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.
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.
Wednesday, May 17th 2006
10.15 - 12.00, Hausdorffraum
Wednesday, May 24th 2006
10.15 - 12.00, Hausdorffraum
Wednesday, May 31st or June 7th 2006
10.15 - 12.00, Hausdorffraum