topological cofiber sequence

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

We discuss in detail the realization of the abstract concept of [[cofiber sequences]] in its explicit incarnation in [[point-set topology]], the way it is traditionally presented in [[topology]] textbooks.

Hence we use the concepts of [[homotopy equivalence]] instead of [[weak homotopy equivalence]]. For discussion using the latter in the context of the [[classical model structure on topological spaces]] see instead at _[[Introduction to Homotopy Theory]]_ the section _[Homotopy fiber sequences](Introduction+to+Homotopy+Theory#HomotopyFiberSequences)_.

## Background

Throughout, write $I \coloneqq [0,1] \subset \mathbb{R}$ for the [[closed interval]] equipped with its [[Euclidean space|Euclidean]] [[metric topology]].

### Homotopy equivalences

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

For $f,g\colon X \longrightarrow Y$ two [[continuous functions]] between [[topological spaces]] $X,Y$, then a **[[homotopy]]**

$$
  \eta \colon f \,\Rightarrow_L\, g
$$

is a [[continuous function]]

$$
  \eta \;\colon\; X \times I \longrightarrow Y
$$

out of the standard [[cylinder object]] over $X$: the [[product space]] of $X$ with the [[Euclidean space|Euclidean]] [[closed interval]], such that this fits into a [[commuting diagram]] of the form

$$
  \array{
     X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     X \times I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     X
  }
  \,.
$$

(graphics grabbed from J. Tauber [here](http://jtauber.com/blog/2005/07/01/path_homotopy/))

=--

+-- {: .num_defn #HomotopyEquivalence}
###### Definition
**([[homotopy equivalence]])**

A [[continuous function]] $f \colon X \to Y$ is called a _[[homotopy equivalence]]_ if there exist

1. a continuous function $g \colon Y \to X$;

1. homotopies (def. \ref{LeftHomotopy}) of the form

   $g \circ f \Rightarrow id_X \phantom{AAAA} f \circ g \Rightarrow id_Y$

=--

+-- {: .num_example #ContractibleTopologicalSpace}
###### Example
**([[contractible topological space]])**

A [[topological space]] $X$ is called _[[contractible topological space|contractible]]_ if the unique map to the [[point space]] $X \to \ast$ is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}).

=--

### Cofibrations

+-- {: .num_defn #HurewiczCofibration}
###### Definition
**([[Hurewicz cofibration]])

A [[continuous function]] $i \colon A \longrightarrow X$ is called a _[[Hurewicz cofibration]]_ or just _cofibration_ if it satisfies the [[homotopy extension property]] in that:

* for any [[topological space]] $Y$,

* for all continuous functions $ f \colon A\to Y$, $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$

  $$
    \array{
       A &\stackrel{f}{\to}& Y
       \\
       {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}}
       \\
       X
    }
  $$

* and any [[left homotopy]] $F \colon A\times I\to Y$ such that $F(-,0)=f$

there is a homotopy $\tilde{F} \colon X\times I\to Y$ such that

* $\tilde{F}\circ(i\times id_I)=F$

  $$
    \array{
       A \times I &\overset{F}{\longrightarrow}& Y
       \\
       {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}}
       \\
       X \times I
    }
  $$

* and $\tilde{F}(-,0)=\tilde{f}$

  $$
    \array{
       A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y
       \\
       {}^{\mathllap{i \times id_I}}\downarrow
       && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}}
       \\
       X &\underset{id_X \times const_0}{\longrightarrow}& X \times I
    }
    \phantom{AAA}
      =
    \phantom{AAA}
    \array{
    \array{
       A &\stackrel{f}{\to}& Y
       \\
       {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}}
       \\
       X
    }cco
    }
  $$

If the [[image]] $i(A) \subset X$ is a [[closed subset]], then $i$ is called a _[[closed Hurewicz cofibration]]_.

=--

+-- {: .num_example #RelativeCellComplexInclusionsAreClosedCofibrations}
###### Example
**([[retracts]] of [[relative cell complex]] inclusions are [[closed Hurewicz cofibrations]])

If $A$ is a [[topological space]] and $i \colon A \hookrightarrow X$ is a [[relative cell complex]] inclusion, then $i$ is a [[closed Hurewicz cofibration]] (def. \ref{HurewiczCofibration}).

Also every [[retract]] of such $i$ ([this sense](Introduction+to+Homotopy+Theory#RetractsOfMorphisms)) is a closed Hurewicz cofibration.

=--

### Pushouts / space attachments

Cofiber sequences ([below](#CofiberSequences)) are constructed by iterated _[[pushouts]]_ in the [[category]] [[Top]] of [[topological spaces]] with [[continuous functions]] between them ([[space attachments]]), see at _[Top -- Universal constructions](Top#UniversalConstructions)_.

+-- {: .num_example #QuotientBySubspace}
###### Example
**([[quotient space]] by a [[subspace]])**

Let $X$ be a [[topological space]] and $A \subset X$ a [[inhabited set|non-empty]] [[subset]].
Consider the [[equivalence relation]] on $X$ which identifies all points in $A$ with each other. The resulting [[quotient space]] is denoted $X/A$.

Notice that $X/A$ is canonically a [[pointed topological space]], with base point the [[equivalence class]] $A/A \subset X/A$ of $A$.

If $A = \emptyset$ is the [[empty space]], then one defines

$$
  X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast
$$

to be the [[disjoint union space]] of $X$ with the [[point space]]. This is no longer a quotient space, but both constructions are unified by the _[[pushout]]_ $i \colon A \to X$ along the map $A \to \ast$, equivalently the [[cokernel]] of the inclusion:

$$
  \array{
    A &\overset{i}{\hookrightarrow}& X
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& X/A
  }
  \,.
$$

=--

+-- {: .num_example #ClosedSubspacesGluing}
###### Example
**([[union]] of two [[open subset|open]] or two [[closed subset|closed]] [[subspaces]] is [[pushout]])**

Let $X$ be a [[topological space]] and let $A,B \subset X$ be [[subspaces]] such that

1. $A,B \subset X$ are both [[open subsets]] or are both [[closed subsets]];

1. they constitute a [[cover]]: $X = A \cup B$

Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion [[continuous functions]].

Then the [[commuting square]]

$$
  \array{
     A \cap B &\longrightarrow& A
     \\
     \downarrow &(po)& \downarrow^{\mathrlap{i_A}}
     \\
     B &\underset{i_B}{\longrightarrow}& X
  }
$$

is a [[pushout]] square in [[Top]] (see [there](Top#UniversalConstructions)).

By the [[universal property]] of the [[pushout]] this means in particular that for $Y$ any [[topological space]] then a function of underlying sets

$$
  f \;\colon\; X \longrightarrow Y
$$

is a [[continuous function]] as soon as its two restrictions

$$
  f\vert_A \;\colon\; A \longrightarrow Y
  \phantom{AAAA}
  f\vert_A \;\colon\; B \longrightarrow Y
$$

are continuous.

=--

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

Clearly the underlying diagram of underlying [[sets]] is a pushout in [[Set]]. Therefore (by [this prop.](Top#DescriptionOfLimitsAndColimitsInTop)) we need to show that the [[topological space|topology]] on $X$ is the [[final topology]] induced by the set of functions $\{i_A, i_B\}$, hence that
a [[subset]] $S \subset X$ is an [[open subset]] precisely if the [[pre-images]] (restrictions)

$$
  i_A^{-1}(S) = S \cap A
   \phantom{AAA}
    \text{and}
   \phantom{AAA}
  i_B^{-1}(S) = S \cap B
$$

are open subsets of $A$ and $B$, respectively.

Now by definition of the [[subspace topology]], if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces.

Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open.

Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the [[subspace topology]], that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open.  Similarly $B \cap S$. Therefore

$$
  \begin{aligned}
    S
     & =
    S \cap X
    \\
    & = S \cap (A \cup B)
    \\
    & =  (S \cap A) \cup (S \cap B)
  \end{aligned}
$$

is the union of two open subsets and therefore open.

Now consider the case that $A,B \subset X$ are both closed subsets.

Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open.

Now observe that (by [[de Morgan duality]])

$$
  \begin{aligned}
    S
      & = X \setminus (X \setminus S)
    \\
      & =
    X \setminus ( (A \cup B) \setminus S )
    \\
      & =
    X \setminus ( (A \setminus S) \cup (B \setminus S) )
    \\
     & =
    (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S))
    \,.
  \end{aligned}
$$

This exhibits $S$ as the intersection of two open subsets, hence as open.

=--

A general abstract fact about pushouts which we will use repeatedly in the proofs below is the following:

+-- {: .num_prop #PastingLaw}
###### Proposition
**([[pasting law]])**

Consider a [[diagram]] in [[Top]] (or in any other [[category]]) of the following form:

$$
  \array{
     &\longrightarrow& &\longrightarrow&
    \\
    \downarrow &(po)& \downarrow && \downarrow
    \\
     &\longrightarrow& &\longrightarrow&
  }
 \,,
$$

where the left square is a [[pushout]].

Then: The total rectangle is a [[pushout]] precisely if the right square is.

=--

We need the following two facts (prop. \ref{CofibrantHomotopyEquivalencePushout} prop. \ref{ContractibleSubcomplexQuotient}) regarding the stability of cofibrations under pushout.

+-- {: .num_prop #CofibrantHomotopyEquivalencePushout}
###### Proposition
**([[pushout]] of cofibrations)

Let $A$ be a [[topological space]] and let
$A \hookrightarrow X$ be a [[closed Hurewicz cofibration]].

Then for every [[continuous function]] $f \colon A \to Y$, the [[pushout]] $f_\ast i$ in

$$
  \array{
     A &\overset{f}{\longrightarrow}& Y
     \\
     {}^{\mathllap{i}}\downarrow &(po)& \downarrow^{\mathrlap{f_\ast i}}
     \\
     X &\longrightarrow& X \underset{A}{\sqcup} Y
  }
$$

is also a [[closed Hurewicz cofibration]]. Moreover, if in addition

1. $i$ is a [[retract]] of a [[relative cell complex]] inclusion, then so is $f_\ast i$,

1. $i$ is a [[homotopy equivalence]], then so is $f_\ast i$.

