Thom's theorem

> This entry is about Thom's theorem in [[cobordism theory]]. For the isomorphism in [[cohomology]] induced by [[Thom classes]] see at _[[Thom isomorphism]]_.

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###Context###
#### Cobordism theory
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[[!include cobordism theory -- contents]]
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#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

What is called _Thom's theorem_ or the _Pontrjagin-Thom isomorphism_ (due to [Thom 54](#Thom54), [Pontrjagin 55](#Pontrjagin55)) states that for a given universal [[tangential structure]] the [[homotopy group of a spectrum|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](Pontrjagin-Thom+collapse+map#ForEmbeddingsIntoAnNSphere)).

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
 {#Ingredients}

### Tangential structure on the Stable normal bundle

+-- {: .num_defn #ClassifyingMapOfNormalBundle}
###### 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 $X$:

$$
  N_i
    \coloneqq
  \left\{
    x\in X, v \in T_{i(x)}\mathbb{R}^k
    \;\vert\;
    v \,\perp\, i_\ast T_x X \subset T_{i(x)}\mathbb{R}^k
  \right\}
  \,.
$$

=--

+-- {: .num_defn #BfStructure}
###### Definition

A **[[tangential structure|(B,f)-structure]]** is

1. for each $n\in \mathbb{N}$ a [[pointed topological space|pointed]] [[CW-complex]] $B_n \in Top_{CW}^{\ast/}$

1. equipped with a pointed [[Serre fibration]]

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

   to the [[classifying space]] $B O(n)$ ([def.](classifying+space#EOn));

1. 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 [[commuting square|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. \ref{InclusionOfBOnIntoBOnPlusOne}.

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.](classifying+space#WhitneySumMapOnClassifyingSpaces))

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

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 $g_{n_1,n_2}$:

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

=--

+-- {: .num_example #ExamplesOfBfStructures}
###### Example

Examples of $(B,f)$-structures (def. \ref{BfStructure}) include the following:

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

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

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

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

Examples of $S^2$-$(B,f)$-structures (def. \ref{BfStructure}) 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]]**).

1. $B_{2n} = B Sp(2n) = E O(2n)/Sp(2n)$ 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. \ref{BfStructure}) 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]]**.

=--

+-- {: .num_defn #ManifoldWithBfStructure}
###### Definition

Given a [[smooth manifold]] $X$ of [[dimension]] $n$, and given a $(B,f)$-structure as in def. \ref{BfStructure}, 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 $k \in \mathbb{N}$;

1. a [[homotopy class]] of a  [[lift]] $\hat g$ of the classifying map $g$ of the normal bundle (def. \ref{ClassifyingMapOfNormalBundle})

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

The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_2, \hat g_2)$ if

1. $k_2 \geq k_1$

1. the second inclusion factors through the first as

   $$
     (i_X)_2
       \;\colon\;
     X
       \overset{(i_X)_1}{\hookrightarrow}
     \mathbb{R}^{k_1}
       \hookrightarrow
     \mathbb{R}^{k_2}
   $$

1. 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.](Introduction+to+Stable+homotopy+theory+--+S#BfStructure)).

+-- {: .num_defn #NegativeOfManifoldWithBfStructure}
###### Definition

Write $I \coloneqq [0,1]$ for the standard interval, regarded as a [[smooth manifold]] [[manifold with boundary|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.](Introduction+to+Stable+homotopy+theory+--+S#ExamplesOfBfStructures)) 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.](Introduction+to+Stable+homotopy+theory+--+S#BfStructure)).  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.](Introduction+to+Stable+homotopy+theory+--+S#ManifoldWithBfStructure)), 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 $t = 1$.

=--

+-- {: .num_defn #BordismRelation}
###### 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.](Introduction+to+Stable+homotopy+theory+--+S#ManifoldWithBfStructure)) 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. \ref{NegativeOfManifoldWithBfStructure}

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

=--

+-- {: .num_prop #BordismIsAnEquivalenceRelation}
###### Proposition

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

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

=--

+-- {: .num_prop #BordismGroupAndBordismRing}
###### Proposition

Under [[disjoint union]] of manifolds, then the set of $\mathcal{B}$-bordism equivalence classes of def. \ref{BordismIsAnEquivalenceRelation} 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.](Introduction+to+Stable+homotopy+theory+--+S#BfStructure)), 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](#Kochmann96))

### Thom spaces

+-- {: .num_defn #ThomSpace}
###### Definition

Let $X$ be a [[topological space]] and let $V \to X$ be a [[vector bundle]] over $X$ of [[rank]] $n$, which is [[associated bundle|associated]] to an [[orthogonal group|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. \ref{EOn}) 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]] $\leq 1$;

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

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

1. the **[[Thom space]]** $Th(V)$ is the [[cofiber]] (formed in [[Top]] ([prop.](Top#DescriptionOfLimitsAndColimitsInTop))) 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.

=--

+-- {: .num_remark #ThomSpaceForRankZeroBundle}
###### Remark

If the [[rank]] of $V$ is positive, then $S(V)$ is non-empty and then the Thom space (def. \ref{ThomSpace}) 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.

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

=--

+-- {: .num_prop #ThomSpaceOfDirectSumAsQuotientOfThomSpacesOfPullbacks}
###### Proposition

Let $V_1,V_2 \to X$ be two real [[vector bundles]]. Then the Thom space (def. \ref{ThomSpace}) 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}
###### Proof

Notice that

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

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

(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 $V_1$.)

=--

+-- {: .num_prop #SuspensionOfThomSpaces}
###### Proposition

For $V$ a [[vector bundle]] then the Thom space (def. \ref{ThomSpace}) of $\mathbb{R}^n \oplus V$, the [[direct sum of vector bundles]] with the trivial rank $n$ vector bundle, is [[homeomorphism|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}
###### Proof

Apply prop. \ref{ThomSpaceOfDirectSumAsQuotientOfThomSpacesOfPullbacks} 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

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

=--

+-- {: .num_example}
###### Example

By prop. \ref{SuspensionOfThomSpaces} and remark \ref{ThomSpaceForRankZeroBundle}  the Thom space (def. \ref{ThomSpace}) 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](Thom+spectrum#ForInfinityModuleBundles)_ for more on this.

=--

+-- {: .num_prop #ThomSpaceOfExternalProductOfVectorBundles}
###### 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. \ref{ThomSpace}) 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 $M G$

+-- {: .num_prop #PullbackOfUniversalOnBundleUnderCoordinateRestriction}
###### 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.](classifying+space#InclusionOfBOnIntoBOnPlusOne)).

=--

(e.g. [Kochmann 96, p. 25](#Kochmann96))

+-- {: .proof}
###### 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.](classifying+space#InclusionOfBOnIntoBOnPlusOne)).

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.](Introduction+to+Stable+homotopy+theory+--+P#kTop)). 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. \ref{CWComplexStructure}. Together this implies (via [this lemma](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes)) 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.

=--

+-- {: .num_defn #UniversalThomSpectrum}
###### Definition

The **universal real [[Thom spectrum]]** $M O$ is the [[spectrum]], which is represented by the [[sequential prespectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1#SequentialSpectra)) whose $n$th component space is the [[Thom space]] (def. \ref{ThomSpace})

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

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. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction}, via the homeomorphisms of prop. \ref{SuspensionOfThomSpaces}:

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

=--

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.:

+-- {: .num_defn #VectorBundleAssociatedWithBfStructure}
###### Definition

Given a [[(B,f)-structure]] $\mathcal{B}$ (def. \ref{BfStructure}),  write $V^\mathcal{B}_n$ for the [[pullback]] of the [[universal vector bundle]] (def. \ref{EOn}) 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. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction} still holds:

+-- {: .num_prop #PullbackOfUniversalBfBundleUnderCoordinateRestriction}
###### Proposiiton

Given a [[(B,f)-structure]] $\mathcal{B}$ (def. \ref{BfStructure}), then the pullback of its rank-$(n+1)$ vector bundle $V^{\mathcal{B}}_{n+1}$ (def. \ref{VectorBundleAssociatedWithBfStructure}) 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}
###### 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. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction}.

=--

+-- {: .num_defn #UniversalThomSpectrumForBfStructure}
###### Definition

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

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

and with structure maps given via prop. \ref{SuspensionOfThomSpaces} by the top maps in prop. \ref{PullbackOfUniversalBfBundleUnderCoordinateRestriction}:

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

Similarly for an $S^k-(B,f)$-structure indexed on every $k$th natural number (such as [[almost complex structure]], [[almost quaternionic structure]], example \ref{ExamplesOfBfStructures}), there is the corresponding Thom spectrum as a sequential $S^k$ spectrum ([def.](Introduction+to+Stable+homotopy+theory+--+1#SequentialTSpectra)).

