# nLab Thom's theorem

Contents

This entry is about Thom’s theorem in cobordism theory. For the isomorphism in cohomology induced by Thom classes see at Thom isomorphism.

### Context

#### Cobordism theory

Concepts of cobordism theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

relative bordism theories:

algebraic:

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

What is called Thom’s theorem or the Pontrjagin-Thom isomorphism (due to Thom 54, Pontrjagin 55) states that for a given universal tangential structure the stable homotopy groups of the universal Thom spectrum $M G$, with its canonical ring spectrum structure, form the cobordism ring

$\Omega^G_\bullet \simeq \pi_\bullet(M G)$

of manifolds with that tangential structure; and that this isomorphism is exhibited by the Pontryagin-Thom construction (see there).

More generally, for $X$ a topological space, the group of $G$-bordism classes of $G$-manifolds in $X$ is isomorphic to the generalized homology of $X$ with coefficients in $M G$:

$\Omega^G_\bullet(X) \simeq \pi_\bullet( M G \wedge X_+) \simeq M G_\bullet(X)$

## Ingredients

### Tangential structure on the Stable normal bundle

###### Definition

Given a smooth manifold $X$ of dimension $n$ and equipped with an embedding

$i \;\colon\; X \hookrightarrow \mathbb{R}^k$

for some $k \in \mathbb{N}$, then the classifying map of its normal bundle is the function

$g_i \;\colon\; X \to Gr_{k-n}(\mathbb{R}^k) \hookrightarrow B O(k-n)$

which sends $x \in X$ to the normal of the tangent space

$N_x X = (T_x X)^{\perp} \hookrightarrow \mathbb{R}^k$

regarded as a point in $G_{k-n}(\mathbb{R}^k)$.

The normal bundle of $i$ itself is the subbundle of the tangent bundle

$T \mathbb{R}^k \simeq \mathbb{R}^k \times \mathbb{R}^k$

consisting of those vectors which are orthogonal to the tangent vectors of $X$:

$N_i \coloneqq \left\{ x\in X, v \in T_{i(x)}\mathbb{R}^k \;\vert\; v \,\perp\, i_\ast T_x X \subset T_{i(x)}\mathbb{R}^k \right\} \,.$
###### Definition
1. for each $n\in \mathbb{N}$ a pointed CW-complex $B_n \in Top_{CW}^{\ast/}$

2. equipped with a pointed Serre fibration

$\array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) }$

to the classifying space $B O(n)$ (def.);

3. for all $n_1 \leq n_2$ a pointed continuous function

$g_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

which is the identity for $n_1 = n_2$;

such that for all $n_1 \leq n_2 \in \mathbb{N}$ these squares commute

$\array{ B_{n_1} &\overset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,$

where the bottom map is the canonical one from def. .

The $(B,f)$-structure is multiplicative if it is moreover equipped with a system of maps $\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}$ which cover the canonical multiplication maps (def.)

$\array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }$

and which satisfy the evident associativity and unitality, for $B_0 = \ast$ the unit, and, finally, which commute with the maps $g$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

$\array{ B_{n_1} \times B_{n_2} &\overset{id \times g_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }$

and

$\array{ B_{n_1} \times B_{n_2} &\overset{g_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{g_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.$

Similarly, an $S^2$-$(B,f)$-structure is a compatible system

$f_{2n} \colon B_{2n} \longrightarrow B O(2n)$

indexed only on the even natural numbers.

Generally, an $S^k$-$(B,f)$-structure for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system

$f_{k n} \colon B_{k n} \longrightarrow B O(k n)$

for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.

We write $V^\mathcal{B}_n$ for the universal vector bundle pulled back to the corresponding space of the $(B,f)$-structure and with

$\array{ V^{\mathcal{B}} &\overset{}{\longrightarrow}& E O(n) \underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{f_n}{\longrightarrow}& B O(n) }$

and we write $e_{n_1,n_2}$ for the maps of total space of vector bundles over the $g_{n_1,n_2}$:

$\array{ V^{\mathcal{B}}_{n_1} &\overset{e_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_2} \\ \downarrow &(pb)& \downarrow \\ B_{n_1} &\underset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} } \,.$
###### Example

Examples of $(B,f)$-structures (def. ) include the following:

1. $B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);

2. $B_n = E O(n)$ and $f_n$ the universal principal bundle-projection is framing-structure;

3. $B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the special orthogonal group and $f_n$ the canonical projection is orientation structure;

4. $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the spin group and $f_n$ the canonical projection is spin structure.

Examples of $S^2$-$(B,f)$-structures (def. ) include

1. $B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the unitary group, and $f_{2n}$ the canonical projection is almost complex structure (or rather: almost Hermitian structure).

2. $B_{2n} = B Sp(2n) = E O(2n)/Sp(2n)$ the classifying space of the symplectic group, and $f_{2n}$ the canonical projection is almost symplectic structure.