=--

Prop. \ref{CofibrantHomotopyEquivalencePushout} is a consequence of the existence of the [[Strøm model structure]] $Top_{Strom}$ and the [[classical model structure on topological spaces]] $Top_{Quillen}$ and of the fact that the identity [[functors]] $Top_{Strom} \underoverset{\underset{id}{\longrightarrow}}{\overset{id}{\longleftarrow}}{\bot} Top_{Quillen}$ constitute a [[Quillen adjunction]].

+-- {: .num_prop #ContractibleSubcomplexQuotient}
###### Proposition
**([[quotient space|quotient]] by [[contractible topological space|contractible]] closed subspace)**

Let $i \colon A \longrightarrow X$ be a [[closed Hurewicz cofibration]].

If $f \colon A \to Y$ is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}) then its [[pushout]] $i_\ast f$ in

$$
  \array{
     A &\overset{i}{\longrightarrow}& X
     \\
     {}^{\mathllap{f}}\downarrow &(po)& \downarrow^{\mathrlap{i_\ast(f)}}
     \\
     Y &\longrightarrow& Y \underset{A}{\sqcup} X
  }
$$

is also a homotopy equivalence.

In particular if $A$ is a [[contractible topological space]] (example \ref{ContractibleTopologicalSpace}), then the coprojection

$$
  X \longrightarrow X/A
$$

to the [[quotient space]] (example \ref{QuotientBySubspace}) is a [[homotopy equivalence]].

=--

Prop. \ref{ContractibleSubcomplexQuotient} is again a consequence of the existence of the [[Strøm model structure]] $Top_{Strom}$: It is the statement that $Top_{Strom}$ is a [[left proper model category]] which follows ([this cor.](proper%20model%20category#AllObjectsFibrantImpliesRightProper)) since all its objects are evidently cofibrant. An elementary proof in [[point-set topology]] is offered in ([Hatcher, prop. 0.17](algebraic+topology#Hatcher))

## Mapping cones

+-- {: .num_defn #ConeCyclinder}
###### Definition
**(topological [[cylinder]] and cone)**

Let $X$ be a [[topological space]]. Then

1. the _standard [[cylinder]]_ on $X$ is the [[product topological space]]

   $$
     Cyl(X) \coloneqq X \times [0,1]
   $$

1. the _standard [[cone]]_ on $X$ is the [[quotient space]] (example \ref{QuotientBySubspace})

   $$
     Cone(X) \coloneqq Cyl(X) / (X \times \{0\})
   $$

   of the standard cylinder by the [[subspace]] $X \times \{0\} \subset X \times [0,1]$.

   Equivalently this is the following [[pushout]] in [[Top]]

   $$
     \array{
        X \times \{0\} &\hookrightarrow& Cyl(X)
        \\
        \downarrow &(po)& \downarrow
        \\
        \ast &\longrightarrow& Cone(X)
     }
   $$

=--

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

Let $X$ be a topological space. Then the canonical inclusions

$$
  X \longrightarrow Cyl(X)
  \phantom{AAAA}
  \ast \longrightarrow Cone(X)
$$

are [[homotopy equivalences]]. Hence
every [[cone]] is a [contractible topological space]] (example \ref{ContractibleTopologicalSpace}).

=--

+-- {: .num_defn #MappingConeCylinder}
###### Definition
**([[mapping cylinder]] and [[mapping cone]])**

Let $f \colon X \to Y$ be a [[continuous function]] between [[topological spaces]]. Then

1. the _[[mapping cylinder]]_ of $f$ is the [[space attachment]]

   $$
     Cyl(f) \coloneqq Y \cup_f Cyl(X)
   $$

   of $Y$ with the [[cylinder]] on $X$, according to def. \ref{ConeCyclinder}, hence the following [[pushout]] in [[Top]]

   $$
     \array{
       X &\overset{f}{\longrightarrow}& Y
       \\
       \downarrow &(po)& \downarrow
       \\
       Cyl(X) &\longrightarrow& Cyl(f)
     }
   $$

1. the _[[mapping cone]]_ of $f$ is the [[space attachment]]

   $$
     Cone(f) \coloneqq Y \cup_f Cone(X)
   $$

   hence the following [[pushout]] in [[Top]]:

   $$
     \array{
       X &\overset{f}{\longrightarrow}& Y
       \\
       \downarrow &(po)& \downarrow
       \\
       Cone(X) &\longrightarrow& Cone(f)
     }
   $$

=--

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

In summary, def. \ref{ConeCyclinder} and def. \ref{MappingConeCylinder} say that for $f \colon X \to Y$ a [[continuous function]] then we have a [[pasting]] of [[pushout]] diagrams in [[Top]] of the following form:

$$
  \array{
    && X &\stackrel{f}{\to}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow
    \\
    X &\stackrel{i_0}{\to}& Cyl(X) &\to & Cyl(f)
    \\
    \downarrow &(po)& \downarrow &(po)& \downarrow
    \\
    {*} &\to& Cone(X) &\to& Cone(f)
  }
  \,.
$$

Since $X \to Cone(X)$ is a [[closed Hurewicz cofibration]] , the [[pasting law]] together with prop. \ref{CofibrantHomotopyEquivalencePushout} therefore implies that also $Y \to Cone(f)$ is a closed Hurwicz cofibration.

=--

+-- {: .num_example #Suspension}
###### Example
**([[suspension]])

For $X$ a [[topological space]], then the [[mapping cone]] (def. \ref{MappingConeCylinder})
of the unique function $X \to \ast$ to the [[point space]] is

$$
  S X
    \coloneqq
  Cone(X \to \ast)
   \simeq
  Cone(X) \underset{X}{\sqcup} Cone(X)
  \,.
$$

This is called the _[[suspension]]_ of $X$.

=--

The mapping cone of a map $X \to Y$ is to be thought of as the homotopy-quotient of the $Y$ by $X$, as opposed to be the naive quotient. This is made precise by the following two statements, lemma \ref{RelativeCellComplexInclusionMappingCone} and lemma \ref{ForClosedImageQuotientOfMappingConeByCone}:

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

If $f \colon X \to Y$ is a [[closed Hurewicz cofibration]] (def. \ref{HurewiczCofibration}), then the coprojection

$$
  Cone(f) \longrightarrow Y/f(X)
$$

from its [[mapping cone]] (def. \ref{MappingConeCylinder}) to the naive [[quotient space]] (example \ref{QuotientBySubspace})  is a [[homotopy equivalence]].

=--

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

Consider the diagram

$$
  \array{
     X &\overset{f}{\longrightarrow}& Y
     \\
     \downarrow &(po)& \downarrow
     \\
     Cone(X) &\longrightarrow& Cone(f)
     \\
     \downarrow &(po)& \downarrow
     \\
     \ast &\longrightarrow& Cone(f)/Cone(X)
  }
$$

Since $f$ is a [[closed Hurewicz cofibration]], so is $Cone(X) \to Cone(f)$, by prop. \ref{CofibrantHomotopyEquivalencePushout}. Since $Cone(X) \to \ast$ is a [[homotopy equivalence]] (lemma \ref{InclusionIntoCylinderIsHomotopyEquivalence}), so is
$Cone(f) \to Cone(f)/Cone(X)$, by prop. \ref{ContractibleSubcomplexQuotient}. But by lemma \ref{ForClosedImageQuotientOfMappingConeByCone} there is also a homeomorphism $Cone(f)/Cone(X) \simeq Y/f(X)$.

=--

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

Let $f \colon X \to Y$ be a [[continuous function]] such that the [[image]] $f(X) \subset Y$ is a [[closed subset]].

Then there is a [[homeomorphism]]

$$
  Cone(f)/Cone(X) \simeq Y / f(X)
$$

between the [[quotient space]] (example \ref{QuotientBySubspace}) of the [[mapping cone]] of $f$ (def. \ref{MappingConeCylinder}) by the [[cone]] of $X$ (def. \ref{ConeCyclinder}) and the [[quotient space]] of $Y$ be the [[image]] of $X$.

=--

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

Consider the following [[diagram]] in [[Top]]:

$$
  \array{
     f(X) &\longrightarrow& Y
     \\
     \downarrow &(po)& \downarrow
     \\
     f(Cone(X)) &\longrightarrow& Cone(f)
     \\
     \downarrow &(po)& \downarrow
     \\
     \ast &\longrightarrow& Cone(f)/Cone(X)
  }
$$

Here the top square is a [[pushout]] by example \ref{ClosedSubspacesGluing}, while the bottom square is a pushout by def. \ref{MappingConeCylinder}. Hence the total rectangle is also a pushout, by the [[pasting law]] (prop. \ref{PastingLaw}). But that total rectangle is the defining pushout for $Y/f(X)$ (by example \ref{QuotientBySubspace}). Hence the statement follows by the [[universal property]] of the pushout.

=--

## Cofiber sequences
 {#CofiberSequences}

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

Let $f \colon X \to Y$ be a [[continuous function]] with [[closed subset|closed]] [[image]] $f(X) \subset Y$. Write $Cone(g)$ in

$$
  \array{
    X
      &\overset{f}{\longrightarrow}&
    Y
      &\overset{g}{\longrightarrow}&
    Cone(f)
      &\overset{}{\longrightarrow}&
    Cone(g)
  }
$$

for the [[mapping cone]] (spring) of the inclusion $g$ of $Y$ into the mapping cone of $f$.

Then the canonical quotient coprojection

$$
  Cone(g) \to \Sigma X
$$

to the [[suspension]] of $X$ (example \ref{Suspension}) is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}).

