Let and be two manifolds and let
The collapse map (or the Pontrjagin-Thom construction) associated to and the choice of tubular neighbourhood is
Since every point of is associated to a particular point of , this map can be refined to a map
If for some , then this refined Thom collapse map induces a stable map , where denotes the sphere spectrum. This stable map is the unit which exhibits the suspension spectrum as a dualizable object in the stable homotopy category. See n-duality and fixed point index?.
Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold with a chosen trivialization of the normal bundle in some one has where is the union of with a disjoint base point. Identify a sphere with a one-point compactification . Then the Pontrjagin-Thom construction is the map obtained by collapsing the complement of the interior of the unit disc bundle to the point corresponding to and by mapping each point of to itself. Thus to a framed manifold one associates the composition
and its homotopy class defines an element in .
with the abstract dual morphisms
The image of this under the -cohomology functor produces
that pushes the -cohomology of to the -cohomology of the point. Analogously if instead of the terminal map we start with a more general map .
More generally a Thom isomorphism may not exists, but may still be equivalent to a twisted cohomology-variant of , namely to , where is an (flat) -(∞,1)-module bundle on and and is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.
The following terms all refer to essentially the same concept:
An illustration is given on slide 15 of
More details are in
Victor Snaith, Stable homotopy around the arf-Kervaire invariant, Birkhauser 2009
The general abstract formulation in stable homotopy theory is in sketched in section 9 of
and is in section 10 of
with an emphases on parameterized spectra.