Examples of $S^4$-$(B,f)$-structures (def. ) include

1. $B_{4n} = B U_{\mathbb{H}}(n) = E O(4n)/U_{\mathbb{H}}(n)$ the classifying space of the quaternionic unitary group, and $f_{4n}$ the canonical projection is almost quaternionic structure.
###### Definition

Given a smooth manifold $X$ of dimension $n$, and given a $(B,f)$-structure as in def. , then a $(B,f)$-structure on the stable normal bundle of the manifold is an equivalence class of the following structure:

1. an embedding $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;

2. a homotopy class of a lift $\hat g$ of the classifying map $g$ of the normal bundle (def. )

$\array{ && B_{k-n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_{k-n}}} \\ X &\overset{g}{\longrightarrow}& B O(k-n) } \,.$

The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if

1. $k_2 \geq k_1$

2. the second inclusion factors through the first as

$(i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}$
3. the lift of the classifying map factors accordingly (as homotopy classes)

$\hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{k_1-n} \overset{g_{k_1-n,k_2-n}}{\longrightarrow} B_{k_2-n} \,.$

### Bordism

Throughout, let $\mathcal{B}$ be a multiplicative (B,f)-structure (def.).

###### Definition

Write $I \coloneqq [0,1]$ for the standard interval, regarded as a smooth manifold with boundary. For $c \in \mathbb{R}_+$ Consider its embedding

$e \;\colon\; I \hookrightarrow \mathbb{R}\oplus \mathbb{R}_{\geq 0}$

as the arc

$e \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2 \,,$

where $(e_1, e_2)$ denotes the canonical linear basis of $\mathbb{R}^2$, and equipped with the structure of a manifold with normal framing structure (def.) by equipping it with the canonical framing

$fr \;\colon\; t \mapsto \cos(\pi t) \cdot e_1 + \sin(\pi t) \cdot e_2$

of its normal bundle.

Let now $\mathcal{B}$ be a (B,f)-structure (def.). Then for $X \overset{i}{\hookrightarrow}\mathbb{R}^k$ any embedded manifold with $\mathcal{B}$-structure $\hat g \colon X \to B_{k-n}$ on its normal bundle (def.), define its negative or orientation reversal $-(X,i,\hat g)$ of $(X,i, \hat g)$ to be the restriction of the structured manifold

$(X \times I \overset{(i,e)}{\hookrightarrow} \mathbb{R}^{k+2}, \hat g \times fr)$

to $t = 1$.

###### Definition

Two closed manifolds of dimension $n$ equipped with normal $\mathcal{B}$-structure $(X_1, i_1, \hat g_1)$ and $(X_2,i_2,\hat g_2)$ (def.) are called bordant if there exists a manifold with boundary $W$ of dimension $n+1$ equipped with $\mathcal{B}$-strcuture $(W,i_W, \hat g_W)$ if its boundary with $\mathcal{B}$-structure restricted to that boundary is the disjoint union of $X_1$ with the negative of $X_2$, according to def.

$\partial(W,i_W,\hat g_W) \simeq (X_1, i_1, \hat g_1) \sqcup -(X_2, i_2, \hat g_2) \,.$
###### Proposition

The relation of $\mathcal{B}$-bordism (def. ) is an equivalence relation.

Write $\Omega^\mathcal{B}_{\bullet}$ for the $\mathbb{N}$-graded set of $\mathcal{B}$-bordism classes of $\mathcal{B}$-manifolds.

###### Proposition

Under disjoint union of manifolds, then the set of $\mathcal{B}$-bordism equivalence classes of def. becomes an $\mathbb{Z}$-graded abelian group

$\Omega^{\mathcal{B}}_\bullet \in Ab^{\mathbb{Z}}$

(that happens to be concentrated in non-negative degrees). This is called the $\mathcal{B}$-bordism group.

Moreover, if the (B,f)-structure $\mathcal{B}$ is multiplicative (def.), then Cartesian product of manifolds followed by the multiplicative composition operation of $\mathcal{B}$-structures makes the $\mathcal{B}$-bordism ring into a commutative ring, called the $\mathcal{B}$-bordism ring.

$\Omega^{\mathcal{B}}_\bullet \in CRing^{\mathbb{Z}} \,.$

e.g. (Kochmann 96, prop. 1.5.3)

### Thom spaces

###### Definition

Let $X$ be a topological space and let $V \to X$ be a vector bundle over $X$ of rank $n$, which is associated to an O(n)-principal bundle. Equivalently this means that $V \to X$ is the pullback of the universal vector bundle $E_n \to B O(n)$ (def. ) over the classifying space. Since $O(n)$ preserves the metric on $\mathbb{R}^n$, by definition, such $V$ inherits the structure of a metric space-fiber bundle. With respect to this structure:

1. the unit disk bundle $D(V) \to X$ is the subbundle of elements of norm $\leq 1$;

2. the unit sphere bundle $S(V)\to X$ is the subbundle of elements of norm $= 1$;

$S(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V$;

3. the Thom space $Th(V)$ is the cofiber (formed in Top (prop.)) of $i_V$

$Th(V) \coloneqq cofib(i_V)$

canonically regarded as a pointed topological space.

