# zbMATH — the first resource for mathematics

Grothendieck groups of sesquilinear forms over a ring with involution. (English) Zbl 0616.57017
For any ring with unit R equipped with an involution, we consider the sets FP(R) and F(R) of isomorphism classes of unimodular sesquilinear forms defined on finitely generated projective, respectively free R- modules. These are monoids with respect to the orthogonal sum operation. We also define a natural notion of exactness for triples of elements of these sets. In the case of F(R) it has the following form: the triple $$(B_ 1,B_ 2,B_ 3)$$ of elements of F(R) is exact if there is a matrix X such that $$B_ 2$$ is congruent to $$\left[\begin{matrix} B_ 1&0\\ X&B_ 3\end{matrix} \right]$$.
Our aim is to compute the corresponding Grothendieck groups KF(R) and KFP(R). We prove that there is an exact sequence connecting KF(R), KFP(R) and a subgroup of the projective class group of the ring R. We denote by KF(R) the kernel of the rank map $$\rho$$ : KF(R)$$\to {\mathbb{Z}}$$. We show that KF(R) is naturally isomorphic to $$K_ 1(R)/NK_ 1(R)$$, where $$NK_ 1(R)$$ denotes the subgroup of $$K_ 1(R)$$ of elements of the form $$X\cdot X^ *$$, where $$X^ *$$ is the transpose-$$conjugate$$ of X.
We prove that the set $$\Sigma$$ (R) of related stable equivalence classes of matrices over R is an abelian group with respect to block sum. The group $$\Sigma$$ (R) turns out to be also a quotient of $$K_ 1(R)$$. In the commutative case, $$\Sigma$$ (R) can be naturally identified with $$SK_ 1(R)/NSK_ 1(R)$$. The group $$\Sigma$$ (R) depends on the way the transpose-$$conjugation$$ acts on $$K_ 1(R)$$, and using topological K-theory we give different instances of this action.
The fact that $$\Sigma$$ ($${\mathbb{Z}})$$ is trivial has a geometric interpretation in high-dimensional knot theory.
##### MSC:
 57R67 Surgery obstructions, Wall groups 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
Full Text:
##### References:
 [1] Alperin, R.C., Dennis, R.K., Oliver, R., Stein, M.R.,SK 1 of finite abelian groups II. Invent. Math.87, 253-302 (1987) · Zbl 0605.18006 [2] Bak, A.: The involution on Whitehead torsion. Gen. Topology Appl.7, 201-206 (1977) · Zbl 0367.18011 [3] Bass, H.: Some problems in classical algebraicK-theory. AlgebraicK-theory II. Batelle Institute Conf. 1972 (Lect. Notes Mathematics, Vol. 342, pp. 3-73.) Berlin Heidelberg New York: Springer 1973 [4] Bourbaki, N.: Algèbre. Paris: Hermann 1970 · Zbl 0211.02401 [5] Cohen, M.M.: A course in simple-homotopy theory. (Graduate Texts Mathematics, Vol. 10) Berlin Heidelberg New York: Springer 1973 · Zbl 0261.57009 [6] Dayton, B.H.:SK 1 of commutative normed algebras. AlgebraicK-theory. Evanston 1976 (Lect. Notes Mathematics, Vol. 551, pp. 30-43) Berlin Heidelberg New York: Springer 1976 [7] Evans, E.G., Jr.: Projective modules as fibre bundles. Proc. Am. Math. Soc.27, 623-626 (1971) · Zbl 0213.29801 [8] Fossum, R.: Vector bundles over spheres are algebraic. Invent. Math.8, 222-225 (1969) · Zbl 0176.52903 [9] Fujii, M.:K 0-groups of projective spaces. Osaka J. Math.4, 141-149 (1967) · Zbl 0153.25201 [10] Geramita, A.V., Roberts, L.C.: Algebraic vector bundles on projective space. Invent. Math.10, 298-304 (1970) · Zbl 0199.55601 [11] Heller, A.: Some exact sequences in algebraicK-theory. Topology4, 389-408 (1965) · Zbl 0161.01507 [12] Hilton, P.J.: An introduction to homotopy theory. Cambridge Tracts 43. Cambridge University Press 1961 [13] Husemoller, D.: Fibre bundles (Graduate Texts Mathematics, Vol. 20) Berlin Heidelberg New York: Springer 1966 · Zbl 0144.44804 [14] Lines, D.: On odd-dimensional fibred knots obtained by plumbing and twisting. J. London Math. Soc. (2)32, 557-571 (1985) · Zbl 0579.57014 [15] Lines, D.: Stable plumbing for high odd-dimensional fibred knots. Canad. Math. Bull.30, 429-435 (1987) · Zbl 0633.57011 [16] Milnor, J.: Introduction to algebraicK-theory. Ann. Math. Stud., Vol. 72 Princeton University Press 1971 · Zbl 0237.18005 [17] Silvester, J.R.: Introduction to algebraicK-theory. London New York: Chapman and Hall 1981 · Zbl 0468.18006 [18] Swan, R.: Vector bundles and projective modules. Trans. Am. Math. Soc.105, 264-277 (1962) · Zbl 0109.41601 [19] Swan, R.: AlgebraicK-theory (Lect. Notes Mathematics, Vol. 76). Berlin Heidelberg New York: Springer 1968 · Zbl 0193.34601 [20] Toda, H., Saito, Y., Yokota, I.: Note on the generator of ?7 SO(n). Mem. Coll. Sci. Univ. Kyoto Ser. A, Math.30, 227-230 (1957) · Zbl 0089.18101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.