Absolute Whitehead torsion

Korzeniewski, Andrew¹

¹ School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

Geometry & topology, Tome 11 (2007) no. 1, pp. 215-249.

Résumé

We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\mathbb{A}$ from an element of ${\tilde{K}}_1^{iso}\left(\mathbb{A}\right)$ to an element of the absolute group $K_1^{iso}\left(\mathbb{A}\right)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invariants. In the companion paper (joint with Ian Hambleton and Andrew Ranicki) this new invariant is applied to obtain the multiplicativity of the signature of fibre bundles mod 4.

Détail

DOI : 10.2140/gt.2007.11.215

Keywords: Whitehead torsion

Bibliographie

[1] I Hambleton, A Korzeniewski, A Ranicki, The signature of a fibre bundle is multiplicative mod 4, Geom. Topol. 11 (2007) 251

[2] I Hambleton, A Ranicki, L Taylor, Round $L$–theory, J. Pure Appl. Algebra 47 (1987) 131

[3] M Kervaire, Courbure intégrale généralisée et homotopie, Math. Ann. 131 (1956) 219

[4] G Lusztig, J Milnor, F P Peterson, Semi-characteristics and cobordism, Topology 8 (1969) 357

[5] A Ranicki, The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. $(3)$ 40 (1980) 87

[6] A Ranicki, The algebraic theory of surgery. II. Applications to topology, Proc. London Math. Soc. $(3)$ 40 (1980) 193

[7] A Ranicki, The algebraic theory of torsion. I. Foundations, from: "Algebraic and geometric topology (New Brunswick, N.J., 1983)", Lecture Notes in Math. 1126, Springer (1985) 199

[8] A Ranicki, Additive $L$–theory, K–Theory 3 (1989) 163

[9] A A Ranicki, Algebraic $L$–theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge University Press (1992)

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