nLab
Pontrjagin-Thom collapse map

Contents

Idea

Given an embedding of manifolds i:XYi : X \hookrightarrow Y, the Thom collapse map is a useful approximation to its would-be left inverse.

Definition

Traditional definition in components

All topological spaces in the following are taken to be compact.

Let XX and YY be two manifolds and let

i:XY i : X \hookrightarrow Y

be an embedding. Write N iXN_i X for the normal bundle i *TY/TXi^* T Y/ T X of the immersion ii of XX and let f:N iXYf : N_i X \to Y be any tubular neighbourhood of ii. Finally write Th(N iX)Th(N_i X) for the Thom space of the normal bundle.

Definition

The collapse map (or the Pontrjagin-Thom construction) associated to ii and the choice of tubular neighbourhood ff is

c i:YY/(Yf(N iX))Th(N iX), c_i : Y \to Y/(Y - f(N_i X)) \stackrel{\simeq}{\to} Th(N_i X) \,,

where the first morphism is the projection onto the quotient topological space and the second is the canonical homeomorphism to the Thom space of the normal bundle.

Since every point of N iXN_i X is associated to a particular point of XX, this map can be refined to a map

YX +Th(N iX) Y \to X_+ \wedge Th(N_i X)

If Y=S nY = S^n for some nn\in\mathbb{N}, then this refined Thom collapse map induces a stable map SΣ + XΣ nTh(N iX)S \to \Sigma_+^\infty X \wedge \Sigma^{-n} Th(N_i X), where SS denotes the sphere spectrum. This stable map is the unit which exhibits the suspension spectrum Σ + X\Sigma_+^\infty X 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 M nM^n with a chosen trivialization of the normal bundle N i(M n)N_i (M^n) in some R n+r\mathbf{R}^{n+r} one has TN i(M n)Σ r(M + n)T N_i(M^n)\cong \Sigma^r(M^n_+) where M + nM^n_+ is the union of M nM^n with a disjoint base point. Identify a sphere S n+rS^{n+r} with a one-point compactification R n+r{}\mathbf{R}^{n+r}\cup \{\infty\}. Then the Pontrjagin-Thom construction is the map S n+rTh(N iX)S^{n+r}\to Th(N_i X) obtained by collapsing the complement of the interior of the unit disc bundle D(N iM n)D(N_i M^n) to the point corresponding to S(N iM n)S(N_i M^n) and by mapping each point of D(N iM n)D(N_i M^n) to itself. Thus to a framed manifold M nM^n one associates the composition

S n+rTh(N iX)Σ rM + nS r S^{n+r}\to Th(N_i X)\cong \Sigma^r M^n_+\to S^r

and its homotopy class defines an element in π n+r(S r)\pi_{n+r}(S^r).

Abstract definition in terms of duality

The following is a more abstract description of Pontryagin-Thom collapse in the stable homotopy theory of sphere spectrum-(∞,1)-module bundles.

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. 2.

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).

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)

Properties

Proposition

For given ii all collapse maps for different choices of tubular neighbourhood ff are homotopic.

Proof

By the fact that the space of tubular neighbourhoods (see there for details) is contractible.

The following terms all refer to essentially the same concept:

References

An illustration is given on slide 15 of

More details are in

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.

Revised on July 19, 2013 14:16:43 by Urs Schreiber (89.204.138.251)