Contents

cohomology

# Contents

## 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.

###### 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

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

### 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.

## Examples

The following terms all refer to essentially the same concept:

## 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

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 06:08:13. See the history of this page for a list of all contributions to it.