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. \ref{BfStructure}), 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.](Thom+space#ThomSpaceOfExternalProductOfVectorBundles)) maps

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

which are [[associativity|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 \ref{ThomSpaceForRankZeroBundle}.

=--

+-- {: .num_example}
###### Example

The universal [[Thom spectrum]] (def. \ref{UniversalThomSpectrumForBfStructure}) for _[[framing]] structure_ ([exmpl.](tangential+structure#ExamplesOfBfStructures)) is equivalently the [[sphere spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1#StandardSphereSpectrum))

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

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

+-- {: .num_defn #TubularNeighbourhood}
###### 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. \ref{ClassifyingMapOfNormalBundle}) given by

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

is a [[diffeomorphism]].

=--

+-- {: .num_prop}
###### Proposition
**([[tubular neighbourhood theorem]])**

For every [[embedding]] of [[smooth manifolds]], there exists a [[tubular neighbourhood]] according to def. \ref{TubularNeighbourhood}.

=--

+-- {: .num_remark #IngredientsOfPontrjaginThomConstruction}
###### Remark

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

1. the [[Thom space]] (def. \ref{ThomSpace}) of the [[normal bundle]] (def. \ref{ClassifyingMapOfNormalBundle}) is [[homeomorphism|homeomorphic]] to the [[quotient topological space]] of the [[topological closure]] of the tubular neighbourhood by its [[boundary]]:

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

1. 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.

=--

+-- {: .num_defn #PontrjaginThomConstruction}
###### 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. \ref{TubularNeighbourhood}) 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 [[n-sphere|k-sphere]] as the [[one-point compactification]] of $\mathbb{R}^k$; and where the second and third maps are those of remark \ref{IngredientsOfPontrjaginThomConstruction}.

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. \ref{ClassifyingMapOfNormalBundle}).

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.](Introduction+to+Stable+homotopy+theory+--+1#StableHomotopyGroups)) of the [[Thom spectrum]] $M O$ (def. \ref{UniversalThomSpectrum}).

More generally, for $X$ a smooth manifold with normal [[(B,f)-structure]] $(X,i,\hat g_i)$ according to def. \ref{ManifoldWithBfStructure}, 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

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

=--

+-- {: .num_prop #PontrjaginThomConstructionGivesWellDefinedStableHomotopyGroup}
###### Proposition

The [[Pontrjagin-Thom construction]] (def. \ref{PontrjaginThomConstruction}) respects the equivalence classes entering the definition of manifolds with stable normal $\mathcal{B}$-structure (def. \ref{ManifoldWithBfStructure}) 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}
###### Proof

It is clear that the homotopies of classifying maps of $\mathcal{B}$-structures that are devided out in def. \ref{ManifoldWithBfStructure} 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. \ref{PontrjaginThomConstruction}). 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. \ref{PullbackOfUniversalBfBundleUnderCoordinateRestriction} 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. \ref{SuspensionOfThomSpaces}) 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. \ref{UniversalThomSpectrumForBfStructure}):

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

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.](Introduction+to+Stable+homotopy+theory+--+1#StableHomotopyGroups)). 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
 {#ThomTheorem}

Recall that the [[Pontrjagin-Thom construction]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#PontrjaginThomConstruction)) associates to an embbeded manifold $(X,i,\hat g)$ with normal $\mathcal{B}$-structure ([def.](Introduction+to+Stable+homotopy+theory+--+S#ManifoldWithBfStructure)) 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.

+-- {: .num_lemma #PontrjaginThomIsRingHomomorphims}
###### Lemma

For $\mathcal{B}$ be a multiplicative [[(B,f)-structure]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#BfStructure)),
the $\mathcal{B}$-[[Pontrjagin-Thom construction]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#PontrjaginThomConstruction)) 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. \ref{BordismGroupAndBordismRing}) 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.](Introduction+to+Stable+homotopy+theory+--+S#UniversalThomSpectrumForBfStructure)).

=--

+-- {: .proof}
###### Proof

By prop. \ref{PontrjaginThomConstructionGivesWellDefinedStableHomotopyGroup} the underlying function of sets is well-defined before dividing out the bordism relation (def. \ref{BordismRelation}). 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;

1. 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 [[n-cube|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.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyGroupsOftopologicalSpaces)). 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 \ref{ThomSpaceForRankZeroBundle}) 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. \ref{UniversalThomSpectrumForBfStructure}) 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. \ref{ThomSpaceOfExternalProductOfVectorBundles} 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 $\pi_\bullet(M\mathcal{B})$.

=--

+-- {: .num_theorem #MorphismFromCobordismRingToStableHomotopyGroupsOfThomSpectrumIsIso}
###### Theorem

The ring homomorphsim in lemma \ref{PontrjaginThomIsRingHomomorphims} is an [[isomorphism]].

=--

Due to ([Thom 54](#Thom54), [Pontrjagin 55](#Pontrjagin55)). See for instance ([Kochmann 96, theorem 1.5.10](#Kochmann96)).

+-- {: .proof}
###### 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](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes)), 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 $i$;

1. even if it is smooth around the image of $i$, it may not be [[transversal map|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]].

(...)

=--

## Related entries

* [[Pontryagin-Thom collapse map]]

* [[Thom's transversality theorem]]

* [[Cohomotopy cohomology theory]], [[Cohomotopy charge]]

## References

[[!include Pontryagin-Thom construction -- references]]

[[!redirects Thom theorem]]

[[!redirects Pontrjagin-Thom theorem]]
[[!redirects Pontrjagin-Thom isomorphism]]

[[!redirects Pontryagin-Thom theorem]]
[[!redirects Pontryagin-Thom isomorphism]]


Thom's transversality theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .un_theorem}
###### Theorem

Let $P$ be a [[smooth manifold]] and let $i \colon M \hookrightarrow E$ be a smooth [[submanifold]]. Then the [[topological subspace]]

$$
  C^{\infty}_{tr i}(P,E) \subset C^\infty(P,E)
$$

of [[smooth functions]] that are [[transverse maps]] to $i$ is a [[dense subspace]].

=--

This is due to [Thom 54](#Thom54), see e.g. [Kosinski 93, IV (2.5)](#Kosinski93).

## Relate entries

* [[Thom's theorem]]

* [[Sard's theorem]]

## References

The result is due to (Theorem I.5?)

* {#Thom54} [[René Thom]], _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

Textbook account:

* {#Kosinski93} [[Antoni Kosinski]], chapter IV.2 of _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

Lecture notes:

* [[John Francis]], _Topology of manifolds_ course notes (2010) ([web](http://math.northwestern.edu/~jnkf/classes/mflds/))

  {#Francis10Lecture4} Lecture 4 _Transversality_ (notes by I. Bobkova) ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf))


Thom-Gysin sequence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Thom-Gysin sequence_ is a type of [[long exact sequence in cohomology]] induced by a [[spherical fibration]] and expressing the [[cohomology groups]] of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the [[Serre spectral sequence]] for the given fibration. It induces, and is induced by, the [[Thom isomorphism]].

## Statement
 {#Statement}

+-- {: .num_prop #ThomGysinSequence}
###### Proposition

Let $R$ be a [[commutative ring]] and let

$$
  \array{
    S^n &\longrightarrow& E
    \\
    && \downarrow^{\mathrlap{\pi}}
    \\
    && B
  }
$$

be a [[Serre fibration]] over a [[simply connected topological space|simply connected]] [[CW-complex]] with typical [[fiber]] ([exmpl.](Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations)) the [[n-sphere]].

Then there exists an element $c \in H^{n+1}(B; R)$ (in the [[ordinary cohomology]] of the total space with [[coefficients]] in $R$, called the _[[Euler class]]_ of $\pi$) such that the [[cup product]] operation $c \cup (-)$ sits in a [[long exact sequence]] of [[cohomology groups]] of the form

$$
  \cdots
   \to
   H^k(B; R)
     \stackrel{\pi^\ast}{\longrightarrow}
   H^k(E; R)
    \stackrel{}{\longrightarrow}
   H^{k-n}(B;R)
   \stackrel{c \cup (-)}{\longrightarrow}
   H^{k+1}(B; R)
   \to
   \cdots
  \,.
$$

=--

(e.g. [Switzer 75, section 15.30](#Switzer75), [Kochman 96, corollary 2.2.6](#Kochmann96))

+-- {: .proof #ProofOfThomGysinSequence}
###### Proof

Under the given assumptions there is the corresponding [[Serre spectral sequence]]

$$
  E_2^{s,t}
    \;=\;
  H^s(B; H^t(S^n;R))
    \;\Rightarrow\;
  H^{s+t}(E; R)
  \,.
$$

Since the [[ordinary cohomology]] of the [[n-sphere]] [[fiber]] is concentrated in just two degees

$$
  H^t(S^n; R)
  =
  \left\{
    \array{
      R & for \; t= 0 \; and \; t = n
      \\
      0 & otherwise
    }
  \right.
$$

the only possibly non-vanishing terms on the $E_2$ page of this spectral sequence, and hence on all the further pages, are in bidegrees $(\bullet,0)$ and $(\bullet,n)$:

$$
  E^{\bullet,0}_2 \simeq H^\bullet(B; R)
  \,,
  \;\;\;\;
  and
  \;\;\;
  E^{\bullet,n}_2 \simeq H^\bullet(B; R)
  \,.
$$

As a consequence, since the differentials $d_r$ on the $r$th page of the Serre spectral sequence have bidegree $(r+1,-r)$, the only possibly non-vanishing differentials are those on the $(n+1)$-page of the form

$$
  \array{
     E_{n+1}^{\bullet,n} & \simeq & H^\bullet(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow
     \\
     E_{n+1}^{\bullet+n+1,0} & \simeq &  H^{\bullet+n+1}(B;R)
  }
  \,.
$$

Now since the [[coefficients]] $R$ is a [[ring]], the [[Serre spectral sequence]] is [[multiplicative spectral sequence|multiplicative]] under [[cup product]] and the [[differential]] is a [[derivation]] (of total degree 1) with respect to this product. (See at _[multiplicative spectral sequence -- Examples -- AHSS for multiplicative cohomology](multiplicative+spectral+sequence#AHSSForMultiplicativeCohomology)_.)

