Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K–theory

Ranicki, Andrew¹ ; Sheiham, Desmond²

¹ School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
²

Geometry & topology, Tome 10 (2006) no. 3, pp. 1761-1853.

Résumé

The classification of high-dimensional $\mu$–component boundary links motivates decomposition theorems for the algebraic $K$–groups of the group ring $A\left[F_{\mu }\right]$ and the noncommutative Cohn localization ${\Sigma }^{-1}A\left[F_{\mu }\right]$, for any $\mu \ge 1$ and an arbitrary ring $A$, with $F_{\mu }$ the free group on $\mu$ generators and $\Sigma$ the set of matrices over $A\left[F_{\mu }\right]$ which become invertible over $A$ under the augmentation $A\left[F_{\mu }\right]\to A$. Blanchfield $A\left[F_{\mu }\right]$–modules and Seifert $A$–modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for $A\left[F_{\mu }\right]$–module chain complexes is used to establish a long exact sequence relating the algebraic $K$–groups of the Blanchfield and Seifert modules, and to obtain the decompositions of $K_{\ast }\left(A\left[F_{\mu }\right]\right)$ and $K_{\ast }\left({\Sigma }^{-1}A\left[F_{\mu }\right]\right)$ subject to a stable flatness condition on ${\Sigma }^{-1}A\left[F_{\mu }\right]$ for the higher $K$–groups.

Détail

DOI : 10.2140/gt.2006.10.1761

Keywords: Boundary link, algebraic $K$–theory, Blanchfield module, Seifert module

Bibliographie

[1] P Ara, W Dicks, Universal localizations embedded in power-series rings, Forum Mathematicum (to appear)

[2] P Balmer, M Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001) 819

[3] H Bass, Projective modules over free groups are free, J. Algebra 1 (1964) 367

[4] H Bass, Algebraic $K$–theory, W A Benjamin (1968)

[5] H Bass, A Heller, R G Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. (1964) 61

[6] A J Berrick, M E Keating, The $K$–theory of triangular matrix rings, from: "Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983)", Contemp. Math., Amer. Math. Soc. (1986) 69

[7] F Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

[8] S E Cappell, J L Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. $(2)$ 99 (1974) 277

[9] S E Cappell, J L Shaneson, Link cobordism, Comment. Math. Helv. 55 (1980) 20

[10] P M Cohn, Free rings and their relations, Academic Press (1971)

[11] W Dicks, E D Sontag, Sylvester domains, J. Pure Appl. Algebra 13 (1978) 243

[12] J Duval, Forme de Blanchfield et cobordisme d'entrelacs bords, Comment. Math. Helv. 61 (1986) 617

[13] M Farber, Hermitian forms on link modules, Comment. Math. Helv. 66 (1991) 189

[14] M Farber, Noncommutative rational functions and boundary links, Math. Ann. 293 (1992) 543

[15] M Farber, Stable-homotopy and homology invariants of boundary links, Trans. Amer. Math. Soc. 334 (1992) 455

[16] M Farber, P Vogel, The Cohn localization of the free group ring, Math. Proc. Cambridge Philos. Soc. 111 (1992) 433

[17] P Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962) 323

[18] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse series 35, Springer (1967)

[19] S M Gersten, On class groups of free products, Ann. of Math. $(2)$ 87 (1968) 392

[20] H Gillet, Riemann–Roch theorems for higher algebraic $K$–theory, Adv. in Math. 40 (1981) 203

[21] D R Grayson, Exact sequences in algebraic $K$–theory, Illinois J. Math. 31 (1987) 598

[22] M A Gutiérrez, Boundary links and an unlinking theorem, Trans. Amer. Math. Soc. 171 (1972) 491

[23] A Haghany, K Varadarajan, Study of modules over formal triangular matrix rings, J. Pure Appl. Algebra 147 (2000) 41

[24] B Keller, Derived categories and their uses, from: "Handbook of algebra, Vol. 1", North-Holland (1996) 671

[25] J Levine, An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970) 185

[26] J Levine, Knot modules I, Trans. Amer. Math. Soc. 229 (1977) 1

[27] C C Liang, An algebraic classification of some links of codimension two, Proc. Amer. Math. Soc. 67 (1977) 147

[28] W Lück, A Ranicki, Chain homotopy projections, J. Algebra 120 (1989) 361

[29] J W Milnor, Infinite cyclic coverings, from: "Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, 1967)", Prindle, Weber Schmidt, Boston (1968) 115

[30] A Neeman, A Ranicki, Noncommutative localisation and chain complexes I: Algebraic $K$– and $L$–theory

[31] A Neeman, A Ranicki, Noncommutative localisation in algebraic $K$–theory I, Geom. Topol. 8 (2004) 1385

[32] D Quillen, Higher algebraic $K$–theory I, from: "Algebraic $K$–theory I: Higher $K$–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)", Lecture Notes in Math. 341, Springer (1973) 85

[33] A Ranicki, High-dimensional knot theory, Springer Monographs in Mathematics, Springer (1998)

[34] A Ranicki, Blanchfield and Seifert algebra in high-dimensional knot theory, Mosc. Math. J. 3 (2003) 1333

[35] A Ranicki, Algebraic and combinatorial codimension–1 transversality, from: "Proceedings of the Casson Fest" (editors C Gordon, R Y), Geom. Topol. Monogr. 7 (2004) 145

[36] A Ranicki, Noncommutative localization in topology, from: "Non-commutative localization in algebra and topology", London Math. Soc. Lecture Note Ser. 330, Cambridge Univ. Press (2006) 81

[37] N Sato, Free coverings and modules of boundary links, Trans. Amer. Math. Soc. 264 (1981) 499

[38] M Schlichting, Negative $K$–theory of derived categories, Math. Z. 253 (2006) 97

[39] A H Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series 92, Cambridge University Press (1985)

[40] D Sheiham, Whitehead groups of localizations and the endomorphism class group, J. Algebra 270 (2003) 261

[41] D Sheiham, Invariants of boundary link cobordism II: The Blanchfield–Duval form, from: "Non-commutative localization in algebra and topology (Edinburgh 2002)", LMS Lecture Note Ser. 330, Cambridge Univ. Press (2006) 143

[42] D Sheiham, Universal localization of triangular matrix rings, Proc. Amer. Math. Soc 134 (2006) 3465

[43] N Smythe, Boundary links, from: "Wisconsin Topology Seminar", Ann. of Maths. Studies 60, Princeton UP (1966) 69

[44] J Stallings, Whitehead torsion of free products, Ann. of Math. $(2)$ 82 (1965) 354

[45] R W Thomason, T Trobaugh, Higher algebraic $K$–theory of schemes and of derived categories, from: "The Grothendieck Festschrift, Vol. III", Progr. Math. 88, Birkhäuser (1990) 247

[46] H F Trotter, Knot modules and Seifert matrices, from: "Knot theory (Proc. Sem., Plans-sur-Bex, 1977)", Lecture Notes in Math. 685, Springer (1978) 291

[47] F Waldhausen, Whitehead groups of generalized free products, from: "Algebraic $K$–theory II: “Classical” algebraic $K$–theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)", Springer (1973)

[48] F Waldhausen, Algebraic $K$–theory of generalized free products, Part 1, Ann. of Math. (2) 108 (1978) 135

[49] F Waldhausen, Algebraic $K$–theory of generalized free products, Part 2, Ann. of Math. (2) 108 (1978) 205

[50] F Waldhausen, Algebraic $K$–theory of spaces, from: "Algebraic and geometric topology (New Brunswick, N.J., 1983)", Lecture Notes in Math. 1126, Springer (1985) 318