$\array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.$

If $V \to X$ is a general real vector bundle, then there exists an isomorphism to an $O(n)$-associated bundle and the Thom space of $V$ is, up to based homeomorphism, that of this orthogonal bundle.

###### Remark

If the rank of $V$ is positive, then $S(V)$ is non-empty and then the Thom space (def. ) is the quotient topological space

$Th(V) \simeq D(V)/S(V) \,.$

However, in the degenerate case that the rank of $V$ vanishes, hence the case that $V = X\times \mathbb{R}^0 \simeq X$, then $D(V) \simeq V \simeq X$, but $S(V) = \emptyset$. Hence now the pushout defining the cofiber is

$\array{ \emptyset &\overset{i_V}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) \simeq X_* } \,,$

which exhibits $Th(V)$ as the coproduct of $X$ with the point, hence as $X$ with a basepoint freely adjoined.

$Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+ \,.$
###### Proposition

Let $V_1,V_2 \to X$ be two real vector bundles. Then the Thom space (def. ) of the direct sum of vector bundles $V_1 \oplus V_2 \to X$ is expressed in terms of the Thom space of the pullbacks $V_2|_{D(V_1)}$ and $V_2|_{S(V_1)}$ of $V_2$ to the disk/sphere bundle of $V_1$ as

$Th(V_1 \oplus V_2) \simeq Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)}) \,.$
###### Proof

Notice that

1. $D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1)$;

2. $S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)})$.

(Since a point at radius $r$ in $V_1 \oplus V_2$ is a point of radius $r_1 \leq r$ in $V_2$ and a point of radius $\sqrt{r^2 - r_1^2}$ in $V_1$.)

###### Proposition

For $V$ a vector bundle then the Thom space (def. ) of $\mathbb{R}^n \oplus V$, the direct sum of vector bundles with the trivial rank $n$ vector bundle, is homeomorphic to the smash product of the Thom space of $V$ with the $n$-sphere (the $n$-fold reduced suspension).

$Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V) \,.$
###### Proof

Apply prop. with $V_1 = \mathbb{R}^n$ and $V_2 = V$. Since $V_1$ is a trivial bundle, then

$V_2|_{D(V_1)} \simeq V_2\times D^n$

(as a bundle over $X\times D^n$) and similarly

$V_2|_{S(V_1)} \simeq V_2\times S^n \,.$
###### Example

By prop. and remark the Thom space (def. ) of a trivial vector bundle of rank $n$ is the $n$-fold suspension of the base space

\begin{aligned} Th(X \times \mathbb{R}^n) & \simeq S^n \wedge Th(X\times \mathbb{R}^0) \\ & \simeq S^n \wedge (X_+) \end{aligned} \,.

Therefore a general Thom space may be thought of as a “twisted suspension”, with twist encoded by a vector bundle (or rather by its underlying spherical fibration). See at Thom spectrum – For infinity-module bundles for more on this.

###### Proposition

For $V_1 \to X_1$ and $V_2 \to X_2$ to vector bundles, let $V_1 \boxtimes V_2 \to X_1 \times X_2$ be the direct sum of vector bundles of their pullbacks to $X_1 \times X_2$. The corresponding Thom space (def. ) is the smash product of the individual Thom spaces:

$Th(V_1 \boxtimes V_2) \simeq Th(V_1) \wedge Th(V_2) \,.$

### Universal Thom spectra $M G$

###### Proposition

For each $n \in \mathbb{N}$ the pullback of the rank-$(n+1)$ universal vector bundle to the classifying space of rank $n$ vector bundles is the direct sum of vector bundles of the rank $n$ universal vector bundle with the trivial rank-1 bundle: there is a pullback diagram of topological spaces of the form

$\array{ \mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n) &\longrightarrow& E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1} \\ \downarrow &(pb)& \downarrow \\ B O(n) &\longrightarrow& B O(n+1) } \,,$

where the bottom morphism is the canonical one (def.).

(e.g. Kochmann 96, p. 25)

###### Proof

For each $k \in \mathbb{N}$, $k \geq n$ there is such a pullback of the canonical vector bundles over Grassmannians

$\array{ \left\{ {V_{n}\subset \mathbb{R}^k, } \atop {v \in V_n, v_{n+1} \in \mathbb{R}} \right\} &\longrightarrow& \left\{ {V_{n+1} \subset \mathbb{R}^{k+1}}, \atop v \in V_{n+1} \right\} \\ \downarrow && \downarrow \\ Gr_n(\mathbb{R}^k) &\longrightarrow& Gr_{n+1}(\mathbb{R}^{k+1}) }$

where the bottom morphism is the canonical inclusion (def.).

Now we claim that taking the colimit in each of the four corners of this system of pullback diagrams yields again a pullback diagram, and this proves the claim.

To see this, remember that we work in the category $Top_{cg}$ of compactly generated topological spaces (def.). By their nature, we may test the universal property of a would-be pullback space already by mapping compact topological spaces into it. Now observe that all the inclusion maps in the four corners of this system of diagrams are relative cell complex inclusions, by prop. . Together this implies (via this lemma) that we may test the universal property of the colimiting square at finite stages. And so this implies the claim by the above fact that at each finite stage there is a pullback diagram.