To make use of this, write

$$
  \iota \coloneqq 1 \in H^0(B;R) \stackrel{\simeq}{\longrightarrow} E_{n+1}^{0,n}
$$

for the unit in the [[cohomology ring]] $H^\bullet(B;R)$, but regarded as an element in bidegree $(0,n)$ on the $(n+1)$-page of the spectral sequence. (In particular $\iota$ does _not_ denote the unit in bidegree $(0,0)$, and hence $d_{n+1}(\iota)$ need not vanish; while by the [[derivation]] property, it does vanish on the actual unit $1 \in H^0(B;R) \simeq E_{n+1}^{0,0}$.)

Write

$$
  c
    \coloneqq
  d_{n+1}(\iota)
   \;\;
  \in E_{n+1}^{n+1,0}
    \stackrel{\simeq}{\longrightarrow}
  H^{n+1}(B; R)
$$

for the image of this element under the differential. We will show that this is the Euler class in question.

To that end, notice that every element in $E_{n+1}^{\bullet,n}$ is of the form $\iota \cdot b$ for $b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R)$.

(Because the [[multiplicative spectral sequence|multiplicative structure]] gives a group homomorphism $\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R)$, which is an isomorphism because the product in the spectral sequence does come from the [[cup product]] in the [[cohomology ring]], see for instance [Kochman 96, first equation in the proof of prop. 4.2.9](#Kochmann96), and since hence $\iota$ does act like the unit that it is in $H^\bullet(B;R)$).

Now since $d_{n+1}$ is a graded [[derivation]] and vanishes on $E_{n+1}^{\bullet,0}$ (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with $c$:

$$
  \begin{aligned}
    d_{n+1}(\iota \cdot b)
      & =
    d_{n+1}(\iota) \cdot b + (-1)^{n}\, \iota \cdot \underset{= 0}{\underbrace{d_{n+1}(b)}}
    \\
    & = c \cdot b
  \end{aligned}
  \,.
$$

This shows that $d_{n+1}$ is identified with the cup product operation in question:

$$
  \array{
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
  }
  \,.
$$

In summary, the non-vanishing entries of the $E_\infty$-page of the spectral sequence sit in [[exact sequences]] like so

$$
  \array{
     0
     \\
     \downarrow
     \\
     E_\infty^{s,n}
     \\
     {}^{\mathllap{ker(d_{n+1})}}\downarrow
     \\
     E_{n+1}^{s,n} & \simeq & H^s(B;R)
     \\
     {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}}
     \\
     E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R)
     \\
     {}^{\mathllap{coker(d_{n+1})}}\downarrow
     \\
     E_\infty^{s+n+1,0}
     \\
     \downarrow
     \\
     0
  }
  \,.
$$

Finally observe (lemma \ref{ImplicationsOfSparesnessOfSSSForSphericalFibration}) that due to the sparseness of the $E_\infty$-page, there are also [[short exact sequences]] of the form

$$
  0 \to E_\infty^{s,0} \longrightarrow H^s(E; R) \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
$$

Concatenating these with the above exact sequences yields the desired [[long exact sequence]].

=--

+-- {: .num_lemma #ImplicationsOfSparesnessOfSSSForSphericalFibration}
###### Lemma

Consider a cohomology [[spectral sequence]] converging to some [[filtered object|filtered]] [[graded abelian group]] $F^\bullet C^\bullet$ such that

1. $F^0 C^\bullet = C^\bullet$;

1. $F^{s} C^{\lt s} = 0$;

1. $E_\infty^{s,t} = 0$ unless $t = 0$ or $t = n$,

for some $n \in \mathbb{N}$, $n \geq 1$. Then there are [[short exact sequences]] of the form

$$
  0 \to E_\infty^{s,0} \overset{}{\longrightarrow} C^s \longrightarrow  E_\infty^{s-n,n} \to 0
  \,.
$$

=--

(e.g. [Switzer 75, p. 356](#Switzer75))

+-- {: .proof}
###### Proof

By definition of convergence of a spectral sequence, the $E_{\infty}^{s,t}$ sit in [[short exact sequences]] of the form

$$
  0 \to F^{s+1}C^{s+t} \overset{i}{\longrightarrow} F^s C^{s+t} \longrightarrow E_\infty^{s,t} \to 0
  \,.
$$

So when $E_\infty^{s,t} = 0$ then the morphism $i$ above is an [[isomorphism]].

We may use this to either shift away the filtering degree

* if $t \geq n$ then $F^s C^{s+t} = F^{(s-1)+1}C^{(s-1)+(t+1)} \underoverset{\simeq}{i^{s-1}}{\longrightarrow} F^0 C^{(s-1)+(t+1)} = F^0 C^{s+t} \simeq C^{s+t}$;

or to shift away the offset of the filtering to the total degree:

* if $0 \leq t-1 \leq n-1$ then $F^{s+1}C^{s+t} = F^{s+1}C^{(s+1)+(t-1)}
  \underoverset{\simeq}{i^{-(t-1)}}{\longrightarrow} F^{s+t}C^{(s+1)+(t-1)} = F^{s+t}C^{s+t}$

Moreover, by the assumption that if $t \lt 0$ then $F^{s}C^{s+t} = 0$, we also get

$$
  F^{s}C^{s} \simeq E_\infty^{s,0}
  \,.
$$

In summary this yields the vertical isomorphisms

$$
  \array{
  0
    &\to&
  F^{s+1}C^{s+n}
    &\longrightarrow&
  F^{s}C^{s+n}
    &\longrightarrow&
  E_\infty^{s,n}
    &\to&
  0
  \\
  &&
    {}^{\mathllap{i^{-(n-1)}}}\downarrow^{\mathrlap{\simeq}}
  &&
    {}^{\mathllap{i^{s-1}}}\downarrow^{\mathrlap{\simeq}}
  &&
    \downarrow^{\mathrlap{=}}
  \\
  0
    &\to&
  F^{s+n}C^{s+n}
  \simeq
  E_\infty^{s+n,0}
    &\longrightarrow&
  C^{s+n}
    &\longrightarrow&
  E_\infty^{s,n}
    &\to&
  0
  }
$$

and hence with the top sequence here being exact, so is the bottom sequence.

=--

## Related concepts

* [[Gysin map]]

* [[Serre long exact sequence]]

* [[Serre spectral sequence]]

* [[long exact sequence in cohomology]]

  * [[long exact sequence in generalized cohomology]]

## References

* {#Switzer75} [[Robert Switzer]], section 15.30 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 ([doi:10.1007/978-3-642-61923-6](https://link.springer.com/book/10.1007/978-3-642-61923-6))

* [[James Milne]], section 16 of _[[Lectures on Étale Cohomology]]_

See also:

* Wikipedia, _[Gysin sequence](http://en.wikipedia.org/wiki/Gysin_sequence)_

In the generality of [[complex oriented cohomology theory]]:

* {#Kochmann96} [[Stanley Kochmann]], section 2.2. of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Dai Tamaki]], [[Akira Kono]], Section 3.8 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Formalization in [[homotopy type theory]]:

* {#Brunerie16} [[Guillaume Brunerie]], _On the homotopy groups of spheres in homotopy type theory_ ([arXiv:1606.05916](http://arxiv.org/abs/1606.05916))

Applications:

* [[Martin Saralegi]], _A Gysin Sequence for Semifree Actions of $S^3$_, Proceedings of the American Mathematical Society Vol. 118, No. 4 (Aug., 1993), pp. 1335-1345 ([jstor](http://www.jstor.org/stable/2160096))

[[!redirects Thom-Gysin sequences]]

