nLab twisted Umkehr map

Idea

For $E$ a cohomology theory, and $f \colon X \to Y$ a map of suitable spaces, an ordinary Umkehr map for the induced map $E^\bullet(f) \colon E^\bullet(Y) \to E^{\bullet}(X)$ is a dual morphism together with self-duality equivalences for $E^\bullet(X)$ and $E^\bullet(Y)$ (orientation/Atiyah duality+Thom isomorphism).

More generally, $E^\bullet(X)$ may not be self-dual, but its dual object may be twisted cohomology $E^{\bullet+ \chi}(X)$ for some twist $\chi$. In this case the Umkehr map goes not between the original spaces and their cohomology, but between twisted cohomology variants of these.

Definition

Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse

Definition

Write

$$D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod$$

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.

(ABG 11, def 10.3).

Proposition

For $X$ a compact manifold, let $X \to \mathbb{R}^n$ be an embedding and write $S^n \to X^{\nu_n}$ for the classical Pontryagin-Thom collapse map for this situation, and write

$$\mathbb{S} \to X^{-T X}$$

for the corresponding looping map from the sphere spectrum to the Thom spectrum of the negative tangent bundle of $X$. Then Atiyah duality produces an equivalence

$$X^{- T X} \simeq D X$$

which identifies the Thom spectrum with the dual object of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a commuting diagram

$$\array{ && X^{- T X} \\ & \nearrow & \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} &\underset{D(X \to \ast)}{\to}& D X }$$

identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. .

More generally, for $W \hookrightarrow X$ an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps

$$\mathbb{S} \to X^{-T X} \to W^{- T W}$$

with the abstract dual morphisms

$$\mathbb{S} \to D X \to D W \,.$$

(ABG 11, prop. 10.5).

Umkehr map

Remark

Given now $E \in CRing_\infty$ an E-∞ ring, then the dual morphism $\mathbb{S} \to D X$ induces under smash product a similar Pontryagin-Thom collapse map, but now not in sphere spectrum-(∞,1)-modules but in $E$-(∞,1)-modules.

$$E \to D X \otimes_{\mathbb{S}} E \,.$$

The image of this under the $E$-cohomology functor produces

$$[D X \otimes_{\mathbb{S}} E, E] \to E \,.$$

If now one has a Thom isomorphism ($E$-orientation) $[D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]$ that identifies the cohomology of the dual object with the original cohomology, then together with the above this produces the Umkehr map

$$[X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E$$

that pushes the $E$-cohomology of $X$ to the $E$-cohomology of the point. Analogously if instead of the terminal map $X \to \ast$ we start with a more general map $X \to Y$.

More generally a Thom isomorphism may not exists, but $[D X \otimes_{\mathbb{S}} E, E]$ may still be equivalent to a twisted cohomology-variant $[X,E]_{\chi}$ of $[X,E]$, namely to $[\Gamma_X(\chi),E]$, where $\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod$ is an (flat) $E$-(∞,1)-module bundle on $X$ and and $\Gamma \simeq \underset{\to}{\lim}$ is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.

(ABG 10, 9.1)

Examples

• fiber integration in ordinary cohomology?

• fiber integration in ordinary differential cohomology

• fiber integration in K-theory

  For a detailed discussion of an example in K-theory see also at Poincaré duality algebra and at Freed-Witten-Kapustin anomaly.

• fiber integration in differential K-theory

Related concepts

The following terms all refer to essentially the same concept:

• fiber integration in generalized cohomology

• push-forward in generalized cohomology

• Umkehr map, twisted Umkehr map

• Gysin map

• Pontrjagin-Thom collapse map

References

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

• Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)

and in section 10 of

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, Geom. Topol. 22(7): 3761-3825 (2018). (arXiv:1112.2203, doi:10.2140/gt.2018.22.3761)

A review and applications to quantization of local prequantum field theory is in

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013

Formalization in dependent linear type theory is discussed

• Urs Schreiber, Quantization via Linear homotopy types