Lectures on the stable parametrized h-cobordism theorem
Smale's h-cobordism theorem, as extended to the non-simply connected case by Barden, Mazur and Stallings, shows that h-cobordisms W on a high-dimensional manifold M are classified up to isomorphism by their simple homotopy type. This amounts to an identification of the set of path components of the space H(M) of h-cobordisms on M with the Whitehead group Wh(pi) of the fundamental group of M, when M is connected. Hatcher extended this to an identification of the whole homotopy type of H(M), in a stable range increasing to infinity with the dimension of M, with a loop space Omega Wh(M), where the Whitehead space Wh(M) is connected space with fundamental group Wh(pi). This can be interpreted in the smooth, piece-wise linear or topological categories, but stably the latter two agree. Hatcher's published proof of this stable parametrized h-cobordism theorem is incomplete. Some years later, Waldhausen developed an independent proof of the stable result, and expressed the Whitehead space Wh(M) in terms of his algebraic K-theory of spaces, i.e., the functor A(X). Not all of Waldhausen's proof was published, and the speaker is working to present a complete argument, jointly with Waldhausen and Jahren. The stability results for H(M), also claimed by Hatcher, were proved in the smooth category by Igusa.
Three introductory lectures, on the geometric and the K-theoretic aspects of the theorem, will be followed by a series of more detailed lectures in the summer semester, on the remaining parts of the proof.
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The Bonn 2006
lecture notes. Last updated February 17th.
- Same in US paper size. Last updated February 17th.
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The BCAT
2002 lecture notes from June 2002 may also be useful.
- Same in US paper size.
1. Introductory lectures
Wednesday, January 25th 2006
10.00 - 12.00, Hausdorffraum
Automorphism groups of manifolds, their moduli spaces, the surgery classification up to block automorphism, concordances = pseudoisotopies, unblocking, the Hatcher spectral sequence and involutions.
Wednesday, February 1st 2006
10.00 - 12.00, Hausdorffraum
Igusa's stability theorem, h-cobordism spaces, the stable parametrized h-cobordism theorem in the PL case, relation to A(X), the manifold approach and the DIFF case.
Wednesday, February 8th 2006
10.00 - 12.00, Hausdorffraum
Computations of A(X), linearization to Quillen K-theory, the cyclotomic trace map to topological cyclic homology. Implications for the automorphisms groups.
2. The manifold part
In about three lectures, I will go through the proof of the manifold part of the stable parametrized h-cobordism theorem, following Waldhausen. The first step is to reduce to a theorem about spaces of stably framed manifolds. A second step is to reduce to a theorem about spaces of thickenings of a polyhedron. The third step is to prove this theorem, by an inductive procedure. Typical ingredients are general position, immersion theory, PL stability and transversality, all in parametrized families.