=--

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

Since $g \colon Y \to Cone(f)$ is a [[closed Hurewicz cofibration]] (by remark \ref{MappingConePasting}), lemma \ref{RelativeCellComplexInclusionMappingCone} gives that

$$
  Cone(g) \to Cone(f)/g(Y)
$$

is a [[homotopy equivalence]].  But then there is the following evident [[homeomorphism]]

$$
  Cone(f)/g(Y)
   =
  (Y \cup_f Cone(X))/Y
   \simeq
  Cone(X)/X
   \simeq
  S X
  \,.
$$

=--

(graphics taken from [Muro 10](http://personal.us.es/fmuro/praha.pdf))

Hence from every $f \colon X \to Y$ with closed image, we get long sequences

$$
  \array{
    X
      &\overset{f}{\longrightarrow}&
    Y
      &\overset{g}{\longrightarrow}&
    Cone(f)
      &\longrightarrow&
    Cone(g)
    \\
    && && &&
    \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}}
    \\
    && && &&
    S X
      &\overset{S f}{\longrightarrow}&
    S Y
      &\longrightarrow& \cdots
  }
$$

## Pointed cofiber sequence

(...)


topological complexity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

The *topological complexity* [[Farber 2001](#Farber01)] of a [[topological space]] is a [[topological invariant]] related to the problem of motion planning and the [[immersion]] problem of [[real projective space]].

## Definition

For a [[topological space]] $X$ its _topological complexity_ $TC(X)$ is the smallest number $n$, so that there is an [[open cover]] $(U_k)_{k=1}^n$ of $X\times X$ by $n$ [[open subsets]] addmitting [[local sections]] $s_k\colon X\times X\rightarrow P X$ of the [[evaluation map]]

$$
  \array{
    P X &\longrightarrow& X\times X
    \\
    \gamma &\mapsto& \big(\gamma(0),\gamma(1)\big)
  }
$$

of the [[path space]] $P X$.

It is also possible to define it using the [[Schwarz genus]] of the path space fibration $\Omega X\hookrightarrow P X\rightarrow X$.

There is also the convention of using the smallest number $n$, so that an [[open cover]] $(U_k)_{k=0}^n$ of $n+1$ [[open sets]] of $X\times X$ with the above property exists. This lowers all topological complexities by one, hence the convention used can be given by giving the topological complexity of the set with one point. ($TC(*)=1$ for the upper convention and $TC(*)=0$ for the lower convention.)

## Properties

\begin{proposition}
A [[topological space]] $X$ is [[contractible]] iff $TC(X)=1$.
\end{proposition}

([Farber 01, Theorem 1](#Farber01))

\begin{proposition}
The [[topological complexity]] is only dependent on the [[homotopy type]] of a [[topological space]].
\end{proposition}

([Farber 01, Theorem 3](#Farber01))

## Special topological complexities

### Spheres and tori

\begin{proposition}
The [[topological complexity]] of a [[sphere]] is
\[
  \label{TopologicalComplexityOfSpheres}
    TC
    \big(
      S^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       2 & n \; odd
       \\
       3 & n \; even
    \end{array}
    \right.
\]
(with convention $\operatorname{TC}(*)=1$)
\end{proposition}

([Farber 01, Theorem 8](#Farber01))

This theorem can be generalized:

\begin{proposition}
The [[topological complexity]] of a product of spheres is
\[
  \label{TopologicalComplexityOfProductOfSpheres}
    TC
    \big(
      (S^m)^n
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       n+1 & m \; odd
       \\
       2n+1 & m \; even
    \end{array}
    \right.
\]
(with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber 01, Theorem 13](#Farber01))

A special case of this proposition is $TC(T^n)=n+1$ for the [[topological complexity]] of the [[torus]].

### Real and complex projective space

\begin{proposition}
For $n\neq 1,3,7$, the smallest [[natural number]] $k\in\mathbb{N}$, so that there exists an [[immersion]] of [[real projective space]] $\mathbb{R}P^n$ into [[euclidean space]] $\mathbb{R}^{k-1}$ is the topologial complexity $\operatorname{TC}(\mathbb{R}P^n)$ (with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02,  Theorem 12](#FarberTabachnikovYuzvinsky02))

\begin{proposition}
For $n=1,3,7$, one has $\operatorname{TC}(\mathbb{R}P^n)=n+1$ (with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02,  Proposition 18](#FarberTabachnikovYuzvinsky02))

\begin{proposition}
For any $n\in\mathbb{N}$, one has $\operatorname{TC}(\mathbb{C}P^n)=2n+1$ (with convention $\operatorname{TC}(*)=1$).
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02,  Corollary 2](#FarberTabachnikovYuzvinsky02))

### $\Sigma$ and $\Xi$ surfaces

\begin{proposition}
The [[topological complexity]] of a $\Sigma$ surface is
\[
  \label{TopologicalComplexityOfSigmaSurface}
    TC
    \big(
      \Sigma_g
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       3 & m \leq 1
       \\
       5 & m \gt 1
    \end{array}
    \right.
\]
(with convention $TC(*)=1$)
\end{proposition}

([Farber 01, Theorem 9](#Farber01))

\begin{proposition}
For $n \geq 2$ and $g \geq 2$ one has
\[
  \label{TopologicalComplexityOfConnectedSumOfRealProjectiveSpace}
    TC
    \big(
      (\mathbb{R}P^n)^g
    \big)
    \;=\;
    2n
\]
for the [[connected sum]] of [[real projective space]] (with convention $TC(*)=0$).
\end{proposition}

([Cohen & Vandembrouq 18, Theorem 1.3.](#CohenVandembroucq18))

### Klein bottle

\begin{proposition}
The topological complexity of the [[Klein bottle]] is $4$ (with convention $TC(*)=0$).
\end{proposition}

([Cohen & Vandembrouq 16, Theorem 1](#CohenVandembroucq16))

## Configuration space

\begin{proposition} The topological complexity of a [[configuration space]] is
\[
  \label{TopologicalComplexityOfConfigurationSpace}
    TC
    \big(
      Conf(\mathbb{R}^m,n)
    \big)
    \;=\;
    \left\{
    \begin{array}{ll}
       2n-1 & m \; odd
       \\
       2n-2 & m \; even
    \end{array}
    \right.
\]
(with convention $TC(*)=1$).
\end{proposition}

([Farber & Grant 08, Theorem 1](#FarberGrant08))

## References

Definition and basic properties of topological complexity:

* {#Farber01} [[Michael Farber]], *Topological complexity of motion planning*, Discrete Comput Geom **29** (2001) 211–221, [[arXiv:math/0111197](https://arxiv.org/abs/math/0111197), [doi:10.1007/s00454-002-0760-9](https://doi.org/10.1007/s00454-002-0760-9)]

See also:

* Wikipedia, *[Topological complexity](https://en.wikipedia.org/wiki/Topological_complexity)*

On topological complexity of [[real projective space]] and connection with their [[immersion]] into [[cartesian space]]:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018);

On topological complexity of [[connected sums]]:

* {#CohenVandembroucq18} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Motion planning in connected sums of real projective spaces_ (2018), [arXiv:1807.09947](https://arxiv.org/abs/1807.09947)

On topological complexity of the [[Klein bottle]]:

* {#CohenVandembroucq16} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Topological Complexity of the Klein bottle_ (2018), [arXiv:1612.03133v3](https://arxiv.org/abs/1612.03133v3)

On topological complexity of [[configuration space]]:

* {#FarberGrant08} [[Michael Farber]], [[Mark Grant]], _Topological Complexity of the Klein bottle_ (2008), [arXiv:0806.4111](https://arxiv.org/pdf/0806.4111.pdf)


topological concrete category

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Topological categories
* table of contents
{: toc}

## Warning

The term 'topological category' is traditional, and comes from the frequent examples in [[topology]].  It does *not* mean an [[internal category]] or [[enriched category]] in [[Top]] (a [[topologically enriched category]])!  (Fortunately the term [[topological groupoid]] is not taken by this tradition; indeed, the only groupoid that is a topological category over $Set$ is [[terminal category|trivial]].  On the other hand, there is use of the term 'topological functor', which  we tend to  avoid other than below.)

## Idea

A _topological category_ is a [[concrete category]] with nice features matching the ability to form [[weak topology|weak]] and [[strong topology|strong]] topologies in [[Top]].

## Definition

### Default version

Most generally, the definition relates to a [[functor]] $U\colon C \to D$ (such as the [[forgetful functor]] from $Top$ to [[Set]]), but one can think of this as giving $C$ as a [[bundle]] over $D$. Sometimes, when $D$ is in fact [[Set]], the category $C$ satisfying the properties described belows is called a _topological construct_ [(Preuss)](#Preuss). Usually $C$ and $D$ will be [[large categories]].

By a _space_ we will mean an object of $C$, and by an _algebra_ we will mean an object of $D$.  By a _map_ we will mean a morphism in $C$, and by a _homomorphism_ we will mean a morphism in $D$.  (The reason is that, typically, $C$ will be a category of spaces with some kind of topological structure while $D$ will be, if not $Set$, then some kind of algebraic category.)

Then $C$ is a __topological category__ over $D$ if, given any algebra $X$ and any (possibly large) family of spaces $S_i$ and homomorphisms $f_i\colon X \to U(S_i)$ (that is, a "$U$-structured" [[sink|source]] from $X$), there exists an [[initial lift]] (think: "smallest topology rendering the $f_i$ continuous"), which is to say

*  a space $T$ such that $U(T)=X$, and maps $m_i\colon T \to S_i$ such that $U(m_i) = f_i$, and

*  given any space $T'$, homomorphism $g'\colon U(T') \to X$, and maps $m'_i\colon T' \to S_i$, if each composite $g' ; f_i$ equals $U(m'_i)$, then there exists a unique map $n\colon T' \to T$ such that $U(n) = g'$ and $n ; m_i = m'_i$.