[[!redirects Thom-Gysin exact sequence]]
[[!redirects Thom-Gysin exact sequences]]

[[!redirects Gysin sequence]]
[[!redirects Gysin sequences]]

[[!redirects Gysin exact sequence]]
[[!redirects Gysin exact sequences]]


Thomas Appelquist

* [institute page](https://physics.yale.edu/people/thomas-appelquist)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Thomas_Appelquist)

## Selected writings

On the [[infinite-temperature thermal field theory]] of [[Yang-Mills theory]]:

* {#AppelquistPisarski81} [[Thomas Appelquist]], [[Robert Pisarski]], _High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics_, Phys. Rev. D **23** 2305 (1981) [[doi:10.1103/PhysRevD.23.2305](https://doi.org/10.1103/PhysRevD.23.2305)]

On [[Kaluza-Klein theory]]:

* [[Thomas Appelquist]], A. Chodos, [[Peter Freund]], *Modern Kaluza-Klein Theories*, Frontiers in Physics, Addison-Wesley (1987) [[spire:255996](https://inspirehep.net/literature/255996)]

category: people


Thomas Athorne

* [Homepage](http://thomasathorne.nfshost.com/)

category:people


Thomas Buchert

* [webpage](http://www.cosmunix.de/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=105890)

## Selected writings

On the [[standard model of cosmology]]:

* {#BCKRW15} [[Thomas Buchert]], Alan A. Coley, [[Hagen Kleinert]], Boudewijn F. Roukema, David L. Wiltshire, _Observational Challenges for the Standard FLRW Model_, Int. J. Mod. Phys. D 25, 1630007 (2016) ([arXiv:1512.03313](https://arxiv.org/abs/1512.03313))

On [[inhomogeneous cosmology]]:

* {#BuchertEhlers95} [[Thomas Buchert]], Juergen Ehlers, _Averaging inhomogeneous Newtonian cosmologies_, Astron. Astrophys.320:1-7, 1997 ([arXiv:astro-ph/9510056](https://arxiv.org/abs/astro-ph/9510056))

* {#Buchert00} [[Thomas Buchert]], _On average properties of inhomogeneous cosmologies_, Gen.Rel.Grav.9:306-321, 2000 ([arXiv:gr-qc/0001056](https://arxiv.org/abs/gr-qc/0001056))

* {#BuchertLarenaAlimi06} [[Thomas Buchert]], Julien Larena, Jean-Michel Alimi, _Correspondence between kinematical backreaction and scalar field cosmologies - the 'morphon field'_, Class. Quant. Grav.23:6379-6408, 2006 ([arXiv:gr-qc/0606020](https://arxiv.org/abs/gr-qc/0606020))

* {#Buchert07} [[Thomas Buchert]], _Dark Energy from structure: a status report_, Gen.Rel.Grav.40:467-527, 2008 ([arXiv:0707.2153](http://xxx.lanl.gov/abs/0707.2153))

* {#BuchertRasanen11} [[Thomas Buchert]], [[Syksy Räsänen]], _Backreaction in late-time cosmology_, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 ([arXiv:1112.5335](https://arxiv.org/abs/1112.5335))

* {#Buchert11} [[Thomas Buchert]], _Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects_, Class.Quant.Grav.28:164007, 2011 ([arXiv:1103.2016](https://arxiv.org/abs/1103.2016))

* {#BuchertEtAl15} [[Thomas Buchert]] et. al, _Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?_, Class. Quantum Grav. 32 215021, 2015 ([arXiv:1505.07800](https://arxiv.org/abs/1505.07800))

  exposition in _[The Universe is inhomogeneous. Does it matter?](https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/)_ CQG+, 2016

* Léo Brunswic, [[Thomas Buchert]], _Gauss-Bonnet-Chern approach to the averaged Universe_, Class. Quantum Grav. 37 (2020) 215022 ([arXiv:2002.08336](https://arxiv.org/abs/2002.08336))

* {#HeinesenBuchert20} Asta Heinesen, [[Thomas Buchert]], _Solving the curvature and Hubble parameter inconsistencies through structure formation-induced curvature_, Class. Quantum Grav. 37 (2020) 164001 ([arXiv:2002.10831](https://arxiv.org/abs/2002.10831))

  p. 14:

  > a natural and consistent explanation of

  > (i) dark energy,

  > (ii) the coincidence problem (here conceptually,not quantitatively),

  > (iii) positive initial curvature,

  > (iv) the small matter density cosmological parameterfound in local probes of the matter density,

  > (v) the large angular diameter distance tothe CMB consistent with JLA supernova sample parameter constraints,

  > and (vi) the local expansion rate measurements (removal of the 'Hubble tension').

  > We believe that this model architecture needs convincing arguments to be rejected as a physically viable show-case, on the basis of which the model ingredients can be improved in order to build a physical cosmology in the future.

Applying [[topological data analysis]]/[[persistent homology]] to the [[CMB]]:

* Pratyush Pranav, [[Thomas Buchert]], *Homology reveals significant anisotropy in the cosmic microwave background* ([arXiv:2308.10738](https://arxiv.org/abs/2308.10738))

## Related entries

* [[inhomogeneous cosmology]]

* [[standard model of cosmology]]

category: people


Thomas Buscher

* [Mathematics Generalogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=99344)

## related $n$Lab entries

* [[T-duality]]

* [[Buscher rules]]

category: people

[[!redirects Thomas Henry Buscher]]


Thomas Church

* [personal page](http://math.stanford.edu/~church/)

* [MathematicsGenealogy page](https://mathgenealogy.org/id.php?id=18914)

## Selected writings

Introducing the notion of *[[FI-modules]]* ([[FinInj|$FinSet_{inj}$]]-[[representations]]):

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]

and the notion of [[representation stability]]:

* {#ChurchFarb13} [[Thomas Church]], [[Benson Farb]], *Representation theory and homological stability*, Advances in Mathematics
**245** (2013) 250-314 [[doi:10.1016/j.aim.2013.06.016](https://doi.org/10.1016/j.aim.2013.06.016)]

category: people


Thomas Creutzig

* [personal page](https://sites.ualberta.ca/~creutzig/)

## Selected writings

On [[logarithmic CFT]]:

*  [[Thomas Creutzig]], [[David Ridout]], *Logarithmic conformal field theory: beyond an introduction*,  J. Phys. A: Math. Theor. **46** (2013) 494006 ([doi:10.1088/1751-8113/46/49/494006](https://iopscience.iop.org/article/10.1088/1751-8113/46/49/494006), [arXiv:1303.0847](https://arxiv.org/abs/1303.0847))

On [[fractional-level WZW models]] as [[logarithmic CFT]]:

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models I*, Nuclear Physics B **865** 1 (2012) 83-114 $[$[arXiv:1205.6513](https://arxiv.org/abs/1205.6513), [doi:10.1016/j.nuclphysb.2012.07.018](https://doi.org/10.1016/j.nuclphysb.2012.07.018)$]$

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models II*, Nuclear Physics B **875** 2 (2013) 423-458 $[$[arXiv:1306.4388](https://arxiv.org/abs/1306.4388), [doi:10.1016/j.nuclphysb.2013.07.008](https://doi.org/10.1016/j.nuclphysb.2013.07.008)$]$

On the [[Verlinde formula]] for [[logarithmic CFTs]]:

* [[Thomas Creutzig]], [[Terry Gannon]], _Logarithmic conformal field theory, log-modular tensor categories and modular forms_, J. Phys. A __50__, 404004 (2017) ([doi:10.1088/1751-8121/aa8538](https://doi.org/10.1088/1751-8121/aa8538), [arXiv:1605.04630](https://arxiv.org/abs/1605.04630))

On [[braided fusion categories]] formed by [[affine Lie algebra]]-[[representations]] at admissible fractional [[level (Chern-Simons theory)|level]]:

* [[Thomas Creutzig]], [[Yi-Zhi Huang]], [[Jinwei Yang]], *Braided tensor categories of admissible modules for affine Lie algebras*, Commun. Math. Phys. **362** (2018) 827–854 [[arXiv:1709.01865](https://arxiv.org/abs/1709.01865)]

On [[braided monoidal category|braided]] [[tensor categories]] arising as [[representation categories]] of [[logarithmic CFTs|logarithmic]] [[vertex operator algebras]]:

* [[Thomas Creutzig]], [[Simon Lentner]], [[Matthew Rupert]], *Characterizing braided tensor categories associated to logarithmic vertex operator algebras* ([arXiv:2104.13262](https://arxiv.org/abs/2104.13262))

category: people


Thomas Curtright

Thomas L. Curtright is professor for theoretical physics at the University of Miami.

