nLab
twisted Umkehr map

Context

Integration theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

More generally, E (X)E^\bullet(X) may not be self-dual, but its dual object may be twisted cohomology E +χ(X)E^{\bullet+ \chi}(X) for some twist χ\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() Σ + L wheTop𝕊Mod 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 XX a compact manifold, let X nX \to \mathbb{R}^n be an embedding and write S nX ν nS^n \to X^{\nu_n} for the classical Pontryagin-Thom collapse map for this situation, and write

𝕊X TX \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 XX. Then Atiyah duality produces an equivalence

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

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

X TX 𝕊 D(X*) DX \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 WXW \hookrightarrow X an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps

𝕊X TXW TW \mathbb{S} \to X^{-T X} \to W^{- T W}

with the abstract dual morphisms

𝕊DXDW. \mathbb{S} \to D X \to D W \,.

(ABG 11, prop. 10.5).

Umkehr map

Remark

Given now ECRing E \in CRing_\infty an E-∞ ring, then the dual morphism 𝕊DX\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 EE-(∞,1)-modules.

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

The image of this under the EE-cohomology functor produces

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

If now one has a Thom isomorphism (EE-orientation) [DX 𝕊E,E][X,E] [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 produces the Umkehr map

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

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

More generally a Thom isomorphism may not exists, but [DX 𝕊E,E][D X \otimes_{\mathbb{S}} E, E] may still be equivalent to a twisted cohomology-variant [X,E] χ[X,E]_{\chi} of [X,E][X,E], namely to [Γ X(χ),E][\Gamma_X(\chi),E], where χ:Π(X)ELineEMod\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod is an (flat) EE-(∞,1)-module bundle on XX and and Γlim\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

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 December 20, 2014 at 14:56:45. See the history of this page for a list of all contributions to it.