Here are some illustrative commutative diagrams (if you can read them):
$$ \array {
T' \\
n \downarrow \downarrow n' & \searrow^{m'_i} \\
T                          & \underset{m_i}\rightarrow & S_i
} \;\;\; \stackrel{U}\mapsto \;\;\; \array {
U(T') \\
U(n) \downarrow \downarrow U(n') & \searrow^{g'}                         &                                  & \searrow^{U(m'_i)} \\
U(T)                             & = & X                                & \underset{f_i}\rightarrow & U(S_i) \\
                                 &                                       & \underset{U(m_i)}\longrightarrow
} $$

It follows by a clever argument that $U\colon C \to D$ must be [[faithful functor|faithful]]; see Theorem 21.3 of [ACC][acc].  That is also often included in the definition, in which case the uniqueness of $n$ can be left out.  Thus we may think of objects of $C$ as objects of $D$ equipped with [[extra structure]].  The idea is then that $T$ is $X$ equipped with the __initial structure__ or __[[weak structure]]__ determined by the requirement that the homomorphisms $f_i$ be structure-preserving maps.

The dual concept could be called a _cotopological category_.  However, this is not actually anything new; $U\colon C \to D$ is topological if and only if $U^op\colon C^op \to D^op$ is.  This is a [[categorification]] of the theorem that any complete [[semilattice]] is a [[complete lattice]].  Thus, every topological category also has __final__ (not usually called _terminal_) or __[[strong structure|strong]]__ structures, each determined by a family of homomorphisms $f_i\colon U(S_i) \to X$ (a $U$-structured [[sink]] to $X$).

Both of these results (faithfulness and self-duality) depend on the fact that we have allowed the family $\{S_i\}$ to be potentially *large*.  Counterexamples are easy to find.  For instance, if $C$ is a large category with all (small) products, then the functor $C \to 1$ to the [[terminal category]] satisfies the above lifting property for small families $\{S_i\}$.  However, it need not satisfy the dual property (unless $C$ also has all small coproducts) nor need it be faithful.

It also follows that $U$ is a [[Grothendieck fibration]] and opfibration.

### Amnestic version

Since initial lifts have a universal property, they are unique up to unique isomorphism.  However, it is traditional in some literature to ask that they be literally unique (this is done for instance in [ACC][acc]). This is tantamount to deciding that $U$ should be an [[amnestic functor]].  A drawback (from an [[nPOV]]) is that this condition violates the [[principle of equivalence]], and arguably doesn't add anything mathematically important.

Thus, although it occurs in the literature, here we will consider it purely optional. (It is possible that some results recorded here about topological categories will depend on this assumption, but only results not respecting the [[principle of equivalence]] could be affected.)

### Weak version

On the other hand, the default definition above does already refer to equality of objects in the condition $U(T)=X$; thus as stated it *already* violates the [[principle of equivalence]], just as the notion of [[Grothendieck fibration]] does.  But (also as for Grothendieck fibrations) this other use of equality of objects is really more of a "typing judgment", which can be made precise by working with [[displayed categories]] instead.  (In the context of [[homotopy type theory]], the amnestic condition is equivalent to "fiberwise [[internal category in homotopy type theory|univalence]]".)

However, if we want to, we can also formulate a "fully isomorphism-invariant" version of the definition, corresponding to the weakened bicategorical notion of [[Street fibration]].  In this case, an initial lift consists of:

*  a space $T$, an [[isomorphism]] $g\colon U(T) \to X$, and maps $m_i\colon T \to S_i$ such that each [[composite]] $g ; f_i$ equals $U(m_i)$ and,

*  given any space $T'$, homomorphism $g'\colon U(T') \to X$, and maps $m'_i\colon T' \to S_i$, if each composite $g' ; f_i$ equals $U(m'_i)$, then there exists a unique map $n\colon T' \to T$ such that $U(n) ; g = g'$ and $n ; m_i = m'_i$.

$$ \array {
T' \\
n \downarrow \downarrow n' & \searrow^{m'_i} \\
T                          & \underset{m_i}\rightarrow & S_i
} \;\;\; \stackrel{U}\mapsto \;\;\; \array {
U(T') \\
U(n) \downarrow \downarrow U(n') & \searrow^{g'}                         &                                  & \searrow^{U(m'_i)} \\
U(T)                             & \overset{\sim}\underset{g}\rightarrow & X                                & \underset{f_i}\rightarrow & U(S_i) \\
                                 &                                       & \underset{U(m_i)}\longrightarrow
} $$

## Examples

*  The name 'topological category' comes from these examples from point-set [[topology]]; these are all topological over [[Set]]:
   *  the category [[Top]] of [[topological spaces]],
   *  the category of [[convergence spaces]] (or of [[pseudotopological space|pseudotopological]] or of [[pretopological space|pretopological]] spaces),
   *  the category of [[uniform spaces]] or of [[Cauchy spaces]],
   *  lots more in this vein.

*  In contrast, the category of [[locales]] is *not* topological over $Set$, apparently not even the category of *spatial* locales (equivalent to the category of [[sober spaces]]), essentially because soberification of a topological space may not preserve the underlying set.

*  Also, the category [[Diff]] of [[smooth manifolds]] is not topological but most categories of [[generalized smooth space]]s are.

*  Outside of topology, the category of [[measurable spaces]] is topological over $Set$.

*  The category of [[topological groups]] is topological over [[Grp]], the category of [[topological vector spaces]] is topological over $\mathbb{R}$-[[Vect]], etc.

## Further properties

*  If $C$ is topological over $D$, then so is any full [[retract]] of $C$, as long as the functors involved live in $Cat/D$.

*  In particular, a [[reflective subcategory|reflective]] or [[coreflective subcategory|coreflective]] subcategory of $C$ is topological, as long as the reflectors or coreflectors become [[identity morphisms]] in $D$.

*  The forgetful functor $U\colon C \to D$ is not only faithful but also (because every algebra must have an initial/indiscrete topology determined by the empty source) [[essentially surjective functor|essentially surjective]] (in fact surjective on the nose for the non-weak definitions).  Thus it is never [[full functor|full]] (except in the trivial case where $U$ is an [[equivalence of categories|equivalence]], of course).

*  If $D$ is [[complete category|complete]] or [[cocomplete category|cocomplete]], then so is $C$.

* If $D$ is [[total category|total]] or cototal, then so is $C$; see [[solid functor]].

* If $D$ is [[mono-complete category|mono-complete]] or epi-cocomplete, then so is $C$.

*  If $D$ is [[well-powered category|well-powered]] or co-well-powered, then so is $C$.

* If $D$ has a [[factorization structure for sinks]] $(E,M)$, then $C$ has one $(E',M')$, where $M'$ is the collection of morphisms in $C$ lying over $M$-morphisms in $D$, and $E'$ the collection of *final* sinks in $C$ lying over $E$-sinks in $D$.  This generalizes the lifting of [[orthogonal factorization systems]] along [[Grothendieck fibrations]].

*  If $D$ is [[concrete category|concrete]], then so is $C$.  More generally, if $D$ has a [[generator]], then $C$ is concrete over $D$.

*  In particular, if $D$ is [[Set]], then $C$ is a concrete category that is complete, cocomplete, well powered, and well copowered.

## Functors

* A functor $F\colon C\to C'$ between topological concrete categories $C/D$, $C'/D$ with the same base category $D$ preserves initial lifts iff it is right adjoint. It preserves final lifts iff it is left adjoint.

* More generally: If a functor $F\colon C\to C'$ between topological concrete categories $C/D$, $C'/D'$ with different base categories lying over a functor $F_0: D\to D'$. If $F$ is right (left) adjoint, then $F_0$ is right (left) adjoint and $F$ preserves initial (final) lifts. A partial converse holds: If $F_0$ is right (left) adjoint to $G_0$ and $F$ preserves initial (final) lifts, then there is functor $G$ lying over $G_0$ so that $F$ is right (left) adjoint to $G_0$.

## Special cases

*  If $X$ is any algebra, then there is a _[[discrete space]]_ over $X$ induced by the empty family of maps.  Similarly, we have an _indiscrete space_ with the final structure induced by no maps.  This defines functors $disc, indisc\colon D \to C$ that are respectively left and right [[adjoint functor|adjoints]] of $U$.

*  Suppose that $D$ has an [[initial object]] $0_D$.  Then the discrete space $0_C$ over $0_D$ is initial in $C$.  Similarly, the indiscrete space over a [[terminal object]] in $D$ is terminal in $C$.

*  More generally, suppose that $D$ has [[products]] or [[coproducts]] (indexed by whichever [[cardinal number|cardinalities]] you may wish to consider).  Then $C$ also has (co)products, lying over the (co)products in $D$, with structures induced by the product projections or coproduct inclusions.

*  More general [[limits]] and [[colimits]] are constructed in a similar way.  However, it is _not_ typically the case that $U$ _[[created limit|creates]]_ (co)limits in $C$ because creation of a limit requires that every preimage of the limiting cone is limiting. This fails for $U: \mathrm{Top} \to \mathrm{Set}$ since we can coarsen the topology on the limit vertex to obtain a counterexample.

*  If a single algebra $X$ has been given the structure of several spaces, then there are a _[[supremum]]_ structure and an _[[infimum]]_ structure on $X$ induced (as the initial and final structures) by the various incarnations of its [[identity morphism|identity]] homomorphism.  Exploiting this shows how to construct final structures out of initial ones and conversely.

*  If $X$ is a [[regular monomorphism|regular]] [[subobject|subalgebra]] of some $U(S)$, then the inclusion homomorphism makes $X$ into a _[[subspace]]_ of $S$, which is also a subobject in $C$.  Every regular subobject of $S$ is of this form; note however that there may be nonregular subobjects in $C$ even if all subobjects in $D$ are regular.

