This expository talk will first explain the definition
of a bordism invariant signature for a singular space,
proceeding along a progression from less singular to more
and more singular spaces, starting out from spaces that
have no odd-codimensional strata and, after having discussed
Goresky-Siegel spaces and Witt spaces, ending up with
general (non-Witt) stratified spaces. We will then discuss
various refinements of the signature to orientation
classes in suitable bordism theories based on singular cycles.
For instance, we will indicate how one may define a
symmetric L-homology orientation for Goresky-Siegel spaces
or a Sullivan orientation for those non-Witt spaces that
still possess generalized Poincaré duality. These classes
are refinements even of the L-class of a singular space.
Along the way, we will also see how to compute twisted
versions of the signature and L-class.

Lecture #12937

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