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The L-Theory of Twisted Quadratic Extensions

Published online by Cambridge University Press:  20 November 2018

Andrew Ranicki
Andrew Ranicki*
Affiliation:
University of Edinburgh, Edinburgh, Scotland
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For surgery on codimension 1 submanifolds with non-trivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN∗(π′ → π), with π a group and π′ a subgroup of index 2, such that there is defined an exact sequence involving the ordinary L-groups of rings with involution

with the superscript w signifying a different involution on Z[π]. Geometry was used in [13] to identify

with (α, u) an antistructure on Z[π′] in the sense of Wall [14]. The main result of this paper is a purely algebraic version of this identification, for any twisted quadratic extension of a ring with antistructure.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 345 - 364
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Browder, W. and Livesay, R., Fixed point free involutions on homotopy spheres, Tohoku J. Math. 25 (1973), 69–88.Google Scholar
2. Cappell, S. and Shaneson, J., Pseudo-free group actions I, Proc. 1978 Arhus Conference on Algebraic Topology, (Springer Lecture Notes 763, 1979), 395–447.Google Scholar
3. Cappell, S., A counterexample on the oozing problem for closed manifolds, ibid. (1979), 627–634.CrossRefGoogle Scholar
4. Hambleton, I., Projective surgery obstructions on closed manifolds, Proc. 1980 Oberwolfach Conference on Algebraic AT-theory (Springer Lecture Notes 967, 1982), 101–131.Google Scholar
5. Hambleton, I., Taylor, L. and Williams, B., An introduction to maps between surgery obstruction groups, Proc. 1982 Arhus Conference on Algebraic Topology (Springer Lecture Notes 1051, 1984), 49–127.Google Scholar
6. Harsiladze, A. F., The construction of discriminants in hermitian K-theory, Uspekhi Mat. Nauk 34 (1919), 204–207. (English translation in Russian Math. Surveys 34:6, 179–183 (1979)).Google Scholar
7. Harsiladze, A. F., Hermitian K-theory and quadratic extensions, Trudy Moskov. Mat. Obshch. 41 (1980), 3–36.Google Scholar
8. Lewis, D. W., Exact sequences of Witt groups of equivariant forms, L'Enseignement Mathématique 29 (1983), 45–51.Google Scholar
9. Lopez de Medrano, , Involutions on manifolds (Springer, 1971).CrossRefGoogle Scholar
10. Ranieki, A. A., Algebraic L-theory I. Foundations, Proe. London Math. Soc. 27 (1973), 101–125.Google Scholar
11. Ranieki, A. A., The algebraic theory of surgery I. Foundations, ibid. 40 (1980), 87–192.Google Scholar
12. Ranieki, A. A., Exact sequences in the algebraic theory of surgery, Mathematical Notes 26 (Princeton, 1981).Google Scholar
13. Wall, C. T. C., Surgery on compact manifolds (Academic Press, 1970).Google Scholar
14. Wall, C. T. C., On the axiomatic foundations of the theory of Hermitian forms, Proc. Camb. Phil. Soc. 67 (1970), 243–250.Google Scholar
15. Warshauer, M., The Witt group of degree k maps and asymmetric inner product spaces (Springer Lecture Notes, 914, 1982).CrossRefGoogle Scholar