Integer-valued polynomials and 𝐾-theory operations

Strong, M-J.; Whitehouse, Sarah

doi:10.1090/S0002-9939-10-10237-8

Integer-valued polynomials and $K$-theory operations
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Proc. Amer. Math. Soc. 138 (2010), 2221-2233 Request permission

Abstract:

This paper provides a unifying approach to recent results linking the fields of integer-valued polynomials and operations in $K$-theory. Following work of Bhargava, we set up a general framework encompassing several examples of rings of integer-valued polynomials. Our main results give bases for the duals of these rings. The rings and their duals all arise in topology as various kinds of cooperations and operations in complex $K$-theory. We show how several previously understood examples fit into this framework and we present some new examples.

References

J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
J. F. Adams and F. W. Clarke, Stable operations on complex $K$-theory, Illinois J. Math. 21 (1977), no. 4, 826–829. MR 454977
Manjul Bhargava, $P$-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101–127. MR 1468927, DOI 10.1515/crll.1997.490.101
Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly 107 (2000), no. 9, 783–799. MR 1792411, DOI 10.2307/2695734
J. Michael Boardman, Stable operations in generalized cohomology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 585–686. MR 1361899, DOI 10.1016/B978-044481779-2/50015-8
J. Michael Boardman, David Copeland Johnson, and W. Stephen Wilson, Unstable operations in generalized cohomology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 687–828. MR 1361900, DOI 10.1016/B978-044481779-2/50016-X
Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, vol. 48, American Mathematical Society, Providence, RI, 1997. MR 1421321, DOI 10.1090/surv/048
Francis Clarke, Self-maps of $B\textrm {U}$, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 3, 491–500. MR 602302, DOI 10.1017/S0305004100058382
Francis Clarke, M. D. Crossley, and Sarah Whitehouse, Bases for cooperations in $K$-theory, $K$-Theory 23 (2001), no. 3, 237–250. MR 1857208, DOI 10.1023/A:1011820819598
Francis Clarke, Martin Crossley, and Sarah Whitehouse, Algebras of operations in $K$-theory, Topology 44 (2005), no. 1, 151–174. MR 2104006, DOI 10.1016/j.top.2004.05.003
Ève Helsmoortel, Module de continuité de polynômes d’interpolation. Applications à l’étude du comportement local des fonctions continues sur un compact régulier d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1168–A1171 (French). MR 244212
Keith Johnson, The invariant subalgebra and anti-invariant submodule of $K_*K_{(p)}$, J. K-Theory 2 (2008), no. 1, 123–145. MR 2434169, DOI 10.1017/is008004002jkt042
Lionel Schwartz, Opérations d’Adams en $K$-homologie et applications, Bull. Soc. Math. France 109 (1981), no. 2, 237–257 (French, with English summary). MR 623792
M.-J. Strong. Additive unstable operations in complex $K$-theory and cobordism. Ph.D. thesis, Department of Pure Mathematics, University of Sheffield, September 2008.
M.-J. Strong and Sarah Whitehouse. Infinite sums of unstable Adams operations and cobordism, arXiv:0812.1481v1, to appear in J. Pure Appl. Alg.