Whitehead-Torsion und Faserungen

TL;DR

This paper investigates conditions under which a smooth map between closed manifolds can be homotoped to a smooth fiber bundle, introducing obstructions in Whitehead torsion groups and analyzing specific cases like the 1-sphere.

Contribution

It defines new obstructions in Whitehead torsion groups that determine when a map can be homotoped to a fiber bundle, extending previous work with Farrell.

Findings

First obstruction in H^1(B;Wh((E)))

Second obstruction in Wh((E))

Converse holds for B=S^1 and dim(M)>5

Abstract

This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(\pi_1(E))) and, provided that this obstruction vanishes and one additional condition is verified, a second obstruction in Wh(\pi_1(E)) >. Both elements vanish if the answer to the above question is positive. In the case where B is the 1-sphere and the dimension of M exceeds five, we show that the converse is also true, using a relationship with two obstructions defined by Farrell in this particular situation.