## Familiarly fibrations

The theory of topological functors can be developed along the lines of Grothendieck's theory of fibrations, where  _cartesian morphisms_  are replaced by  _cartesian families_. In this way just as by definition  "A functor is a _fibration_ if it creates cartesian morphisms and cartesian morphism compose", there is the definition "A functor is _topological_  if it creates cartesian families and cartesian families compose".

## References

* [[Jiří Adámek]], Horst Herrlich, & George E. Strecker; 1990; Abstract and Concrete Categories; originally published John Wiley & Sons ISBN 0-471-60922-6; [free on-line edition][acc] (4.2MB PDF).

[acc]: http://katmat.math.uni-bremen.de/acc "Abstract and Concrete Categories"

* Gerhard Preuss; 2002; _Foundations of Topology: An Approach to Convenient Topology_; Kluwer ISBN 1-4020-0891-0.
 {#Preuss}

* [[Richard Garner]], *Topological functors as total categories*, [TAC](http://www.tac.mta.ca/tac/volumes/29/15/29-15abs.html)

* [[Eduardo J. Dubuc]], Luis Español, *Topological functors as familiarly fibrations*
[arXiv](https://arxiv.org/abs/math/0611701) (2006)

* J. Martin Harvey, _Topological functors from factorization_, Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978. Springer Berlin Heidelberg, 1979.

[[!redirects topological concrete category]]
[[!redirects topological concrete categories]]

[[!redirects topological construct]]
[[!redirects topological constructs]]

[[!redirects topological category]]
[[!redirects topological categories]]

[[!redirects topological functor]]
[[!redirects topological functors]]


topological crystalline insulator

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### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], a *topological crystalline insulator* is a [[crystal|crystalline]] [[quantum material]] in a [[topological phase of matter]] like a [[topological insulator]], which as such is "[[symmetry protected topological phase|protected]]" by (possibly [[time-reversal symmetry]] and) a [[crystallographic group|crystallographic point group]]-symmetry of the underlying [[crystal]].

This means that a topological crystalline insulator phase observed in the [[bulk]] of some [[crystal]] whose [[dynamics]] (or rather that of its [[electrons]]) respects a given [[crystallographic group|point group]] may disappear for instance at [[boundary]] regions where the dynamics [[symmetry breaking|breaks]] this symmetry.

## References

### Original articles

* [[Liang Fu]], *Topological Crystalline Insulators*, Phys. Rev. Lett. **106** (2011) 106802 ([doi:10.1103/PhysRevLett.106.106802](https://doi.org/10.1103/PhysRevLett.106.106802))

* [[Robert-Jan Slager]], Andrej Mesaros, Vladimir Juricic, [[Jan Zaanen]], *The space group classification of topological band insulators*, Nature Physics **9** 98 (2013) $[$[doi:10.1038/nphys2513](https://doi.org/10.1038/nphys2513), [arXiv:1209.2610](https://arxiv.org/abs/1209.2610)$]$

* Jie Shen, Judy J. Cha,  *Topological Crystalline Insulator Nanostructures*, Nanoscale **6** 23 (2014) 14133-14140 $[$[doi:10.1039/c4nr05124f](https://doi.org/10.1039/c4nr05124f)$]$

* [[Jorrit Kruthoff]], [[Jan de Boer]], Jasper van Wezel, [[Charles L. Kane]], [[Robert-Jan Slager]], *Topological Classification of Crystalline Insulators through Band Structure Combinatorics*, Phys. Rev. X **7** (2017) 041069 $[$[doi:10.1103/PhysRevX.7.041069](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.041069)$]$

  > (explicit relation to [[equivariant K-theory]] in appendix B)

### Review

* [[Yoichi Ando]] and [[Liang Fu]], *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Annual Review of Condensed Matter Physics **6** (2015) 361-381 [[doi:10.1146/annurev-conmatphys-031214-014501](https://doi.org/10.1146/annurev-conmatphys-031214-014501)]

* [[Titus Neupert]], [[Frank Schindler]], *Lecture Notes on Topological Crystalline Insulators*, in: *Topological Matter* Springer Series in Solid-State Sciences **190**, Springer (2018) [[arXiv:1810.03484](https://arxiv.org/abs/1810.03484), [doi:10.1007/978-3-319-76388-0_2](https://doi.org/10.1007/978-3-319-76388-0_2)]

  > (emphasis on [[Wilson loops]] of the [[Berry connection]])

* Chieh-Wen Liu, Zhenhua Wang, Richard L J Qiu, Xuan P A Gao, *Development of topological insulator and topological crystalline insulator nanostructures*,  Nanotechnology **31** (2020) 192001 [[doi:10.1088/1361-6528/ab6dfc](https://iopscience.iop.org/article/10.1088/1361-6528/ab6dfc)]

* [[M. Zahid Hasan]], *[Topological Crystalline Insulators](https://physics.princeton.edu/zahidhasangroup/index_TCI.html)* (research group page)

### Experimental realization
 {#ReferencesExperimentalRealization}

* Y. Tanaka, Zhi Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, Kouji Segawa, [[Yoichi Ando]], *Experimental realization of a topological crystalline insulator in SnTe*, Nature Physics **8** (2012) 800–803 $[$[doi:10.1038/nphys2442](https://doi.org/10.1038/nphys2442)$]$

* [[M. Zahid Hasan]] et al., *Observation of a topological crystalline insulator phase and topological phase transition in Pb1−xSnxTe*, Nature Communications **3** 1192 (2012) $[$[doi:10.1038/ncomms2191](https://doi.org/10.1038/ncomms2191)$]$

* Timothy H Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, [[Liang Fu]], *Topological crystalline insulators in the SnTe material class*,
Nature Communications **3** (2012) 982 $[$[doi:10.1038/ncomms1969](https://doi.org/10.1038/ncomms1969)$]$

* [[M. Zahid Hasan]] et al., *Observation of Dirac Node Formation and Mass Acquisition in a Topological Crystalline Insulator*, Science **341** 1496 (2013) $[$[doi:10.1126/science.1239451](https://www.science.org/doi/10.1126/science.1239451)$]$

* [[M. Zahid Hasan]] et al., *Mapping the unconventional orbital texture in topological crystalline insulators*, Nature Physics **10** (2014) 572–577 $[$[doi:10.1038/nphys3012](https://doi.org/10.1038/nphys3012)$]$

* Wenhui Fan, Simin Nie, Cuixiang Wang, Binbin Fu, Changjiang Yi, Shunye Gao, Zhicheng Rao, Dayu Yan, Junzhang Ma, Ming Shi, Yaobo Huang, Youguo Shi, Zhijun Wang, Tian Qian, Hong Ding,

  *Discovery of Ĉ2 rotation anomaly in topological crystalline insulator SrPb*, Nature Communications **12** 2052 (2021) $[$[doi:10.1038/s41467-021-22350-6](https://doi.org/10.1038/s41467-021-22350-6)$]$

[[!redirects topological crystalline insulators]]

[[!redirects crystalline topological insulator]]
[[!redirects crystalline topological insulators]]


topological cyclic homology

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Topological Hochschild homology__ (resp. __topological cyclic homology__)  (see the survey in ([May](#May))) is a refinement of [[Hochschild homology]]/[[cyclic homology]] from [[commutative rings]]/algebras to the [[higher algebra]] of [[ring spectra]]/[[E-∞ rings]]/[[E-∞ algebras]].

One motivation for their study: Comes from computational considerations, as well in certain cases, these invariants are easier to compute than [[algebraic K-theory]], while there is a natural morphism of [[spectra]]

$$
  \array{
     && \mathbf{TC}(R)
     \\
      & {}^{\mathllap{cyclotomic \atop trace}}\nearrow & \downarrow
     \\
     \mathbf{K}(R) &\underset{}{\longrightarrow}& \mathbf{THH}(R)
  }
$$

from the [[algebraic K-theory]] spectrum to the [[topological Hochschild homology]] spectrum, called the [[Dennis trace]] map, whose [[homotopy fiber|fiber]] is relatively well-understood. Since Hochschild homology spectra are naturally [[cyclotomic spectra]], this map factors through the topological cyclic homology spectrum via a map called the _[[cyclotomic trace]]_, which acts much like a [[Chern character]] map for [[algebraic K-theory]].

The spectra $THH(R)$ and $TC(R)$ are typically easier to analyze than $K(R)$. Moreover, the difference between them and $K(R)$ is "locally constant" ([Dundas-Goodwillie-McCarthy13](#DundasGoodwillieMcCarthy13)) and often otherwise bounded in complexity. Accordingly, $THH$ and $TC$ are in practice computationally useful approximations to $K$.

There are various generalizations:

1. Just as for basic Hochschild homology, there is _higher topological Hochschild homology_ ([Carlsson-Douglas-Dundas 08](#CarlssonDouglasDundas08)) given not just by [[derived loop spaces]] but by derived mapping spaces out of higher dimensional [[tori]].

