## Products and duality in Waldhausen categories

HTML articles powered by AMS MathViewer

- by Michael S. Weiss and Bruce Williams PDF
- Trans. Amer. Math. Soc.
**352**(2000), 689-709 Request permission

## Abstract:

The natural transformation $\Xi$ from $\mathbf {L}$–theory to the Tate cohomology of $\mathbb {Z} /2$ acting on $\mathbf {K}$–theory commutes with external products. Corollary: The Tate cohomology of $\mathbb {Z} /2$ acting on the $\mathbf {K}$–theory of any ring with involution is a generalized Eilenberg–Mac Lane spectrum, and it is 4–periodic.## References

- M. F. Atiyah, R. Bott, and A. Shapiro,
*Clifford modules*, Topology**3**(1964), no. suppl, suppl. 1, 3–38. MR**167985**, DOI 10.1016/0040-9383(64)90003-5 - J. F. Adams,
*Stable homotopy and generalised homology*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR**0402720** - J. F. Adams,
*Vector fields on spheres*, Ann. of Math. (2)**75**(1962), 603–632. MR**139178**, DOI 10.2307/1970213 - A. Adem, R. L. Cohen, and W. G. Dwyer,
*Generalized Tate homology, homotopy fixed points and the transfer*, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 1–13. MR**1022669**, DOI 10.1090/conm/096/1022669 - M. F. Atiyah,
*Immersions and embeddings of manifolds*, Topology**1**(1962), 125–132. MR**145549**, DOI 10.1016/0040-9383(65)90020-0 - M. F. Atiyah and G. B. Segal,
*Equivariant $K$-theory and completion*, J. Differential Geometry**3**(1969), 1–18. MR**259946** - Gunnar Carlsson,
*Equivariant stable homotopy and Segal’s Burnside ring conjecture*, Ann. of Math. (2)**120**(1984), no. 2, 189–224. MR**763905**, DOI 10.2307/2006940 - Edward B. Curtis,
*Simplicial homotopy theory*, Advances in Math.**6**(1971), 107–209 (1971). MR**279808**, DOI 10.1016/0001-8708(71)90015-6 - J. P. C.Greenlees and J.P.May,
*Generalized Tate, Borel and CoBorel Cohomology*, University of Chicago Preprint, 1992. - D. W. Lewis,
*Forms over real algebras and the multisignature of a manifold*, Advances in Math.**23**(1977), no. 3, 272–284. MR**424687**, DOI 10.1016/S0001-8708(77)80030-3 - Wen Hsiung Lin,
*On conjectures of Mahowald, Segal and Sullivan*, Math. Proc. Cambridge Philos. Soc.**87**(1980), no. 3, 449–458. MR**556925**, DOI 10.1017/S0305004100056887 - Andrew Ranicki,
*The algebraic theory of surgery. I. Foundations*, Proc. London Math. Soc. (3)**40**(1980), no. 1, 87–192. MR**560997**, DOI 10.1112/plms/s3-40.1.87 - A. A. Ranicki,
*Algebraic $L$-theory and topological manifolds*, Cambridge Tracts in Mathematics, vol. 102, Cambridge University Press, Cambridge, 1992. MR**1211640** - Graeme Segal,
*Categories and cohomology theories*, Topology**13**(1974), 293–312. MR**353298**, DOI 10.1016/0040-9383(74)90022-6 - 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** - Wolrad Vogell,
*The involution in the algebraic $K$-theory of spaces*, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 277–317. MR**802795**, DOI 10.1007/BFb0074448 - C. T. C. Wall,
*Classification of Hermitian Forms. VI. Group rings*, Ann. of Math. (2)**103**(1976), no. 1, 1–80. MR**432737**, DOI 10.2307/1971019 - Friedhelm Waldhausen,
*Algebraic $K$-theory of spaces*, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR**802796**, DOI 10.1007/BFb0074449 - Michael Weiss and Bruce Williams,
*Automorphisms of manifolds and algebraic $K$-theory. II*, J. Pure Appl. Algebra**62**(1989), no. 1, 47–107. MR**1026874**, DOI 10.1016/0022-4049(89)90020-0 - M. Weiss and B. Williams,
*Automorphisms of manifolds*, to appear in one of two C.T.C. Wall 60’th Birthday celebration volumes, 1999. - Michael Weiss and Bruce Williams,
*Duality in Waldhausen categories*, Forum Math.**10**(1998), no. 5, 533–603. MR**1644309**, DOI 10.1515/form.10.5.533

## Additional Information

**Michael S. Weiss**- Affiliation: Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, U.K.
- MR Author ID: 223956
- Email: m.weiss@maths.abdn.ac.uk
**Bruce Williams**- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- Email: williams.4@nd.edu
- Received by editor(s): January 9, 1997
- Published electronically: October 5, 1999
- Additional Notes: Both authors supported in part by NSF grant.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 689-709 - MSC (1991): Primary 57N99, 57R50, 19D10
- DOI: https://doi.org/10.1090/S0002-9947-99-02552-0
- MathSciNet review: 1694381