. Lecture Note Series 197, Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
11. Jajodia, S. and Magurn, B. A., Surjective stability of units and simple homotopy type, J. Pure Appl. Algebra 18 (1980), 45–58.CrossRefGoogle Scholar
12. Latiolais, M. P., Simple homotopy type of finite 2-complexes with finite abelian fundamental group, Trans. Amer. Math. Soc. 293 (1986), 655–662.CrossRefGoogle Scholar
13. Latiolais, M. P., When homology implies homotopy equivalence for 2-complexes, J. Pure Appl. Algebra 76 (1991), 155–165.CrossRefGoogle Scholar
14. Luft, E. and Sjerve, D., On regular coverings of 3-manifolds by homology 3-spheres, Pacific J. Math. 152 (1992), 151–163.CrossRefGoogle Scholar
15. Lustig, M., Nielsen equivalence and simple homotopy type, Proc. London Math. Soc. 62 (1991), 537–562.CrossRefGoogle Scholar
16. Magurn, B., Oliver, R. and Vaserstein, L., Units in Whitehead groups of finite groups, J. Algebra 84 (1983), 324–360.CrossRefGoogle Scholar
17. Metzler, W., Über den Homotopietyp zweidimensionaler CW-Komplexe und Elementartransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen, J. Reine Angew. Math. 285 (1976), 7–23.Google Scholar
18. Metzler, W., Die Unterscheidung von Homotopietyp und einfachem Homotopietyp bei zweidimensionalen Komplexen, J. Reine Angew. Math. 403 (1990), 201–219.Google Scholar
19. Milnor, J., Groups which act on Sn without fixed points, Amer. J. Math. 79 (1957), 623–631.CrossRefGoogle Scholar
20. Oliver, R., SK 1 for finite group rings, I, Invent. Math. 57 (1980), 183–204.CrossRefGoogle Scholar
21. Oliver, R., SK 1 for finite group rings, III, in Algebraic K-Theory (Evanston 1980, Lecture Notes in Math. 854, Springer-Verlag, Berlin, 1981), 299–337.Google Scholar
22. Oliver, R., Whitehead Groups of Finite Groups (London Mathematical Soc. Lecture Note Series, 132, Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
23. Quinn, F., Topological quantum invariants and the Andrews-Curtis conjecture (progress report), Virginia Polytechnic Institute & State University, Blacksburg 1993 (pre-print).Google Scholar
24. Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.CrossRefGoogle Scholar
25. Serre, J.-P., Trees (Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
26. Swan, R. G., Periodic resolutions for finite groups, Ann. of Math. 72 (1960), 267–291.CrossRefGoogle Scholar
27. Swan, R. G., Projective modules over binary polyhedral groups, J. Reine Angew. Math. 342 (1983), 66–172.Google Scholar
28. Threlfall, W. and Seifert, H., Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Ann. 104 (1931), 1–70 (1930).CrossRefGoogle Scholar
29. Threlfall, W. and Seifert, H., Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes (Schluß), Math. Ann. 107 (1933), 543–586(1932).CrossRefGoogle Scholar
30. Whitehead, J. H. C., Simple homotopy type, Amer. J. Math. 72 (1950), 1–57.CrossRefGoogle Scholar