Abstract. This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.

for sharing their ideas, time and knowledge with me. I am indebted to many graduate students at MIT, for friendship and mathematical discussions, particularly to Charles Rezk, Nitya Kitchloo, Dan Christiensen and, last but not least, to my soulmate and sister in crime, Jean Strachan.The Studienstiftung des deutschen Volkes and the MIT deserve thanks for a generous financial support.This work is dedicated to Christina in appreciation for her constant faith and love.

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincaré bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's functor from bordism-type theories to spectra lifts to the category of symmetric spectra. We also give a new account of the foundations.