1. Just as algebraic K-theory generalizes from [[E-∞ rings]] to [[stable ∞-categories]], so do $TC$ and the cyclotomic trace map ([Blumberg-Gepner-Tabuada 11](#BlumbergGepnerTabuada11))

## Related concepts

* [[cyclic homology]], [[dihedral homology]]

* [[Chern character]]

* [[norm cofibration]]

* [[prismatic cohomology]]

## References

### General

The original references are

* [[Marcel Bökstedt]], _Topological Hochschild homology_, Bielefeld, 1985, 1988

* [[Marcel Bökstedt]], W.C. Hsiang, [[Ib Madsen]], _The cyclotomic trace and algebraic K-theory of spaces_, Invent. Math. __111__ (1993), 463-539, [MR94g:55011](http://www.ams.org/mathscinet-getitem?mr=1202133), [doi](http://dx.doi.org/10.1007/BF01231296)

* [[Marcel Bökstedt]], [[Ib Madsen]], _Topological cyclic homology of the integers_, $K$-theory (Strasbourg, 1992). Astérisque __226__ (1994), 7–8, 57–143.

and a further generalization is defined in

* [[Bjørn Ian Dundas]], Randy McCarthy, _Topological Hochschild homology of ring functors and exact categories_, J. Pure Appl. Algebra __109__ (1996), no. 3, 231–294, [MR97i:19001](http://www.ams.org/mathscinet-getitem?mr=1388700), doi

Higher topological Hochschild Homology is discussed in

* {#CarlssonDouglasDundas08} [[Gunnar Carlsson]], [[Christopher Douglas]], [[Bjørn Ian Dundas]], _Higher topological cyclic homology and the Segal conjecture for tori_,
Adv. Math. __226__ (2011), no. 2, 1823–1874, ([arXiv:0803.2745](http://arxiv.org/abs/0803.2745), [MR2737802](http://www.ams.org/mathscinet-getitem?mr=2737802), [doi](http://dx.doi.org/10.1016/j.aim.2010.08.016))

A [[(infinity,1)-category theory|$(\infty,1)$-category theoretic]] construction:

* {#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)]

A relation to integral [[p-adic Hodge theory]] is discussed in

* {#BhattMorrowScholze18}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild Homology and Integral $p$-adic Hodge Theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]

Review and exposition includes

* {#May} [[Peter May]], _Topological Hochschild and Cyclic Homology and Algebraic K-theory_ ([pdf](http://www.math.uchicago.edu/~may/TALKS/THHTC.pdf))

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter IX of _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], _The local structure of algebraic K-theory_, Springer 2013 ([pdf](http://math.mit.edu/~nrozen/juvitop/dundas.pdf))

* {#Gerhardt14} [[Teena Gerhardt]], _Computations in algebraic K-theory_, talk at [CUNY Workshop on differential cohomologies 2014](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html) ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1800/in/channel/55/))

* Ricardo Andrade, _THH notes_, MIT juvitop seminar [pdf](http://math.mit.edu/~nrozen/juvitop/ra%20-%20juvitop.pdf), babytop seminar [pdf](http://math.mit.edu/~nrozen/juvitop/ra%20-%20babytop.pdf)

* Anatoly Preygel, _Hochschild homology notes_, juvitop seminar, [pdf](http://math.mit.edu/~nrozen/juvitop/note_hh_talk.pdf)

* Thomas Geisser, _Motivic Cohomology, K-Theory and Topological Cyclic Homology_, Handbook of K-theory II.1, [pdf](http://www.math.uiuc.edu/K-theory/handbook/1-193-234.pdf)

* Ib Madsen, _Algebraic K-theory and traces_, [pdf](https://math.mit.edu/juvitop/old/notes/2009_Fall/madsen.pdf)

Abstract characterization of the [[Dennis trace]] and cyclotomic trace is discussed in

* {#BlumbergGepnerTabuada11} [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _Uniqueness of the multiplicative cyclotomic trace_, Advances in Mathematics 260 (2014) 191-232 ([arXiv:1103.3923](http://arxiv.org/abs/1103.3923))

See also

* T. Pirashvili, F. Waldhausen, _Mac Lane homology and topological Hochschild homology_, J. Pure Appl. Algebra __82__ (1992), 81-98, [MR96d:19005](http://www.ams.org/mathscinet/search/mathscinet-getitem?mr=520492), doi

* [[T. Pirashvili]], _On the topological Hochschild homology of $\mathbf{Z}/p^k\mathbf{Z}$_, Comm. Algebra __23__ (1995), no. 4, 1545–1549, [MR97h:19007](http://www.ams.org/mathscinet-getitem?mr=1317414), [doi](http://dx.doi.org/10.1080/00927879508825293)

* Z. Fiedorowicz, T. Pirashvili, R. Schwänzl, R. Vogt, F. Waldhausen, _Mac Lane homology and topological Hochschild homology_, Math. Ann. __303__ (1995), no. 1, 149–164, [MR97h:19007](http://www.ams.org/mathscinet-getitem?mr=1348360), [doi](http://dx.doi.org/10.1007/BF01460984)

* [[Bjørn Ian Dundas]], _Relative K-theory and topological cyclic homology_, Acta Math. __179__ (1997), 223-242, ([publisher](http://link.springer.com/article/10.1007%2FBF02392744))

* Thomas Geisser, Lars Hesselhoft, _Topological cyclic homology of schemes_, in: Algebraic $K$-theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. __67__, Amer. Math. Soc. 1999, [MR2001g:19003](http://www.ams.org/mathscinet-getitem?mr=1743237); [K-theory archive](http://www.math.uiuc.edu/K-theory/0231/)

* R. McCarthy, _Relative algebraic K-theory and topological cyclic homology_, Acta Math. __179__ (1997), 197-222.

* J. McClure, R. Staffeldt, _On the topological Hochschild homology of $b u$, I_, [pdf](http://math.mit.edu/~nrozen/juvitop/mcclure.pdf)

* Daniel Joseph Vera, _Topological Hochschild homology of twisted group algebra_, MIT Ph. D. thesis 2006, [pdf](http://dspace.mit.edu/bitstream/handle/1721.1/34615/71329129.pdf?sequence=1)

* V. Angeltveit, A. Blumberg, T. Gerhardt, M. Hill, T. Lawson, M. Mandell, _Topological cyclic homology via the norm_ ([arXiv:1401.5001](http://arxiv.org/abs/1401.5001))

### Examples
 {#ReferencesExamples}

THH and TC specifically of [[KU|ku]] and [[KO|ko]] is discussed in

* [[Christian Ausoni]], [[John Rognes]], _Algebraic K-theory of topological K-theory_, Acta Mathematica March 2002, Volume 188, Issue 1, pp 1-39 ([KTheory 0405](http://www.math.uiuc.edu/K-theory/0405/))

* [[Vigleik Angeltveit]], [[Michael Hill]], [[Tyler Lawson]], _Topological Hochschild homology of $\ell$ and $ko$_ ([arXiv:0710.4368](http://arxiv.org/abs/0710.4368))

* [[Andrew Blumberg]], [[Michael Mandell]], _Localization for $THH(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories_ ([arXiv:1111.4003](http://arxiv.org/abs/1111.4003))

and of [[tmf]] in

* [[Robert Bruner]], [[John Rognes]], *Topological Hochschild homology of topological modular forms*, talk at *Nordic Topology Meeting* NTNU (2008) [[[Bruner-Rognes_THHtmf.pdf:file]]]

On the topological Hochschild homology of the [[Lubin-Tate theory|Lubin-Tate ring spectrum]] via [[factorization homology]]:

* [[Geoffroy Horel]], *Higher Hochschild homology of the Lubin-Tate ring spectrum*, Algebr. Geom. Topol. **15** (2015) 3215-3252 [[arXiv:1311.2805](https://arxiv.org/abs/1311.2805), [doi:10.2140/agt.2015.15.3215](https://doi.org/10.2140/agt.2015.15.3215)]

[[!redirects topological Hochschild homology]]

[[!redirects THH]]
[[!redirects TC]]


topological data analysis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Topological data analysis
* table of contents
{: toc}

## Idea

\begin{imagefromfile}
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    }
\end{imagefromfile}

*Topological data analysis* (TDA) is *qualitative* [[data analysis]] with tools from [[topology]], in particular with tools from *[[algebraic topology]]*, aiming to extract (hidden) structure in large datasets which is *robust* against uncertainties and noise.

This notably includes tools from [[ordinary homology|ordinary]] ([[cohomology theory|co-]])[[homology]]-theory, which in the guise of *[[persistent homology]]* has become the signature method in TDA; but it also includes more general tools of [[homotopy theory]] and [[differential topology]], which have more recently found their way into TDA in the guise of *[[persistent homotopy theory]]* and *[[persistent cohomotopy theory]]*. $\;\;\;\;\;\;\;$(graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

Typically, TDA deals with *large* data sets modeled as ([[subsets]] of) [[topological spaces]]. Collections of *data points* appear as *[[cycles]]* in the topological space of data, and *values* of data appear as *[[cocycles]]*.

### Strategy of persistent topology/homotopy
 {#StrategyOfPersistentHomology}

\begin{imagefromfile}
    "file_name": "SomewhatPersistentCycle-220523.gif",
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\end{imagefromfile}

The strategy of [[persistent homology|persistent homology]]/[[persistent homotopy|homotopy]] in TDA  is to see which chunks ([[subspaces]]) of the data appear to be ([[n-connected topological space|higher]]) [[connected topological space|connected]] when viewed at some resolution (technically: at some [[filtered topological space|filter stage]]) and how much these apparent chunks *persist* as the resolution changes.
This may be recorded in *[[persistence diagrams]]* also known as "[[barcodes]]".

The fundamental theorem of the field -- the *[[stability of persistence diagrams|stability theorem]]* -- says that [[persistence diagram|persistent]] ([[cocycle|co-]])[[cycle]]-[[equivalence class|classes]] are indeed a good invariant of data, in that they remain stable under small perturbations of the initial data (e.g. under [[noise]], uncertainty, measurement errors, etc.).

The idea then is that those  ([[cocycle|co-]])[[cycles]] which persist for longer (appear as longer bars in the [[barcode]]) reflect relevant structure hidden in the (large) data set.

