Controlled Mather–Thurston theorems

Michael H. Freedman¹

¹Microsoft Corporation; University of California, Santa Barbara, USA

EMS surveys in mathematical sciences, Tome 9 (2022) no. 2, pp. 415-445

Résumé

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor–Wood inequality is for circle bundles over surfaces, whereas the Mather–Thurston theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern–Weil theory and other smooth obstructions such as the Bott classes and the Godbillion–Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g., PSL(2,R) versus U(1) in the former case and C1 versus C2 in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather–Thurston cobordism to a semi-s-cobordism (ssc), and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one.

Détail

DOI : 10.4171/emss/63

Classification : 57-XX
Mots-clés : Dynamics, perfect groups, bilipschitz, foliation