* [webpage](http://www.physics.miami.edu/~curtright/home.html)

* [wikipedia entry](https://en.wikipedia.org/wiki/Thomas_Curtright)

## Selected writings

On [[conformal field theory|conformal]] [[quantization]] of [[Liouville theory]]:

* [[Thomas L. Curtright]], [[Charles B. Thorn]], *Conformally Invariant Quantization of the Liouville Theory*, Phys. Rev. Lett. **48** (1982) 1309; Erratum Phys. Rev. Lett. **48**,  (1982) 1768 [[doi:10.1103/PhysRevLett.48.1309](https://doi.org/10.1103/PhysRevLett.48.1309)]

On [[Nambu mechanics]]:

* [[Thomas L. Curtright]], [[Cosmas K. Zachos]], _Classical and Quantum Nambu Mechanics_, Phys. Rev. D **68** 085001 (2003) [[arXiv:hep-th/0212267](http://arxiv.org/abs/hep-th/0212267), [doi:10.1103/PhysRevD.68.085001](https://doi.org/10.1103/PhysRevD.68.085001)]

On [[Weyl quantization]]:

* [[Thomas L. Curtright]], [[David B. Fairlie]], [[Cosmas K. Zachos]], *A Concise Treatise on Quantum Mechanics in Phase Space*, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]

## Related entries

* [[dual graviton]]

* [Curtright field](https://en.wikipedia.org/wiki/Curtright_field)

category: people

[[!redirects Thomas L. Curtright]]


Thomas Dove

* [institute page](https://ecampus.uni-goettingen.de/sb/rds;jsessionid=58BEFA402F45405DC3A1A35CFC4D5D11.s46?state=verpublish&status=init&vmfile=no&moduleCall=webInfo&publishConfFile=webInfoPerson&publishSubDir=personal&keep=y&purge=y&personal.pid=141615)

## Selected writings

On [[Tate K-theory]] and [[equivariant elliptic cohomology]]:

* {#Dove19} [[Thomas Dove]], _Twisted Equivariant Tate K-Theory_ ([arXiv:1912.02374](https://arxiv.org/abs/1912.02374))

On [[topological T-duality]] in the [[equivariant cohomology|equivariant]] case, as an equivalence of [[twisted equivariant K-theory]] groups:

* [[Tom Dove]], [[Thomas Schick]], *Equivariant Topological T-Duality* [[arXiv:2310.06064](https://arxiv.org/abs/2310.06064)]

category: people

[[!redirects Tom Dove]]


Thomas Dubé

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=88012)

## Selected writings

On [[exact real computer arithmetic]]:

* [[Chee-Keng Yap]], [[Thomas Dubé]], *The exact computation paradigm*,  in: *Computing in Euclidean Geometry*, Lecture Notes Series on Computing, World Scientific (1995) 452-492  [[doi:10.1142/9789812831699_0011](https://doi.org/10.1142/9789812831699_0011), [[YapDube-ExactComputationParadigm.pdf:file]]]

category: people

[[!redirects Thomas Dube]]


Thomas Dumitrescu

* [webpage](https://hetg.physics.harvard.edu/people/thomas-dumitrescu)

## Selected writings

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] in [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]] from a [[higher gauge theory]]-perspective:

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], _2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories_ ([arXiv:2009.00138](https://arxiv.org/abs/2009.00138))

On [[generalized global symmetries]]:

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], [[Shu-Heng Shao]], *Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond* [[arXiv:2205.09545](https://arxiv.org/abs/2205.09545)]

## Related entries

* [[higher gauge theory]]

* [[Green-Schwarz mechanism]]

category: people

[[!redirects Thomas T. Dumitrescu]]


Thomas Ehrhard

Thomas Ehrhard is a theoretical computer scientist and logician based in [[IRIF]]. He is mainly interested in logic, and especially  Proof Theory and its connections with Computer Science through the Curry-Howard Correspondence.

* [Webpage](https://www.irif.fr/~ehrhard/)

## Selected writings

On [[differential linear logic]]:

* {#EhrhardtRegnier09} [[Thomas Ehrhard]], [[Laurent Regnier]]: _The differential lambda-calculus_, Theor. Comput. Sci. **309** 1 (2003) 1–41  (doi:10.1016/S0304-3975(03)00392-X)

* [[Thomas Ehrhard]], [[Laurent Regnier]]: _Differential interaction nets,_ Theor. Comput. Sci. **364** 2 (2006) 166–195 ([doi:10.1016/j.tcs.2006.08.003](https://doi.org/10.1016/j.tcs.2006.08.003))

* [[Thomas Ehrhard]], *An introduction to Differential Linear Logic: proof-nets, models and antiderivatives*, Mathematical Structure in Computer Science **28** 7 (2018) 995-1060 ([arXiv:1606.01642](https://arxiv.org/abs/1606.01642), [doi:10.1017/S0960129516000372](https://doi.org/10.1017/S0960129516000372))

## Related $n$Lab entries

* [[differential linear logic]]

* [[differential category]]

[[!redirects T. Ehrhard]]

category: people


Thomas Faulkner

* [Institute page](https://physics.illinois.edu/people/directory/profile/tomf)

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Thomas_Faulkner_(physicist))

## Selected writings

On [[holographic entanglement entropy]] formulated via [[local nets of observables]] in [[algebraic quantum field theory]]:

* [[Thomas Faulkner]], _The holographic map as a conditional expectation_ ([arXiv:2008.04810](https://arxiv.org/abs/2008.04810))

category: people


Thomas Fiore

* [homepage](http://www-personal.umd.umich.edu/~tmfiore/)

## Selected writings

On a [[Thomason model structure]] on [[n-categories]]:

* [[Thomas M. Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*, Algebr. Geom. Topol. **10** 4 (2010) 1933-2008 [[arXiv:0808.4108](http://arxiv.org/abs/0808.4108), [doi:10.2140/agt.2010.10.1933](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-10/issue-4/A-Thomason-model-structure-on-the-category-of-small-nfold/10.2140/agt.2010.10.1933.full)]

On [[model structures on DblCat]]:

* {#FPPModel} [[Thomas M. Fiore]], [[Simona Paoli]], [[Dorette A. Pronk]], *Model structures on the category of small double categories*, _Algebraic & Geometric Topology_ **8** (2008) 1855–1959 [[doi:10.2140/agt.2008.8.1855](https://doi.org/10.2140/agt.2008.8.1855)]

* {#FPThomason} [[Thomas Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*_Algebraic & Geometric Topology_ **10** (2010) 1933–2008 [[doi:10.2140/agt.2010.10.1933](http://doi.org/10.2140/agt.2010.10.1933)]

On [[Euler characteristics]] of [[categories]] and [[homotopy colimits]]:

* [[Thomas M. Fiore]], [[Wolfgang Lück]], [[Roman Sauer]], *Euler Characteristics of Categories and Homotopy Colimits* [[arXiv:1007.3868](http://arxiv.org/abs/1007.3868)]

See also:

* Thomas M. Fiore, Nicola Gambino, and Joachim Kock. Monads in Double Categories.
To appear in the _Journal of Pure and Applied Algebra_. 31 pages.
[arXiv:1006.0797](http://arxiv.org/abs/1006.0797)

* Thomas M. Fiore, Wolfgang Lück, and Roman Sauer. Finiteness obstructions and Euler characteristics of categories. 87 pages. [arxiv:0908.3417](http://arxiv.org/abs/0908.3417)

* Thomas M. Fiore and Igor Kriz. What is the Jacobian of a Riemann Surface with Boundary?. _Proceedings of the 2007 Max-Planck Workshop on Hochschild and Cyclic (co-)homology and Applications to Geometry and Physics_, Aspects of Mathematics, Viehweg, Braunschweig. 27 pages. [arXiv:math/0610463](http://arxiv.org/abs/math.CT/0610463)

* Thomas M. Fiore, Simona Paoli, and Dorette Pronk. Model Structures on the Category of Small Double Categories. _Algebraic and Geometric Topology_, Volume 8, Number 4, pages 1855--1959, 2008. 103 pages.

* Thomas M. Fiore, Po Hu, and Igor Kriz. Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras. _Advances in Mathematics_, Volume 218, Number 6, pages 1705-1722, 2008. 17 pages.

* Alissa Crans, Thomas M. Fiore, and Ramon Satyendra.  Musical Actions of Dihedral Groups. _American Mathematical Monthly_, Volume 116, Number 6, June-July 2009, pp. 479-495. 17 pages.

* Thomas M. Fiore. Pseudo Algebras and Pseudo Double Categories. _Journal of Homotopy and Related Structures_, Volume 2, Number 2, pages 119-170, 2007. 51 pages.

* Thomas M. Fiore. On the Cobordism and Commutative Monoid with Cancellation Approaches to Conformal Field Theory. _Journal of Pure and Applied Algebra_, Volume 209 Number 3, pages 583-620, 2007. 37 pages.