###### Definition

The universal real Thom spectrum $M O$ is the spectrum, which is represented by the sequential prespectrum (def.) whose $n$th component space is the Thom space (def. )

$(M O)_n \coloneqq Th(E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$

of the rank-$n$ universal vector bundle, and whose structure maps are the image under the Thom space functor $Th(-)$ of the top morphisms in prop. , via the homeomorphisms of prop. :

$\sigma_n \;\colon\; \Sigma (M O)_n \simeq Th(\mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)) \stackrel{}{\longrightarrow} Th(E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}) = (M O)_{n+1} \,.$

More generally, there are universal Thom spectra associated with any other tangent structure (“(B,f)-structure”), notably for the orthogonal group replaced by the special orthogonal groups $SO(n)$, or the spin groups $Spin(n)$, or the string 2-group $String(n)$, or the fivebrane 6-group $Fivebrane(n)$,…, or any level in the Whitehead tower of $O(n)$. To any of these groups there corresponds a Thom spectrum (denoted, respectively, $M SO$, MSpin, $M String$, $M Fivebrane$, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.:

###### Definition

Given a (B,f)-structure $\mathcal{B}$ (def. ), write $V^\mathcal{B}_n$ for the pullback of the universal vector bundle (def. ) to the corresponding space of the $(B,f)$-structure and with

$\array{ V^{\mathcal{B}} &\overset{}{\longrightarrow}& V O(n) \underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{f_n}{\longrightarrow}& B O(n) }$

and we write $e_{n_1,n_2}$ for the maps of total space of vector bundles over the $g_{n_1,n_2}$:

$\array{ V^{\mathcal{B}}_{n_1} &\overset{e_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_2} \\ \downarrow &(pb)& \downarrow \\ B_{n_1} &\underset{g_{n_1,n_2}}{\longrightarrow}& B_{n_2} } \,.$

Observe that the analog of prop. still holds:

###### Proposiiton

Given a (B,f)-structure $\mathcal{B}$ (def. ), then the pullback of its rank-$(n+1)$ vector bundle $V^{\mathcal{B}}_{n+1}$ (def. ) along the map $g_{n,n+1} \colon B_n \to B_{n+1}$ is the direct sum of vector bundles of the rank-$n$ bundle $V^{\mathcal{B}}_n$ with the trivial rank-1-bundle: there is a pullback square

$\array{ \mathbb{R} \oplus V^{\mathcal{B}}_n &\overset{e_{n,n+1}}{\longrightarrow}& V^{\mathcal{B}}_{n+1} \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{g_{n,n+1}}{\longrightarrow}& B_{n+1} } \,.$
###### Proof

Unwinding the definitions, the pullback in question is

\begin{aligned} (g_{n,n+1})^\ast V^{\mathcal{B}}_{n+1} & = (g_{n,n+1})^\ast f_{n+1}^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq (g_{n,n+1} \circ f_{n+1})^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq ( f_n \circ i_n )^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast i_n^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast (\mathbb{R} \oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^{n})) \\ &\simeq \mathbb{R} \oplus V^{\mathcal{B}_n} \,, \end{aligned}

where the second but last step is due to prop. .

###### Definition

Given a (B,f)-structure $\mathcal{B}$ (def. ), its universal Thom spectrum $M \mathcal{B}$ is, as a sequential prespectrum, given by component spaces being the Thom spaces (def. ) of the $\mathcal{B}$-associated vector bundles of def.

$(M \mathcal{B})_n \coloneqq Th(V^{\mathcal{B}}_n)$

and with structure maps given via prop. by the top maps in prop. :

$\sigma_n \;\colon\; \Sigma (M \mathcal{B})_n = \Sigma Th(V^{\mathcal{E}}_n) \simeq Th(\mathbb{R}\oplus V^{\mathcal{E}}_n) \overset{Th(e_{n,n+1})}{\longrightarrow} Th(V^{\mathcal{E}}_{n+1}) = (M \mathcal{B})_{n+1} \,.$

Similarly for an $S^k-(B,f)$-structure indexed on every $k$th natural number (such as almost complex structure, almost quaternionic structure, example ), there is the corresponding Thom spectrum as a sequential $S^k$ spectrum (def.).

If $B_n = B G_n$ for some natural system of groups $G_n \to O(n)$, then one usually writes $M G$ for $M \mathcal{B}$. For instance $M SO$, MSpin, MU, MSp etc.

