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COMPUTATION OF TWISTED CHARACTERISTIC CLASSES OF SINGULAR SPACES.
In ongoing joint work with Sylvain Cappell and Julius Shaneson, we establish
characteristic
class formulae for both
the twisted signature and the twisted L-classes of stratified
spaces. The local system
is assumed to extend into the singularities. For the twisted signature
of manifolds, this formula reduces to a prototypical
formula due to Atiyah and W. Meyer.
We obtain applications to stratified maps by combining our formula
with the Cappell-Shaneson
signature formula. The current focus of our
efforts is the geometrically dual situation
of (nonlocally flat PL) embeddings of stratified spaces.
Banagl showed that the above
formulae hold even when the stratified space does not
satisfy the Witt condition.
This required the use of a homology theory called
signature homology, constructed
by Augusto Minatta and Matthias Kreck.
If the local systems do not
extend into the singularities, then we have examples that show
that the above formulae become
false. Building on recent results of Banagl, we obtain formulae
for nonlocally flat embeddings
that involve rho-invariants, even when the local systems do not
extend.
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