* Thomas M. Fiore. Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory. _Memoirs of the American Mathematical Society_ 182 (2006), no. 860. 171 pages. [math.CT/0408298](http://arxiv.org/abs/math.CT/0408298)

* Thomas M. Fiore and Ramon Satyendra. Generalized Contextual Groups. _Music Theory Online_, Volume 11, Number 3, September 2005. 29 pages. [Free article on MTO website](http://mto.societymusictheory.org/issues/mto.05.11.3/toc.11.3.html)

category: people

[[!redirects Thomas Fiore]]
[[!redirects Tom Fiore]]
[[!redirects Thomas M. Fiore]]
[[!redirects T. M. Fiore]]


Thomas Fox

Category theorist who pioneered [[category theory|category theoretic]] methods in [[deformation theory]].

* [Homepage](https://www.math.mcgill.ca/fox/index.html)

## Selected writings

Characterization of [[cartesian monoidal categories]] as [[symmetric monoidal categories]] with [[extra structure]]:

* {#Fox76} [[Thomas Fox]], *Coalgebras and cartesian categories*, Communications in Algebra **4** 7 (1976) 665-667 ([doi:10.1080/00927877608822127](https://doi.org/10.1080/00927877608822127), [pdf](https://www.tandfonline.com/doi/pdf/10.1080/00927877608822127))

category: people


Thomas Friedrich

* [webpage](http://www.mathematik.hu-berlin.de/forschung/forschungsgebiete/differentialgeometrie-und-globale-analysis/tf)

## Related entries

* [[Dirac operator]]

category: people


Thomas Goodwillie

* [website](https://www.math.brown.edu/tgoodwil/)

## Selected writings

Introducing [[Goodwillie calculus]]:

* {#Goodwillie90} [[Tom Goodwillie]], _The differential calculus of homotopy functors_, Proceedings of the International Congress of Mathematicians in Kyoto 1990, Vol. I, Math. Soc. Japan, 1991, pp. 621–6 ([article pdf](https://math.mit.edu/juvitop/old/notes/2009_Fall/goodwillie-icm.pdf), [full proceedings Vol I pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.2/ICM1990.2.ocr.pdf), [[GoodwillieICM1990.pdf:file]])

* [[Thomas Goodwillie]], _Calculus. I. The first derivative of pseudoisotopy theory_, K-Theory __4__ (1990), no. 1, 1-27 ([doi:10.1007/BF00534191](http://dx.doi.org/10.1007/BF00534191), MR:1076523)

* [[Thomas Goodwillie]], _Calculus. II. Analytic functors_, K-Theory __5__ (1991/92), no. 4, 295-332 ([doi:10.1007/BF00535644](http://dx.doi.org/10.1007/BF00535644), MR:1162445)

* [[Thomas Goodwillie]], _Calculus. III. Taylor series_, Geom. Topol. 7 (2003), 645--711 ([euclid:gt/1513883319](https://projecteuclid.org/euclid.gt/1513883319) [doi:10.2140/gt.2003.7.645](http://dx.doi.org/10.2140/gt.2003.7.645), [arXiv:math/0310481](http://arxiv.org/abs/math/0310481))

On [[algebraic K-theory]]:

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], *The local structure of algebraic K-theory*, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]

## Related entries

* [[Goodwillie calculus]]

* [[Goodwillie-Taylor tower]]

* [[excisive (∞,1)-functor]], [[n-excisive (∞,1)-functor]]

* [[Goodwillie differentiation]], [[Goodwillie chain rule]]

* [[Goodwillie derivative of the identity functor]]

* [[Goodwillie spectral sequence]]

* [[Goodwillie-differentiable (∞,1)-category]]

* [[orthogonal calculus]], [[manifold calculus]]

[[!redirects T. Goodwillie]]

[[!redirects Tom Goodwillie]]

category: people


Thomas Grimm

* [webpage](http://www.th.physik.uni-bonn.de/people/grimm/)

## Selected writings

On [[M-theory on Spin(7)-manifolds]], [[F-theory on Spin(7)-manifolds]] and [[Witten's Dark Fantasy]]:

* [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

On [[axion]]-driven [[cosmic inflation]] in [[F-theory]]:

* {#Grimm14} [[Thomas Grimm]], _Axion Inflation in F-theory_ ([arXiv:1404.4268](http://arxiv.org/abs/1404.4268))

On [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]:

* [[Thomas Grimm]], Raffaele Savelli, sections 3.1, 3.2 of _Gravitational Instantons and Fluxes from M/F-theory on Calabi-Yau fourfolds_ ([arXiv:1109.3191](http://arxiv.org/abs/1109.3191))

On [[flux compactifications]] in [[F-theory]]:

* {#GKP12} [[Thomas Grimm]], Denis Klevers, Maximilian Poretschkin, _Fluxes and Warping for Gauge Couplings in F-theory_ ([arXiv:1202.0285](http://arxiv.org/abs/1202.0285))

On [[5d Chern-Simons theory]] in [[5-dimensional supergravity]]:

* {#BonettiGrimmHohenegger13} [[Federico Bonetti]], [[Thomas Grimm]], [[Stefan Hohenegger]], _One-loop Chern-Simons terms in five dimensions_ ([arXiv:1302.2918](http://arxiv.org/abs/1302.2918))

## Related $n$Lab entries

* [[F-theory]]

* [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]

* [[flux compactifications]]

* [[M-theory on Spin(7)-manifolds]]

* [[F-theory on Spin(7)-manifolds]]

* [[Spin(7)-manifolds]])

* [[5d Chern-Simons theory]],

* [[5-dimensional supergravity]])

* [[axion]]-driven [[cosmic inflation]]

category: people


Thomas Hales

* [webpage](http://www.mathematics.pitt.edu/people/hales.php)

## Selected writings

On the [[Flyspeck project]] ([[proof]] of the [[Kepler conjecture]] with help of the [[computer proof assistant]] [[HOL Light]]):

* {#Hales06} [[Thomas Hales]], _Introduction to the Flyspeck Project_, 2006 ([drops:2006/432](http://drops.dagstuhl.de/opus/volltexte/2006/432/), [pdf](https://pdfs.semanticscholar.org/0f72/54bb124913ca58f814dd0c6ab2585dc05c77.pdf))

On [[formal proof]] using [[proof assistants]]:

* [[Thomas Hales]], _Formal proof_, Notices of the AMS **55** 11 (2008) 1370-1379 [[pdf](http://www.ams.org/notices/200811/tx081101370p.pdf)]

category: people

[[!redirects Thomas Hale]]
[[!redirects Thomas Hales]]


Thomas Hartman

* [webpage](https://physics.cornell.edu/thomas-hartman)

## Selected writings

On [[Kerr black holes]] in the context of the [[holographic principle]] and the [[AdS-CFT correspondence]]:

* [[Monica Guica]], [[Thomas Hartman]], Wei Song, [[Andrew Strominger]], _The Kerr/CFT Correspondence_, Phys. Rev. D **80** (2009) 124008 [[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)]

On using the [[conformal bootstrap]] to re-prove the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24, and interpretation of the result, via [[AdS/CFT]] in terms of the [[weak gravity conjecture]]:

* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020)  \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\]

Review:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]

category: people


Thomas Heinzl

* [personal page](https://www.plymouth.ac.uk/staff/tom-heinzl)

## Selected writings

On [[light-cone quantization]]:

* [[Thomas Heinzl]], *Light-Cone Quantization: Foundations and Applications*, in *Methods of Quantization*, Lecture Notes in Physics **572**, Springer (2001) [[arXiv:hep-th/0008096](https://arxiv.org/abs/hep-th/0008096), [doi:10.1007/3-540-45114-5_2](https://doi.org/10.1007/3-540-45114-5_2)]

* [[Stanley Brodsky]], [[Thomas Heinzl]] et al. *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*, White Paper of International Light Cone Advisory Committee, Nuclear Physics B - Proceedings Supplements **251**, **252** (2014) 165-174 [[doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004), [arXiv:1309.6333](https://arxiv.org/abs/1309.6333)]

category: people


Thomas Holder

Is there any way to contact Thomas Holder?!

* [MO page](http://mathoverflow.net/users/46788/thomas-holder)

category: people


Thomas Hunter

I was a PhD student of Frank Peterson at MIT in the 1980's.

I've been a professor at Swarthmore College since 1991.

In Spring semester 2024 I'll be teaching an introduction to category theory for undergraduates (mostly mathematics and computer science majors).

You can [email me](mailto:thunter1@swarthmore.edu) if you want to discuss anything at all.

category: people


Thomas Jech

Tomáš Jech is a Professor Emeritus at Penn State.

He got his PhD degree in 1966 from Charles University in Prague, advised by [[Petr Vopěnka]].

