We give a description of the factorization homology and
topological Hochschild cohomology of Thom spectra arising from
–fold loop
maps
,
where
is
an
–fold
loop space. We describe the factorization homology
as the Thom spectrum associated to a certain map
, where
is the factorization
homology of
with
coefficients in
.
When
is
framed and
is
–connected,
this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping
space
;
in general, this is a Thom spectrum of a virtual bundle over a certain
section space. This can be viewed as a twisted form of the nonabelian
Poincaré duality theorem of Segal, Salvatore and Lurie, which occurs when
is
nullhomotopic. This result also generalizes the results of Blumberg, Cohen
and Schlichtkrull on the topological Hochschild homology of Thom spectra,
and of Schlichtkrull on higher topological Hochschild homology of Thom
spectra. We use this description of the factorization homology of Thom spectra
to calculate the factorization homology of the classical cobordism spectra,
spectra arising from systems of groups and the Eilenberg–Mac Lane spectra
,
and
. We
build upon the description of the factorization homology of Thom spectra to study
the (
and higher) topological Hochschild cohomology of Thom spectra, which enables
calculations and a description in terms of sections of a parametrized spectrum. If
is a closed
manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between
topological Hochschild
homology and
topological Hochschild cohomology, recovering string topology operations when
is
nullhomotopic. In conjunction with the higher Deligne conjecture, this gives
–structures
on a certain family of Thom spectra, which were not previously known to be ring
spectra.
