×

Controlled linear algebra. (English) Zbl 0924.19003

Quinn, Frank (ed.), Prospects in topology. Proceedings of a conference in honor of William Browder, Princeton, NJ, USA, March 1994. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 138, 138-156 (1995).
These notes derive from lectures given at UCSD in 1990. The purpose of those lectures was to elucidate the two main theorems of controlled linear algebra: the vanishing theorem for Whitehead and \(K_0\)-type obstructions when only “simply connected directions” are left uncontrolled and the squeezing principle – that once a threshold level of geometric control is obtained any finer amount of control is also available. These notes present in detail (and with perhaps some new estimates on the optimal relations between \(\varepsilon,\delta\) and dimension) the vanishing theorem of F. Quinn’s section 8 [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0394.57022)] in the context of Whitehead torsion. Quinn’s section 8 is brilliant and difficult. In our efforts to understand the details we developed – with copious advice from Frank – the argument presented here. We have neglected all the important applications to savor the intricacy of the central argument.
For the entire collection see [Zbl 0833.00037].

MSC:

19J99 Obstructions from topology
19D35 Negative \(K\)-theory, NK and Nil
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
PDFBibTeX XMLCite