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On realization of Whitehead torsion

Published online by Cambridge University Press:  24 October 2008

Slawomir Kwasik
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.

Abstract

We prove that the Poincaré conjecture implies that the 4-dimensional version of the realization theorem for Whitehead torsion is false. We show that infinitely many examples may be constructed to demonstrate this.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Cohen, M.. A Course in Simple-Homotopy Theory, Graduate Texts in Mathematics 10 (Springer- Verlag, 1973).CrossRefGoogle Scholar
[2]Hambleton, I. and Milgram, R. J.. The surgery obstruction groups for finite 2-groups. Invent. Math. 61 (1980), 3352.CrossRefGoogle Scholar
[3]Milnor, J.. Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358426.CrossRefGoogle Scholar
[4]Plotnick, S. P.. Vanishing of Whitehead groups for Seifert manifold with infinite fundamental group. Comment. Math. Helv. 55 (1980), 654667.CrossRefGoogle Scholar
[5]Plotnick, S. P.. Homotopy equivalences and free modules. Topology 21 (1982), 9199.CrossRefGoogle Scholar
[6]Oliver, R.. SK1 for finite group rings, III. In Algebraic K-Theory (Evanston 1980), Lecture Notes in Math. vol. 854 (Springer-Verlag, 1981), 299–337.Google Scholar
[7]Orlik, P.. Seifert Manifolds, Lecture Notes in Math. vol. 291 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[8]Ritter, G.. Free actions of Z8 on S3. Trans Amer. Math. Soc. 181 (1973), 795–212.Google Scholar
[9]Rourke, C. P. and Sanderson, B. J.. Introduction to Piecewise-Linear Topology (Springer-Verlag, 1972).CrossRefGoogle Scholar
[10]Rubinstein, J. H.. Free actions of some finite groups on S3:I. Math. Ann. 240 (1979), no. 2, 165175.CrossRefGoogle Scholar
[11]Waldhausen, F.. Algebraic K-theory of generalized free products. Ann. of Math. 108 (1978), 135256.CrossRefGoogle Scholar
[12]Wall, C. T. C.. Norms of units in group rings. Proc. London Math. Soc. (3) 29 (1974), 593632.CrossRefGoogle Scholar
[13]Wall, C. T. C.. Classification of hermitian forms, VI: Group rings. Ann. of Math. 103 (1976), 180.CrossRefGoogle Scholar