The projective class group transfer induced by an \(S^ 1\)-bundle. (English) Zbl 0564.57018

Munkholm, Hans J.; Ranicki, Andrew A.

Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, 2, 461-484 (1982).

[For the entire collection see Zbl 0538.00016.]
Being given an \(S^ 1\)-bundle, \(S^ 1\to E\to B\) with fundamental group exact sequence \({\mathbb{Z}}\to \pi \to \rho \to 1\), one obtains several geometrically defined transfer homomorphisms. If B is a finitely dominated CW complex there is a transfer \(\tilde K({\mathbb{Z}}\pi)\to \tilde K_ 0({\mathbb{Z}}\pi)\) mapping the finiteness obstruction for B to that of E. If B, \(B_ 1\) are finite complexes and \(f: B_ 1\to B\) is a homotopy equivalence, the pullback \(\bar f: E_ 1\to E\) is a homotopy equivalence and there is a transfer \(Wh(\rho)\to Wh(\pi)\) sending the Whitehead torsion of f to that of \(\bar f.\) If B is a (simple) Poincaré duality space and \(f: N\to B\) is a surgery problem, the pullback \(\bar f:M\to E\) is a surgery problem and there is a transfer \(L^{\epsilon}_{\ell}(\rho,w_ B) \to L^{\epsilon}_{\ell+1}(\pi,w_ E)\) mapping the surgery obstruction for f to that of \(\bar f.\) (Here orientations w must be properly chosen, and \(\epsilon =s\) or h.) These papers describe these transfer homomorphisms in a purely algebraic framework, and, in particular, the transfer on Wh is lifted to \(K_ 1\). The two papers are considerably interlaced and represent two approaches to the same problem.

Reviewer: R.E.Stong

MSC:

57Q10	Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R10	Smoothing in differential topology
57R67	Surgery obstructions, Wall groups
57R65	Surgery and handlebodies
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Keywords:

Whitehead group; first algebraic K-group of the group ring of the fundamental group; \(S^ 1\)-bundle; transfer homomorphisms; finitely dominated CW complex; finiteness obstruction; Whitehead torsion; Poincaré duality space; surgery problem; surgery obstruction

Citations:

Zbl 0564.57017; Zbl 0538.00016