Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For a cohomology theory, and a map of suitable spaces, an ordinary Umkehr map for the induced map is a dual morphism together with self-duality equivalences for and (orientation/Atiyah duality+Thom isomorphism).
More generally, may not be self-dual, but its dual object may be twisted cohomology for some twist . In this case the Umkehr map goes not between the original spaces and their cohomology, but between twisted cohomology variants of these.
Write
for the Spanier-Whitehead duality map which sends a topological space first to its suspension spectrum and then that to its dual object in the (∞,1)-category of spectra.
For a compact manifold, let be an embedding and write for the classical Pontryagin-Thom collapse map for this situation, and write
for the corresponding looping map from the sphere spectrum to the Thom spectrum of the negative tangent bundle of . Then Atiyah duality produces an equivalence
which identifies the Thom spectrum with the dual object of in and this constitutes a commuting diagram
identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. .
More generally, for an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps
with the abstract dual morphisms
Given now an E-∞ ring, then the dual morphism induces under smash product a similar Pontryagin-Thom collapse map, but now not in sphere spectrum-(∞,1)-modules but in -(∞,1)-modules.
The image of this under the -cohomology functor produces
If now one has a Thom isomorphism (-orientation) that identifies the cohomology of the dual object with the original cohomology, then together with the above this produces the Umkehr map
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.
fiber integration in ordinary cohomology?
For a detailed discussion of an example in K-theory see also at Poincaré duality algebra and at Freed-Witten-Kapustin anomaly.
The following terms all refer to essentially the same concept:
Twisted Umkehr maps in topological K-theory are discussed (somewhat implicitly sometimes) in the literature on KK-theory. See the references at Poincaré duality algebra.
The general abstract formulation in stable homotopy theory is sketched in section 9 of
and in section 10 of
A review and applications to quantization of local prequantum field theory is in
Formalization in dependent linear type theory is discussed
Last revised on March 15, 2021 at 10:08:13. See the history of this page for a list of all contributions to it.