However, it is often unclear (and certainly not part of the mathematical theory) what significance or *meaning* any persistent cycle has for the practical problem of interpreting data. $\;\;\;\;\;\;$(graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

\linebreak

### Strategy of persistent cohomology/cohomotopy
 {#StrategyOfPersistentCohomotopy}

\begin{imagefromfile}
    "file_name": "IndicatorFunctionOnData-220523.jpg",
    "float": "right",
    "width": 420,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 0,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In contrast, [[persistent cohomotopy]] in TDA is the effective answer to a concrete and common question in [[data analysis]]:

Given a large-[[dimension of a CW-complex|dimensional]] [[topological space|space]] of data, and a small [[natural number|number]] $n$ of ([[real number|real]]) indicator values assigned to each data point with given precision $1/r$, *does _any_ data meet a prescribed target indication precisely?*

{#FundamentalTheoremOfPersistentCohomotopy} A [fundamental theorem](persistent+cohomotopy#GuaranteeThatSolutionsExist) of [[persistent Cohomotopy]] ([Franek,  Krčál & Wagner 2018](persistent+cohomotopy#FranekKrcalWagner18), [Franek & Krčál 2017, p. 5](persistent+cohomotopy#FranekKrcal17), see [here](persistent+cohomotopy#GuaranteeThatSolutionsExist)) shows that (1.) the answer to this question is detected by a certain [[Cohomotopy]]-class and (2.) in a fair range of dimensions, this Cohomotopy class is provably [[computability|computable]], hence the above question is effectively [[decidability|decidable]]. $\phantom{--}$ (graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

> (graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

(Alternatively, with tools from [[persistent homology theory]] an answer to this question is given by the method of *[[well groups]]* -- but (1.) it is known that well groups are in general too coarse to provide a complete answer and (2.) despite effort it remains unknown if well groups are actually [[computability|computable]] in relevant cases, see [Franek & Krčál 2016](persistent+cohomotopy#FranekKrcal16).)

If the [[topological space]] $X$ of data may be assumed to be a [[smooth manifold]] (indeed, in typical examples $X$ is itself a large-dimensional [[Cartesian space]]) then persistent cohomotopy may be understood dually via [[Pontryagin's theorem]] as characterizing [[iso-hypersurfaces]] of data (close to a given target indicator) by [[framed cobordism|framed]] *[[cobordism theory]]* ([Franek & Krčál 2017, p. 8-9](persistent+cohomotopy#FranekKrcal17)). The full implications of this relation for topological data analysis remain to be explored.

## Related concepts

* [[persistent homology]], [[persistent homotopy]]

* [[applied topology]]

* [[computational topology]]

* [[machine learning]]

* [[topological machine learning]]

## References

### General

General introduction and survey:

* [[Robert Ghrist]], _Barcodes: The Persistent Topology of Data_, Bull. Amer. Math. Soc. 45 (2008), 61-75 ([doi:10.1090/S0273-0979-07-01191-3](https://doi.org/10.1090/S0273-0979-07-01191-3), [pdf](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf))

* [[Herbert Edelsbrunner]], [[John Harer]], *Persistent homology -- a survey*, in: *Surveys on Discrete and Computational Geometry: Twenty Years Later*, Contemporary Mathematics *453* (2008) $[$[doi:10.1090/conm/453](http://dx.doi.org/10.1090/conm/453)$]$

* [[Gunnar Carlsson]], *Topology and data*, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308 $[$[doi:10.1090/S0273-0979-09-01249-X](https://doi.org/10.1090/S0273-0979-09-01249-X)$]$

* [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], *Persistent homology: theory and practice*, in: *European Congress of Mathematics Kraków, 2–7 July, 2012* EMS $[$[doi:10.4171/120-1/3](https://www.ems-ph.org/books/show_abstract.php?proj_nr=170&vol=1&rank=3), [pdf](http://mrzv.org/publications/persistent-homology-theory-practice/ecm)$]$

* [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf), [ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/)$]$

* [[Gunnar Carlsson]], *Persistent Homology and Applied Homotopy Theory*, in: [[Handbook of Homotopy Theory]], CRC Press (2019) $[$[arXiv:2004.00738](https://arxiv.org/abs/2004.00738), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)$]$

* [[Donald Pinckney]], *[Topological Data Analysis and Persistent Homology](https://donaldpinckney.com/machine%20learning/2019/05/02/tda.html)* (2019)

Flashy exposition for commercial use:

* [[Gunnar Carlsson]], *Professor Gunnar Carlsson Introduces Topological Data Analysis* $[$[video](https://www.ayasdi.com/resources/professor-gunnar-carlsson-introduces-topological-data-analysis/)$]$

See also:

* Wikipedia, _[Topological data analysis](http://en.wikipedia.org/wiki/Topological_data_analysis)_

and see the references at

* [[persistent homology]]

* [[persistent homotopy]]

* [[persistent cohomotopy]]

### Implementation

Implementation via concrete [[algorithms]], etc.

* {#SinghMemoluCarlsson07} [[Gurjeet Singh]], [[Facundo Mémoli]], [[Gunnar Carlsson]], *Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition*, in: M. Botsch, R. Pajarola (eds.), *Eurographics Symposium on Point-Based Graphics* (2007) $[$[doi:10.2312/SPBG/SPBG07/091-100](http://dx.doi.org/10.2312/SPBG/SPBG07/091-100), [pdf](http://diglib.eg.org/bitstream/handle/10.2312/SPBG.SPBG07.091-100/091-100.pdf?sequence=1&isAllowed=y)$]$

Review:

* [[Stefan Huber]], *Persistent Homology in Data Science* In: *Data Science – Analytics and Applications* Springer (2021) $[$[doi:10.1007/978-3-658-32182-6_13](https://doi.org/10.1007/978-3-658-32182-6_13), [pdf](https://www.sthu.org/research/publications/files/Hub20.pdf)$]$

### Applications

Application of [[topological data analysis]] ([[persistent homology]]) to

analysis of [[quasicrystals]]:

* Pavlo Solokha et al., *New Quasicrystal Approximant in the Sc–Pd System: From Topological Data Mining to the Bench*, Chem. Mater. 2020, 32, 3, 1064–1079 ([doi:10.1021/acs.chemmater.9b03767](https://doi.org/10.1021/acs.chemmater.9b03767))

analysis of [[cosmological structure formation]]:

* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))

analysis of [[phase transitions]]:

* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))

recognition of [[instantons]] and [[confinement]] in [[lattice gauge theory]]:

* Daniel Spitz, Julian M. Urban, Jan M. Pawlowski, *Confinement in non-Abelian lattice gauge theory via persistent homology* [[arXiv:2208.03955](https://arxiv.org/abs/2208.03955)]

analysis of the [[cosmic microwave background]] in search for hints of [[inhomogeneous cosmology]]:

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

Persistent homology has strong computational limits for large data sets and major problems with multifitrations. An alternative propoposal in TDA to persistent homology which is worse in 1 dimension but does not suffer above problems is

* Paweł Dłotko, Davide Gurnari, _Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems_, [arXiv:2212.01666](https://arxiv.org/abs/2212.01666)

### Relation to quantum computation

Potential implementation of TDA on [[quantum computers]]:

* He-Liang Huang, Xi-Lin Wang, Peter P. Rohde, Yi-Han Luo, You-Wei Zhao, Chang Liu, Li Li, Nai-Le Liu, Chao-Yang Lu, Jian-Wei Pan, _Demonstration of Topological Data Analysis on a Quantum Processor_, Optica 5(2), 193 (2018) ([arXiv:1801.06316](https://arxiv.org/abs/1801.06316))

* {#IMD23} Massimiliano Incudini, Francesco Martini, [[Alessandra Di Pierro]], *Higher-order topological kernels via quantum computation*, 2023 IEEE International Conference on Quantum Computing and Engineering, QCE **1** (2023) [[arXiv:2307.07383](https://arxiv.org/abs/2307.07383), [doi:10.1109/QCE57702.2023.00076](https://doi.ieeecomputersociety.org/10.1109/QCE57702.2023.00076)]

[[!include cohomotopy in topological data analysis -- references]]

[[!redirects topological data analysis]]

[[!redirects TDA]]


topological defect

* in [[quantum field theory]] and specifically [[gauge theory]]: a topological [[singularity]] in the [[vacuum]] structure caused by [[spontaneous symmetry breaking]].

  See at _[[QFT with defects]]_ for details of the general mechanism, and see at _[[domain wall]]_, _[[cosmic string]]_ and _[[monopole]]_ (also [[vortex]]) for (classes of) examples.

[[!redirects topological defects]]


topological domain theory

#Contents#
* table of contents
{:toc}

## Idea

Topological domain theory is a generalization of [[domain theory]] that includes a wider collection of [[topological spaces]] than traditional domain theory. The generalization overcomes certain known limitations of domain theory, which is unable to model various awkward combinations of [[computability|computational]] features.

## Related concepts

* [[computable analysis]], [[computable function (analysis)]]

## References

A collection of resources is at

* Website _[Topological Domain Theory](https://web.archive.org/web/20110611035538/http://homepages.inf.ed.ac.uk/als/Research/topological-domain-theory.html)_

A survey is in

* [[Ingo Battenfeld]], _Topological Domain Theory_, PhD thesis, 2009 ([pdf slides](http://ls10-www.cs.uni-dortmund.de/~battenfeld/phdtalk.pdf))


topological dynamics

* Wikipedia, _[Topological dynamics](http://en.wikipedia.org/wiki/Topological_dynamics)_


topological entanglement entropy -- references

### Topological entanglement entropy
 {#ReferencesTopologicalEntanglementEntropy}

#### General

Identification of a contribution to [[entanglement entropy]] at [[absolute zero]] which is independent of the [[subsystem]]'s size ("[[topological entanglement entropy]]", "[[long-range entanglement]]"), reflecting [[topological order]] and proportional to the total [[quantum dimension]] of [[anyon|anyon excitations]]:

* {#KitaevPreskill06} [[Alexei Kitaev]], [[John Preskill]], *Topological entanglement entropy*, Phys. Rev. Lett. 96 (2006) 110404 ([arXiv:hep-th/0510092](https://arxiv.org/abs/hep-th/0510092))

