TITLE: Non-oriented version of Poincaré duality
QUESTION [6 upvotes]: Hi. Is there a result in the spirit of Poincaré duality but for non-oriented manifolds? Thanks.

REPLY [8 votes]: A high level purely homotopical answer that does not require explicit consideration
of homology or cohomology is given in Theorem 19.1.5 of Parametrized Homotopy Theory,
by Johann Sigurdsson and myself.  That result works equivariantly, where orientation
is not as well understood as one would like.  Nonequivariantly, the result gives a 
description of the spectrum $k\wedge M_+$ for any spectrum $k$ and smooth closed 
manifold $M$ as a function spectrum defined in terms of parametrized spectra. That
may sound daunting, but it is really very natural. 
Here $k$ doesn't even have to be a ring spectrum.