If the $(B,f)$-structure is multiplicative (def. ), then the Thom spectrum $M \mathcal{B}$ canonical becomes a ring spectrum: the multiplication maps $B_{n_1} \times B_{n_2}\to B_{n_1 + n_2}$ are covered by maps of vector bundles

$V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} \longrightarrow V^{\mathcal{B}}_{n_1 + n_2}$

and under forming Thom spaces this yields (prop.) maps

$(M \mathcal{B})_{n_1} \wedge (M \mathcal{B})_{n_2} \longrightarrow (M \mathcal{B})_{n_1 + n_2}$

which are associative by the associativity condition in a multiplicative $(B,f)$-structure. The unit is

$(M \mathcal{B})_0 = Th(V^{\mathcal{B}}_0) \simeq Th(\ast) \simeq S^0 \,,$

by remark .

###### Example

The universal Thom spectrum (def. ) for framing structure (exmpl.) is equivalently the sphere spectrum (def.)

$M 1 \simeq \mathbb{S} \,.$

Because in this case $B_n \simeq \ast$ and so $E^{\mathcal{B}}_n \simeq \mathbb{R}^n$, whence $Th(E^{\mathcal{B}}_n) \simeq S^n$.

### Pontrjagin-Thom construction

###### Definition

For $X$ a smooth manifold and $i \colon X \hookrightarrow \mathbb{R}^k$ an embedding, then a tubular neighbourhood of $X$ is a subset of the form

$\tau_i X \coloneqq \left\{ x \in \mathbb{R}^k \;\vert\; d(x,i(X)) \lt \epsilon \right\}$

for some $\epsilon \in \mathbb{R}$, $\epsilon \gt 0$, small enough such that the map

$N_i X \longrightarrow \tau_i X$

from the normal bundle (def. ) given by

$(i(x),v) \mapsto (i(x), \epsilon (1-e^{- {\vert v\vert}}) v )$

is a diffeomorphism.

###### Proposition

(tubular neighbourhood theorem)

For every embedding of smooth manifolds, there exists a tubular neighbourhood according to def. .

###### Remark

Given an embedding $i \colon X \hookrightarrow \mathbb{R}^k$ with a tubuluar neighbourhood $\tau_i X \hookrigtharrow \mathbb{R}^k$ (def. ) then by construction:

1. the Thom space (def. ) of the normal bundle (def. ) is homeomorphic to the quotient topological space of the topological closure of the tubular neighbourhood by its boundary:

$Th(N_i(X)) \simeq \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}$;

2. there exists a continous function

$\mathbb{R}^k \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)}$

which is the identity on $\tau_i(X)\subset \mathbb{R}^k$ and is constant on the basepoint of the quotient on all other points.

###### Definition

For $X$ a smooth manifold of dimension $n$ and for $i \colon X \hookrightarrow \mathbb{R}^k$ an embedding, then the Pontrjagin-Thom collapse map is, for any choice of tubular neighbourhood $\tau_i(X)\subset \mathbb{R}^k$ (def. ) the composite map of pointed topological spaces

$S^k \overset{\simeq}{\to} (\mathbb{R}^k)^\ast \longrightarrow \overline{ \tau_i(X)}/\partial \overline{\tau_i(X)} \overset{\simeq}{\to} Th(N_i X)$

where the first map identifies the k-sphere as the one-point compactification of $\mathbb{R}^k$; and where the second and third maps are those of remark .

The Pontrjagin-Thom construction is the further composite

$\xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(e_i)}{\longrightarrow} Th( E O(k-n) \underset{O(k-n)}{\times} \mathbb{R}^{k-n} ) \simeq (M O)_{k-n}$

with the image under the Thom space construction of the morphism of vector bundles

$\array{ \nu &\overset{e_i}{\longrightarrow}& E O(k-n)\underset{O(k-n)}{\times} \mathbb{R}^{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{g_i}{\longrightarrow}& B O(k-n) }$

induced by the classifying map $g_i$ of the normal bundle (def. ).

This defines an element

$[S^{n+(k-n)} \overset{\xi_i}{\to} (M O)_{k-n}] \in \pi_{n} M O$

in the $n$th stable homotopy group (def.) of the Thom spectrum $M O$ (def. ).

More generally, for $X$ a smooth manifold with normal (B,f)-structure $(X,i,\hat g_i)$ according to def. , then its Pontrjagin-Thom construction is the composite

$\xi_i \;\colon\; S^k \longrightarrow Th(N_i X) \overset{Th(\hat e_i)}{\longrightarrow} Th( V^{\mathcal{B}}_{k-n} ) \simeq (M \mathcal{B})_{k-n}$

with

$\array{ \nu &\overset{\hat e_i}{\longrightarrow}& V^{\mathcal{B}}_{k-n} \\ \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B O(k-n) } \,.$
###### Proposition

The Pontrjagin-Thom construction (def. ) respects the equivalence classes entering the definition of manifolds with stable normal $\mathcal{B}$-structure (def. ) hence descends to a function (of sets)

$\xi \;\colon\; \left\{ {n\text{-}manifolds\;with\;stable} \atop {normal\;\mathcal{B}\text{-}structure} \right\} \longrightarrow \pi_n(M\mathcal{B}) \,.$
###### Proof

It is clear that the homotopies of classifying maps of $\mathcal{B}$-structures that are devided out in def. map to homotopies of representatives of stable homotopy groups. What needs to be shown is that the construction respects the enlargement of the embedding spaces.