[[!redirects Tomáš Jech]]
[[!redirects Thomas J. Jech]]


Thomas Kehrenberg

* [personal page](https://thomkeh.github.io/)

* [blog](https://tm.kehrenberg.net/)

## Selected writings

Gentle exposition of the basic principles of [[dependent type theory]]:

* [[Thomas Kehrenberg]], *Introduction to dependent type theory* (2022-23) [[blog series](https://tm.kehrenberg.net/series/introduction-to-dependent-type-theory/)]

category: people


Thomas Klose

* [webpage](https://qft.physik.hu-berlin.de/research/elementary-particle-physics-group/thomas-klose/)

## Selected writings

On the [[BMN matrix model]] as a [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] on the [[3-sphere]]:

* [[Nakwoo Kim]], [[Thomas Klose]], [[Jan Plefka]], _Plane-wave Matrix Theory from N=4 Super Yang-Mills on $\mathbb{R} \times S^3$_, Nucl. Phys. B671:359-382, 2003 ([arXiv:hep-th/0306054](https://arxiv.org/abs/hep-th/0306054))

category: people


Thomas Lam

* [personal page](https://dept.math.lsa.umich.edu/~tfylam/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=34065)

## Selected writings

Proving the [[Peterson isomorphism]] relating the [[quantum cohomology]] of [[flag varieties]] with a [[Pontryagin ring]]:

* [[Thomas Lam]], [[Mark Shimozono]], *Quantum cohomology of $G/P$ and homology of affine Grassmannian*, Acta Mathematica **204** (2010) 49–90 [[arXiv:0705.1386](https://arxiv.org/abs/0705.1386), [doi:10.1007/s11511-010-0045-8](https://doi.org/10.1007/s11511-010-0045-8)]

The variant for [[Pontryagin products]] not in [[ordinary homology]] but in [[topological K-theory|topological]] [[K-homology]]:

* [[Thomas Lam]], [[Changzheng Li]], [[Leonardo C. Mihalcea]], [[Mark Shimozono]], *A conjectural Peterson isomorphism in K-theory*, Journal of Algebra **513** (2018) 326-343 [[doi:10.1016/j.jalgebra.2018.07.029](https://doi.org/10.1016/j.jalgebra.2018.07.029), [arXiv:1705.03435](https://arxiv.org/abs/1705.03435)]

On [[configuration spaces of points]] in general poisition inside [[complex projective spaces]]:

* [[Nima Arkani-Hamed]], [[Thomas Lam]], [[Marcus Spradlin]], *Positive configuration space*, Commun. Math. Phys. **384** (2021) 909–954  [[arXiv:2003.03904](https://arxiv.org/abs/2003.03904), [doi:10.1007/s00220-021-04041-x](https://doi.org/10.1007/s00220-021-04041-x)]

category: people


Thomas Lamiaux

* [personal page](https://thomas-lamiaux.github.io/)

## Selected writings

Implementation of [[ordinary cohomology|ordinary]]$\;$[[cohomology rings]] in [[cubical agda]]:

* [[Thomas Lamiaux]], [[Axel Ljungström]], [[Anders Mörtberg]], _Computing Cohomology Rings in Cubical Agda_, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [[arxiv:2212.04182] (https://arxiv.org/abs/2212.04182), [doi:10.1145/3573105.3575677](https://doi.org/10.1145/3573105.3575677)]

category: people


Thomas M. Stace

* [Institute page](https://smp.uq.edu.au/profile/170/tom-stace)

* [GoogleScholar page](https://scholar.google.com/citations?user=ajC8aLAAAAAJ&hl=en)

## Selected writings

On [[quantum error correcting codes]]:

* [[Robert J. Harris]], [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], *Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes*, Phys. Rev. A 98, 052301 (2018) ([arXiv:1806.06472](https://arxiv.org/abs/1806.06472))

* [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], [[Robert J. Harris]], Elliot Coupe, *Decoding Holographic Codes with an Integer Optimisation Decoder* ([arXiv:2008.10206](https://arxiv.org/abs/2008.10206))

* [[Terry Farrelly]], [[Robert J. Harris]], [[Nathan A. McMahon]], [[Thomas M. Stace]], *Tensor-network codes* ([arXiv:2009.10329](https://arxiv.org/abs/2009.10329))

category: people


Thomas Mertens

* [webpage](https://biblio.ugent.be/person/802001091991)

## Selected writings

On [[weight systems]] on [[chord diagrams]] as [[observables]] in  [[Jackiw-Teitelboim gravity]] and, under [[AdS2/CFT1]], the [[Sachdev-Ye-Kitaev model]]:

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))

category: people


Thomas Mohaupt

* [webpage](http://www.liv.ac.uk/~mohaupt/)

category: people


Thomas Nikolaus

Thomas Nikolaus is a full Professor at the Fachbereich Mathematik und Informatik of the WWU Münster. He is working on topics in [[homotopy theory]] and [[higher category theory|higher categorial structures]] motivated from [[string theory|string]] [[physics]], such as [[topological T-duality]].

[Homepage](https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/index.html)

## Selected writings

* _[[NikolausHigherCatStructuresInGeometry.pdf:file]]_ ([[NikolausHigherCatStructuresInGeometrySlides.pdf:file]]) PhD thesis (2011)

* _[[AlgModB.pdf:file]]_, 2010 ([blog discussion](http://golem.ph.utexas.edu/category/2010/03/_guest_post_by_thomas_nikolaus.html), [arxiv/1003.1342](http://arxiv.org/abs/1003.1342))

  **Abstract:** We establish a [[model category]] structure on algebraic [[Kan complex]]es. In fact, we introduce the notion of an algebraic fibrant
object in a general model category (obeying certain technical conditions). Based on this construction we propose algebraic Kan complexes as an algebraic model for [[∞-groupoid]]s and algebraic [[quasi-categories]] as an algebraic model for [[(∞,1)-categories]]. (Published as Indag. Math. (N.S.) 21.1- 2 (2011), pp. 52 - 75.)

* ...

On [[T-folds]] via [[principal 2-bundles]] for the [[T-duality 2-group]]:

* _[[T-Duality in K-theory and Elliptic Cohomology]]_, talk at _String Geometry Network Meeting_, Feb 2014, ESI Vienna ([website](http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html))

* {#NikolausWaldorf18} [[Thomas Nikolaus]], [[Konrad Waldorf]], *Higher geometry for non-geometric T-duals*, Commun. Math. Phys. **374** (2020) 317-366 [[arXiv:1804.00677](https://arxiv.org/abs/1804.00677), [doi:10.1007/s00220-019-03496-3](https://doi.org/10.1007/s00220-019-03496-3)]

On [[group completion theorem]] and [[Quillen plus construction]]:

* _The group completion theorem via localizations of ring spectra_, 2017 ([pdf](https://people.mpim-bonn.mpg.de/thoni/Papers/Group_completion.pdf))

On [[cyclotomic spectra]] and [[topological cyclic homology]]:

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_,  Acta Math. **221** 2 (2018) 203-409 [[arXiv:1707.01799](https://arxiv.org/abs/1707.01799), [doi:10.4310/ACTA.2018.v221.n2.a1](https://dx.doi.org/10.4310/ACTA.2018.v221.n2.a1)]

On [[prismatic cohomology]] applied to [[algebraic K-theory]]:

* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)[

category: people


Thomas Quella

* [webpage](https://findanexpert.unimelb.edu.au/display/person754514)

## related $n$Lab entries

* [[permutation D-brane]]

category:people


Thomas Rot

* [webpage](https://www.few.vu.nl/~trt800/)

## Selected writings

On [[Cohomotopy sets]] of [[Thom spaces]]:

* [[Thomas Rot]], _Homotopy classes of proper maps out of vector bundles_,  Arch. Math. 114, 107–117 (2020 ([arXiv:1808.08073](https://arxiv.org/abs/1808.08073), [doi:10.1007/s00013-019-01372-z](https://doi.org/10.1007/s00013-019-01372-z))

category: people


Thomas Schaefer

* [webpage](http://wonka.physics.ncsu.edu/~tmschaef/)

## Selected writings:

On the [[phase diagram]] of [[QCD]]:

* {#Schaefer05} [[Thomas Schaefer]], _Phases of QCD_ ([arXiv:hep-ph/0509068](https://arxiv.org/abs/hep-ph/0509068))

On [[instantons in QCD]]:

* {#SchaeferShuryak98} [[Thomas Schaefer]], [[Edward Shuryak]], _Instantons in QCD_, Rev. Mod. Phys.70:323-426,1998 ([arXiv:hep-ph/9610451](http://arxiv.org/abs/hep-ph/9610451))

* {#Schaefer02} [[Thomas Schaefer]], _QCD and the eta prime Mass: Instantons or Confinement?_, Phys.Rev. D67 (2003) 074502 ([arXiv:hep-lat/0211035](http://arxiv.org/abs/hep-lat/0211035))

On [[colour superconductivity]]

* Mark G. Alford, Krishna Rajagopal, [[Thomas Schaefer]], Andreas Schmitt, _Color superconductivity in dense quark matter_, Rev. Mod. Phys. 80:1455-1515, 2008 ([arXiv:0709.4635](https://arxiv.org/abs/0709.4635))

