Parametrized cobordism categories and the Dwyer–Weiss–Williams index theorem. Issue 3 (12th June 2017)

• Record Type: Journal Article
• Authors: Raptis, George; Steimle, Wolfgang
• Is Part Of: Journal of topology. Volume 10:Issue 3(2017)
• Page Start: 700
• Page End: 719
• Publication Date: 2017-06-12
• Subjects: 19D10 -- 57R90 (primary) -- 55R70 -- 55R12 (secondary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12019
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer–Weiss–Williams family index theorem for the parametrized A ‐theory Euler characteristic of a smooth bundle.