Given a embedded manifold $X \overset{i}{\hookrightarrow}\mathbb{R}^{k_1}$ with normal $\mathcal{B}$-structure

$\array{ && B_{k_1-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) }$

write

$\alpha \;\colon\; S^{n+(k_1-n)} \overset{}{\longrightarrow} Th(E^{\mathcal{B}_{k_1-n}})$

for its image under the Pontrjagin-Thom construction (def. ). Now given $k_2 \in \mathbb{N}$, consider the induced embedding $X \overset{i}{\hookrightarrow} \mathbb{R}^{k_1}\hookrightarrow \mathbb{R}^{k_1 + k_2}$ with normal $\mathcal{B}$-structure given by the composite

$\array{ && B_{k_1-n} &\overset{g_{k_1-n, k_1+ k_2 -n}}{\longrightarrow}& B_{k_1 + k_2-n} \\ & {}^{\mathllap{\hat g_i}}\nearrow & \downarrow^{\mathrlap{f_{k_1 - n} \times f_{k_2}}} && \downarrow^{\mathrlap{f_{k_1 + k_2-n}}} \\ X &\underset{g_i}{\longrightarrow}& B O(k_1-n) &\longrightarrow& B O(k_1 + k_2-n) } \,.$

By prop. and using the pasting law for pullbacks, the classifying map $\hat g'_i$ for the enlarged normal bundle sits in a diagram of the form

$\array{ (\nu_i \oplus \mathbb{R}^{k_2}) &\overset{(\hat e_i \oplus id)}{\longrightarrow}& (V^{\mathcal{B}}_{k_1-n} \oplus \mathbb{R}^{k_2}) &\overset{e_{k_1-n,k_1+k_2-n}}{\longrightarrow}& V^{\mathcal{B}}_{k_1 + k_2 - n} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X &\underset{\hat g_i}{\longrightarrow}& B_{k_1-n} &\underset{g_{k_1-n, k_1 + k_2 - n}}{\longrightarrow}& B_{k_1 +k_2 - n} } \,.$

Hence the Pontrjagin-Thom construction for the enlarged embedding space is (using prop. ) the composite

$\alpha_{k_2} \;\colon\; S^{n + (k_1+ k_2 - n)} \simeq Th(\mathbb{R}^{k_2}) \wedge S^{n + (k_1 - n)} \overset{}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(\nu_i) \overset{Th(id)\wedge Th(\hat e_i)}{\longrightarrow} Th(\mathbb{R}^{k_2}) \wedge Th(E^{\mathcal{B}}_{k_1-n})) \overset{Th(e_{k_1-n, k_1 + k_2 - n})}{\longrightarrow} Th(V^{\mathcal{B}}_{k_1 + k_2 - n}) \,.$

The composite of the first two morphisms here is $S^{k_k}\wedge \alpha$, while last morphism $Th(\hat e_{k_1-n,k_1+k_2-n})$ is the structure map in the Thom spectrum (by def. ):

$\alpha_{k_2} \;\colon\; S^{k_2} \wedge S^{n + (k_1 - n)} \overset{S^{k_2} \wedge \alpha}{\longrightarrow} S^{k_2} \wedge Th(E^{\mathcal{B}}_{k_1 + k_2 - n}) \overset{\sigma^{M \mathcal{B}}_{k_1-n,k_1 + k_2 - n} }{\longrightarrow} Th(V^{\mathcal{B}}_{k_1+k_2 - n})$

This manifestly identifies $\alpha_{k_2}$ as being the image of $\alpha$ under the component map in the sequential colimit that defines the stable homotopy groups (def.). Therefore $\alpha$ and $\alpha_{k_2}$, for all $k_2 \in \mathbb{N}$, represent the same element in $\pi_{\bullet}(M \mathcal{B})$.

## Thom’s theorem

Recall that the Pontrjagin-Thom construction (def.) associates to an embbeded manifold $(X,i,\hat g)$ with normal $\mathcal{B}$-structure (def.) an element in the stable homotopy group $\pi_{dim(X)}(M \mathcal{B})$ of the universal $\mathcal{B}$-Thom spectrum in degree the dimension of that manifold.

###### Lemma

For $\mathcal{B}$ be a multiplicative (B,f)-structure (def.), the $\mathcal{B}$-Pontrjagin-Thom construction (def.) is compatible with all the relations involved to yield a graded ring homomorphism

$\xi \;\colon\; \Omega^{\mathcal{B}}_\bullet \longrightarrow \pi_\bullet(M \mathcal{B})$

from the $\mathcal{B}$-bordism ring (def. ) to the stable homotopy groups of the universal $\mathcal{B}$-Thom spectrum equipped with the ring structure induced from the canonical ring spectrum structure (def.).

###### Proof

By prop. the underlying function of sets is well-defined before dividing out the bordism relation (def. ). To descend this further to a function out of the set underlying the bordism ring, we need to see that the Pontrjagin-Thom construction respects the bordism relation. But the definition of bordism is just so as to exhibit under $\xi$ a left homotopy of representatives of homotopy groups.