## Related $n$Lab entries

* [[instantons in QCD]]

category: people


Thomas Schick

* [website](http://www.uni-math.gwdg.de/schick/)

## Selected writings

On [[topological T-duality]]:

* {#BunkeSchick04} [[Ulrich Bunke]], [[Thomas Schick]], _On the topology of T-duality_, Rev. Math. Phys. **17** (2005) 77-112 [[arXiv:math/0405132](https://arxiv.org/abs/math/0405132), [doi:10.1142/S0129055X05002315](https://doi.org/10.1142/S0129055X05002315)]

* {#BunkeRumpfSchick08} {#BunkeRumpfSchick08} [[Ulrich Bunke]], P. Rumpf, [[Thomas Schick]], _The topology of $T$-duality for $T^n$-bundles_,  Rev. Math. Phys. **18** 1103 (2006) [[arXiv:math.GT/0501487](http://arxiv.org/abs/math.GT/0501487), [doi:10.1142/S0129055X06002875](https://doi.org/10.1142/S0129055X06002875)]

On [[differential K-theory]]:

* {#BunkeSchickReferences} [[Ulrich Bunke]], [[Thomas Schick]], _Smooth K-Theory_, Astérisque 328 (2009), 45-135 ([arXiv:0707.0046](http://arxiv.org/abs/0707.0046), [numdam:AST_2009__328__45_0](http://www.numdam.org/item/?id=AST_2009__328__45_0))

On [[differential cobordism cohomology]], i.e. enhancement of [[MU]] ([[complex cobordism cohomology]]) to [[differential cohomology]]:

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

On [[topological T-duality]] in the [[equivariant cohomology|equivariant]] case, as an equivalence of [[twisted equivariant K-theory]] groups:

* [[Tom Dove]], [[Thomas Schick]], *Equivariant Topological T-Duality* [[arXiv:2310.06064](https://arxiv.org/abs/2310.06064)]

## Related entries

* [[differential cohomology]]

category: people


Thomas Streicher

* [website](https://www2.mathematik.tu-darmstadt.de/~streicher/)

## Selected writings

On [[categorical semantics]] of ([[dependent type theory|dependent]]) [[type theory]] (and proving an [[initiality conjecture]] in Ch. 4), see at *[[categorical model of dependent types]]*:

* {#Streicher91} [[Thomas Streicher]], _Semantics of type theory -- Correctness, completeness and independence results_, Progress in Theoretical Computer Science, Birkhäuser Boston (1991) [ISBN:0-8176-3594-7, [doi:10.1007/978-1-4612-0433-6](https://doi.org/10.1007/978-1-4612-0433-6)]

On [[intensional type theory]]:

* [[Thomas Streicher]], *Investigations into Intensional Type Theory*, Habilitation Thesis, Darmstadt (1993) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf), [[Streicher-IntensionalTT.pdf:file]]]

Introducing the [[homotopy type theory]]-interpretation of [[identity types]] (the "groupoid model")

* [[Martin Hofmann]], [[Thomas Streicher]], *The groupoid model refutes uniqueness of identity proofs*, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994) [[doi:10.1109/LICS.1994.316071](https://doi.org/10.1109/LICS.1994.316071)]

and introducing what came to be known the [[univalence axiom]] (under the name "universe extensionality"):

* {#HofmannStreicher98} [[Martin Hofmann]], [[Thomas Streicher]]  _The groupoid interpretation of type theory_, in: [[Giovanni Sambin]] et al. (eds.), *Twenty-five years of constructive type theory*, Proceedings of a congress, Venice, Italy, October 19-21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. **36** (1998) 83-111  [[ISBN:9780198501275](https://global.oup.com/academic/product/twenty-five-years-of-constructive-type-theory-9780198501275), [ps.gz](http://www.mathematik.tu-darmstadt.de/~streicher/venedig.ps.gz), [[HofmannStreicherGroupoidInterpretation.pdf:file]]]

  see also:

  Ethan Lewis, Max Bohnet, *The groupoid model of type theory*, seminar notes (2017) [[pdf](https://staff.fnwi.uva.nl/b.vandenberg3/Onderwijs/Homotopy_Type_Theory_2017/HoTT-bohnet-lewis-handout.pdf), [[LewisBohnet.GroupoidModelOfTT.pdf:file]]]

On [[category theory]] and [[categorical logic]]:

* [[Thomas Streicher]], *Introduction to Category Theory and Categorical Logic* (2003) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf), [[Streicher-CategoryTheory.pdf:file]]]

On [[denotational semantics]] and [[domain theory]] for [[functional programming|functional]] [[programming languages]]:

* [[Thomas Streicher]], *Domain-Theoretic Foundations of Functional Programming*, World Scientific (2006) [[pdf](https://doc.lagout.org/programmation/Functional%20Programming/Domain-Theoretic%20Foundations%20of%20Functional%20Programming%20%5BStreicher%202006-12-04%5D.pdf), [doi:10.1142/6284](https://www.worldscientific.com/worldscibooks/10.1142/6284)]

On [[first-order set theory]] and [[categorical logic]]:

* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], *Relating first-order set theories, toposes and categories of classes*, Annals of Pure and Applied Logic **165** 2 (2014) 428-502 [[doi:10.1016/j.apal.2013.06.004](https://doi.org/10.1016/j.apal.2013.06.004)]

On [[fibered categories]] following [[Jean Bénabou]]:

* {#Streicher18} [[Thomas Streicher]], _Fibred Categories à la Jean Bénabou_,  [[arXiv:1801.02927](https://arxiv.org/abs/1801.02927)]

A [[2-comonad]] characterizing [[Grothendieck fibrations]]:

* [[Jacopo Emmenegger]], [[Luca Mesiti]], [[Giuseppe Rosolini]], [[Thomas Streicher]], *A comonad for Grothendieck fibrations* [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]

category: people


Thomas Tradler

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=77678)

## Selected writings

On [[BV-algebra]] [[structure]] on [[Hochschild cohomology]]:

* {#Tradler08} [[Thomas Tradler]], *The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products*, Annales de l'institut Fourier **58** 7 (2008) 2351-2379 [[arXiv:math/0210150](http://arxiv.org/abs/math/0210150)]

On higher [[Hochschild homology]] and [[factorization algebras]]:

* [[Gregory Ginot]], [[Thomas Tradler]], [[Mahmoud Zeinalian]], _Derived higher Hochschild homology, topological chiral homology and factorization algebras_, [arxiv/1011.6483](http://arxiv.org/abs/1011.6483)

category: people


Thomas Van Riet

* [webpage](https://www.kuleuven.be/wieiswie/en/person/00055735)

## Selected writings

On [[de Sitter spacetimes]] in the [[landscape of string theory vacua]] (or not), problems with [KKLT 03](landscape+of+string+theory+vacua#KKLT03):

* {#DanielssonVanRiet18} [[Ulf Danielsson]], [[Thomas Van Riet]],  _What if string theory has no de Sitter vacua?_, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018)  ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070))

* {#vanRiet18} [[Thomas van Riet]], _Is dS space in the Swampland_, talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf))

* {#vanRiet18a} [[Thomas van Riet]], _Status of KKLT_, talk at _[Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870)_ ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730))

## Related $n$Lab entries

* [[string phenomenology]]

* [[landscape of string theory vacua]]

* [[de Sitter spacetime]]

* [[inhomogeneous cosmology]]

category: people

[[!redirects Thomas van Riet]]


Thomas Willwacher

* [webpage](http://www.math.ethz.ch/people/wilthoma)

## Selected writings

On [[graph complex]]-models for [[configuration spaces of points]] (and the Grothedieck-Teichmueller group]]):

* {#Willwacher10} [[Thomas Willwacher]], _M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra_, Invent. math. (2015) 200: 671 ([arxiv:1009.1654](http://arxiv.org/abs/1009.1654))

* {#CamposWillwacher16} [[Ricardo Campos]], [[Thomas Willwacher]], _A model for configuration spaces of points_ ([arXiv:1604.02043](https://arxiv.org/abs/1604.02043))

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, [[Najib Idrissi]], [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

category: people


Thomas Winyard

* [personal page](http://www1.maths.leeds.ac.uk/~pmttw/)

* [institute page](https://eps.leeds.ac.uk/maths/staff/5260/dr-thomas-winyard)

## Selected writings

On [[Skyrmions]]:

* [[Chris Halcrow]], [[Thomas Winyard]], _A consistent two-skyrmion configuration space from instantons_ ([arXiv:2103.15669](https://arxiv.org/abs/2103.15669))

* [[Josh Cork]], [[Derek Harland]], [[Thomas Winyard]], *A model for gauged skyrmions with low binding energies* ([arXiv:2109.06886](https://arxiv.org/abs/2109.06886))

category: people