Next we need to show that it is

1. a group homomorphism;

2. a ring homomorphism.

Regarding the first point:

The element 0 in the cobordism group is represented by the empty manifold. It is clear that the Pontrjagin-Thom construction takes this to the trivial stable homotopy now.

Given two $n$-manifolds with $\mathcal{B}$-structure, we may consider an embedding of their disjoint union into some $\mathbb{R}^{k}$ such that the tubular neighbourhoods of the two direct summands do not intersect. There is then a map from two copies of the k-cube, glued at one face

$\Box^k \underset{\Box^{k-1}}{\sqcup} \Box^k \longrightarrow \mathbb{R}^k$

such that the first manifold with its tubular neighbourhood sits inside the image of the first cube, while the second manifold with its tubular neighbourhood sits indide the second cube. After applying the Pontryagin-Thom construction to this setup, each cube separately maps to the image under $\xi$ of the respective manifold, while the union of the two cubes manifestly maps to the sum of the resulting elements of homotopy groups, by the very definition of the group operation in the homotopy groups (def.). This shows that $\xi$ is a group homomorphism.

Regarding the second point:

The element 1 in the cobordism ring is represented by the manifold which is the point. Without restriction we may consoder this as embedded into $\mathbb{R}^0$, by the identity map. The corresponding normal bundle is of rank 0 and hence (by remark ) its Thom space is $S^0$, the 0-sphere. Also $V^{\mathcal{B}}_0$ is the rank-0 vector bundle over the point, and hence $(M \mathcal{B})_0 \simeq S^0$ (by def. ) and so $\xi(\ast) \colon (S^0 \overset{\simeq}{\to} S^0)$ indeed represents the unit element in $\pi_\bullet(M\mathcal{B})$.

Finally regarding respect for the ring product structure: for two manifolds with stable normal $\mathcal{B}$-structure, represented by embeddings into $\mathbb{R}^{k_i}$, then the normal bundle of the embedding of their Cartesian product is the direct sum of vector bundles of the separate normal bundles bulled back to the product manifold. In the notation of prop. there is a diagram of the form

$\array{ \nu_1 \boxtimes \nu_2 &\overset{\hat e_1 \boxtimes \hat e_2}{\longrightarrow}& V^{\mathcal{B}}_{n_1} \boxtimes V^{\mathcal{B}}_{n_2} &\overset{\kappa_{n_1,n_2}}{\longrightarrow}& V^{\mathcal{B}}_{n_1 + n_2} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X_1 \times X_2 &\underset{\hat g_1 \times \hat g_2}{\longrightarrow}& B_{k_1-n_1} \times B_{k_2-n_2} &\underset{\mu_{k_1-n_1,k_2-n_2}}{\longrightarrow}& B_{k_1 + k_2 - n_1 - n_2} } \,.$

To the Pontrjagin-Thom construction of the product manifold is by definition the top composite in the diagram

$\array{ S^{n_1 +n_2 + (k_1 + k_2 - n_1 - n_2)} &\overset{}{\longrightarrow}& Th(\nu_1 \boxtimes \nu_2) &\overset{Th(\hat e_1 \boxtimes \hat e_2)}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1-n_1} \boxtimes V^{\mathcal{B}}_{k_2-n_2}) &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2}) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ S^{n_1 + (k_1 - n_1)} \wedge S^{n_2 + (k_2 - n_2)} &\overset{}{\longrightarrow}& Th(\nu_1) \wedge Th(\nu_2) &\overset{Th(\hat e_1)\wedge Th(\hat e_2)}{\longrightarrow}& Th(V^{\mathcal{B}}_1) \wedge Th(V^{\mathcal{B}}_2) &\overset{\kappa_{k_1-n_1, k_2-n_2}}{\longrightarrow}& Th(V^{\mathcal{B}}_{k_1 + k_2 - n_1 - n_2}) } \,,$

which hence is equivalently the bottom composite, which in turn manifestly represents the product of the separate PT constructions in $\pi_\bullet(M\mathcal{B})$.

###### Theorem

The ring homomorphsim in lemma is an isomorphism.

Due to (Thom 54, Pontrjagin 55). See for instance (Kochmann 96, theorem 1.5.10).

###### Proof idea

Observe that given the result $\alpha \colon S^{n+(k-n)} \to Th(V_{k-n})$ of the Pontrjagin-Thom construction map, the original manifold $X \overset{i}{\hookrightarrow} \mathbb{R}^k$ may be recovered as this pullback:

$\array{ X &\overset{i}{\longrightarrow}& S^{n + (k-n)} \\ {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow^{\mathrlap{\alpha}} \\ B O(k-n) &\longrightarrow& Th(V^{B O}_{k-n}) } \,.$

To see this more explicitly, break it up into pieces:

$\array{ X &\longrightarrow& X_+ &\hookrightarrow& S^{n + (k-n)} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ X &\longrightarrow& X_+ \simeq Th(X) &\overset{Th(0)}{\longrightarrow}& Th(\nu_i) \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ B_{k-n} &\longrightarrow& (B_{k-n})_+ \simeq Th(B_{k-n}) &\underset{Th(0)}{\longrightarrow}& Th(V^{\mathcal{B}}_{k-n}) \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow \\ B O(k-n) &\longrightarrow& (B O(k-n))_+ \simeq Th(B O(k-n)) &\longrightarrow& Th(V^{B O}_{k-n}) } \,.$

Moreover, since the n-spheres are compact topological spaces, and since the classifying space $B O(n)$, and hence its universal Thom space, is a sequential colimit over relative cell complex inclusions, the right vertical map factors through some finite stage (by this lemma), the manifold $X$ is equivalently recovered as a pullback of the form

$\array{ X &\longrightarrow& S^{n + (k-n)} \\ {}^{\mathllap{g_i}}\downarrow &(pb)& \downarrow \\ Gr_{k-n}(\mathbb{R}^k) &\overset{i}{\longrightarrow}& Th( V_{k-n}(\mathbb{R}^k) \underset{O(k-n)}{\times} \mathbb{R}^{k-n}) } \,.$

(Recall that $V^{\mathcal{B}}_{k-n}$ is our notation for the universal vector bundle with $\mathcal{B}$-structure, while $V_{k-n}(\mathbb{R}^k)$ denotes a Stiefel manifold.)

The idea of the proof now is to use this property as the blueprint of the construction of an inverse $\zeta$ to $\xi$: given an element in $\pi_{n}(M \mathcal{B})$ represented by a map as on the right of the above diagram, try to define $X$ and the structure map $g_i$ of its normal bundle as the pullback on the left.

The technical problem to be overcome is that for a general continuous function as on the right, the pullback has no reason to be a smooth manifold, and for two reasons:

1. the map $S^{n+(k-n)} \to Th(V_{k-n})$ may not be smooth around the image of $i$;

2. even if it is smooth around the image of $i$, it may not be transversal to $i$, and the intersection of two non-transversal smooth functions is in general still not a smooth manifold.

The heart of the proof is in showing that for any $\alpha$ there are small homotopies relating it to an $\alpha'$ that is both smooth around the image of $i$ and transversal to $i$.

The first condition is guaranteed by Sard's theorem, the second by Thom's transversality theorem.

(…)

## References

### Pontrjagin-Thom construction

#### Pontrjagin’s construction

##### General

The Pontryagin theorem, i.e. the unstable and framed version of the Pontrjagin-Thom construction, identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge in unstable Borsuk-Spanier Cohomotopy sets, is due to:

(both available in English translation in Gamkrelidze 86),

as presented more comprehensively in:

The Pontrjagin theorem must have been known to Pontrjagin at least by 1936, when he announced the computation of the second stem of homotopy groups of spheres:

• Lev Pontrjagin, Sur les transformations des sphères en sphères (pdf) in: Comptes Rendus du Congrès International des Mathématiques – Oslo 1936 (pdf)

Review:

Discussion of the early history:

##### Twisted/equivariant generalizations

The (fairly straightforward) generalization of the Pontrjagin theorem to the twisted Pontrjagin theorem, identifying twisted Cohomotopy with cobordism classes of normally twisted-framed submanifolds, is made explicit in:

A general equivariant Pontrjagin theorem – relating equivariant Cohomotopy to normal equivariant framed submanifolds – remains elusive, but on free G-manifolds it is again straightforward (and reduces to the twisted Pontrjagin theorem on the quotient space), made explicit in:

• James Cruickshank, Thm. 5.0.6, Cor. 6.0.13 in: Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)
##### In negative codimension

In negative codimension, the Cohomotopy charge map from the Pontrjagin theorem gives the May-Segal theorem, now identifying Cohomotopy cocycle spaces with configuration spaces of points:

• Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)

• Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

c Generalization of these constructions and results is due to

• Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

• Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)

#### Thom’s construction

Thom's theorem i.e. the unstable and oriented version of the Pontrjagin-Thom construction, identifying cobordism classes of normally oriented submanifolds with homotopy classes of maps to the universal special orthogonal Thom space $M SO(n)$, is due to:

Textbook accounts:

#### Lashof’s construction

The joint generalization of Pontryagin 38a, 55 (framing structure) and Thom 54 (orientation structure) to any family of tangential structures (“(B,f)-structure”) is first made explicit in

and the general statement that has come to be known as Pontryagin-Thom isomorphism (identifying the stable cobordism classes of normally (B,f)-structure submanifolds with homotopy classes of maps to the Thom spectrum Mf) is Lashof 63, Theorem C.

Textbook accounts:

Lecture notes:

• John Francis, Topology of manifolds course notes (2010) (web), Lecture 3: Thom’s theorem (pdf), Lecture 4 Transversality (notes by I. Bobkova) (pdf)

• Cary Malkiewich, Section 3 of: Unoriented cobordism and $M O$, 2011 (pdf)

• Tom Weston, Part I of An introduction to cobordism theory (pdf)