* {#LevinWen06} [[Michael Levin]], [[Xiao-Gang Wen]], *Detecting topological order in a ground state wave function*, Phys. Rev. Lett., 96, 110405 (2006) $[$[arXiv:cond-mat/0510613](https://arxiv.org/abs/cond-mat/0510613), [doi:10.1103/PhysRevLett.96.110405](https://doi.org/10.1103/PhysRevLett.96.110405)$]$

  > (in view of [[string-net model|string-net models]])

Review:

* {#Furukawa08} Shunsuke Furukawa, *Entanglement Entropy in Conventional and Topological Orders*, talk at [Topological Aspects of Solid State Physics](http://www.issp.u-tokyo.ac.jp/public/tassp/) 2008 ([pdf](http://www.issp.u-tokyo.ac.jp/public/tassp/pdf_presentation/SY_S_Furukawa.pdf), [[FurukawaEntanglementEntropy.pdf:file]])

* [[Tarun Grover]], *Entanglement entropy and strongly correlated topological matter*,  Modern Physics Letters A **28** 05 (2013)  1330001  $[$[doi:10.1142/S0217732313300012](https://doi.org/10.1142/S0217732313300012)$]$

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 5 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

In terms of [[Renyi entropy]] (it's independent of the Renyi entropy parameter):

* Ulrich Schollwöck, *(Almost) 25 Years of DMRG - What Is It About?* ([pdf](https://confs.physics.ox.ac.uk/pauli2016/files/slides/Schollwoeck.pdf))

and in the example of [[Chern-Simons theory]]:

* Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, [[Vivek Kumar Singh]], *Topological entanglement and hyperbolic volume*, J. High Energy Phys. **21021** 172 (2021) 172 [[arXiv:2106.03396](https://arxiv.org/abs/2106.03396), doi:10.1007/JHEP10(2021)172]

Discussion in the [[dimer model]]:

* Shunsuke Furukawa, Gregoire Misguich, *Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice*, Phys. Rev. B 75, 214407 (2007) ([arXiv:cond-mat/0612227](https://arxiv.org/abs/cond-mat/0612227))

Discussion via [[holographic entanglement entropy]]:

* Ari Pakman, [[Andrei Parnachev]], *Topological Entanglement Entropy and Holography*, JHEP 0807: 097 (2008) ([arXiv:0805.1891](https://arxiv.org/abs/0805.1891))

* [[Andrei Parnachev]], Napat Poovuttikul, *Topological Entanglement Entropy, Ground State Degeneracy and Holography*, Journal of High Energy Physics volume 2015, Article number: 92 (2015) ([arXiv:1504.08244](https://arxiv.org/abs/1504.08244))

See also:

* Tatsuma Nishioka, Tadashi Takayanagi, Yusuke Taki, *Topological pseudo entropy* ([arXiv:2107.01797](https://arxiv.org/abs/2107.01797))

#### Relation to strong interaction
 {#ReferencesLongRangeEntanglementRelationToStrongInteraction}

Relation of [[long-range entanglement]] to strong [[interaction]]:

* {#ZaanenLiuSunSchalm15} [[Jan Zaanen]], Yan Liu, Ya-Wen Sun, [[Koenraad Schalm]], _Holographic Duality in Condensed Matter Physics_, Cambridge University Press 2015 $[$[doi:10.1017/CBO9781139942492](https://doi.org/10.1017/CBO9781139942492)$]$

  > In a way it appears obvious that the strongly interacting bosonic quantum critical state is subject to long-range entanglement. Nonetheless, the status of this claim is conjectural.

  > It is at present impossible to arrive at more solid conclusions that are based on rigorous mathematical procedures. It does illustrate emphatically the central challenge in the pursuit of field-theoretical quantum information: there are as yet not general measures available to precisely enumerate the meaning of long-range entanglement in such seriously quantum field-theoretical systems. $[$p. 527$]$

* {#LuVijay22} Tsung-Cheng Lu, Sagar Vijay, *Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface* $[$[arXiv:2201.07792](https://arxiv.org/abs/2201.07792)$]$

  > strongly interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement

#### Characterizing topological order
 {#CharacterizingTopologicalOrder}

On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:

* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$

* [[Yi Zhang]], [[Tarun Grover]], [[Ashvin Vishwanath]], *General procedure for determining braiding and statistics of anyons using entanglement interferometry*, Phys. Rev. B **91** (2015) 035127 $[$[arXiv:1412.0677](https://arxiv.org/abs/1412.0677), [doi:10.1103/PhysRevB.91.035127](https://doi.org/10.1103/PhysRevB.91.035127)$]$

* [[Zhuan Li]], [[Roger S. K. Mong]], *Detecting topological order from modular transformations of ground states on the torus*, Phys. Rev. B **106** (2022) 235115 [[doi:10.1103/PhysRevB.106.235115](https://doi.org/10.1103/PhysRevB.106.235115), [arXiv:2203.04329](https://arxiv.org/abs/2203.04329)]

#### Simulation and experiment

Experimental observation:

* A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, *Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order*, Phys. Rev. B 77, 155111 (2008) ([doi:10.1103/PhysRevB.77.155111](https://doi.org/10.1103/PhysRevB.77.155111), [arXiv:0705.0026](https://arxiv.org/abs/0705.0026))

* Hong-Chen Jiang, [[Zhenghan Wang]], [[Leon Balents]], *Identifying Topological Order by Entanglement Entropy*, Nature Physics **8** 902-905 (2012) $[$[arXiv:1205.4289](https://arxiv.org/abs/1205.4289)$]$

Detection of long-range entanglement entropy in quantum simulations on [[quantum computers]]:

* *Realizing topologically ordered states on a quantum processor*, Science
**374** 6572 (2021) 1237-1241 $[$[doi:10.1126/science.abi8378](https://doi.org/10.1126/science.abi8378)$]$

* *Probing topological spin liquids on a programmable quantum simulator*, Science **374** 6572 (2021) 1242-1247 $[$[doi:10.1126/science.abi8794](https://doi.org/10.1126/science.abi8794)$]$

exposition in:

*  Isabelle Dumé, *[Long-range quantum entanglement measured at last](https://physicsworld.com/a/long-range-quantum-entanglement-measured-at-last/)*, PhysicsWorld (Jan 2022)


topological field

# Contents #
* table of contents
{:toc}

## Idea

A topological field is a [[field]] equipped with a topology such that all of the field operations are [[continuous functions]].

## Definition

A _topological field_ is a [[topological ring]] whose underlying [[ring]] in $Set$ is a [[field]] $K$ and such that the multiplicative inversion operation $i: K \setminus \{0\} \to K \setminus \{0\}$ is [[continuous function|continuous]] with respect to the [[subspace]] topology inherited from $K$.

+-- {: .num_remark}
###### Remark
Note it is not automatic that inversion is continuous for a field equipped with a ring topology (when it is, we say the ring topology is a *field topology*). The following gives an example of a ring topology that is not a field topology: on the [[rational numbers]] $\mathbb{Q}$, take the set of [[ideals]] $n\mathbb{Z}$, $0 \neq n \in \mathbb{Z}$, to be a filter base for a [[filter]] of [[neighborhoods]] of $0$. That this gives a ring topology follows from three easily verified facts: (1) for any ideal $I$ there is an ideal $J$ such that $J + J \subseteq I$ and $-J \subseteq I$ (e.g., $J = I$), (2) if $I$ is any ideal and $q \in \mathbb{Q}$ is any element, there is an ideal $J$ such that $q J \subseteq I$, and (3) if $I$ is any ideal, there is an ideal $J$ such that $J J \subseteq I$ (again take $J = I$). It's easy to see that this ring topology is [[Hausdorff space|Hausdorff]].

But inversion on the nonzero elements is not continuous. Indeed, for a neighborhood of $1$ not containing $0$, e.g., $1 + 2\mathbb{Z}$, no neighborhood $1 + m\mathbb{Z}$ of $1$ will fit inside the set of reciprocals $\{\frac1{1 + 2n}: n \in \mathbb{Z}\}$.
=--

+-- {: .num_remark}
###### Remark
If $K$ is a topological field, then either $K$ is a [[codiscrete space]] or is a [[Tychonoff space]]. The reason is that the closure of $\{0\}$ in a topological ring $K$ must be an [[ideal]] $I$, and since $K$ is a field, $I$ is either all of $K$ (whence $K$ is codiscrete), or $I = \{0\}$ in which case $K$ is a $T_1$-[[separation axiom|space]]. In the latter case, since a topological ring is a [[uniform space]], the $T_1$-condition implies $K$ is a Tychonoff space.
=--

## Internalization

As discussed at [[field]], the notion of field is not algebraic in the sense of [[algebraic theory]], and there are various inequivalent ways of attempting to [[internal logic|internalize]] the notion inside structured categories. This issue is compounded in $Top$ by the fact that $Top$ has few of the exactness properties one needs to enact the more "traditional" fragments of first-order logic (such as being a [[pretopos]] or [[Heyting category]] or [[exact category]] or [[regular category]]). So it is a matter of interest to give a categorical definition of field that internalizes correctly in $Top$, as well as in other categories of interest.

(Note that $Top$ does enjoy *some* elementary exactness properties: it is a [[lextensive category]] with finite colimits. It also satisfies a strong non-elementary condition: it is $\infty$-extensive and the underlying-set functor $Top \to Set$ is a [[topological functor]]. Curiously, $Top^{op}$ is a [[regular category]].)

### "Classical" internalization

One straightforward approach, at least if we are thinking along lines of [[excluded middle|classical logic]], is to define a field $K$ in terms of the following limit-colimit [[sketch]]:

1. Introduce structure to make $K$ a commutative [[ring object]]: two binary operations $a: K \times K \to K$ (addition) and $m: K \times K \to K$ (multiplication), two constants $0: 1 \to K$ and $e: 1 \to K$ (additive and multiplicative identities), additive inversion $-: K \to K$, all subject to the usual equations for [[commutative rings]];

1. Letting $i: U \to K \times K$ denote the [[equalizer]] of $m: K \times K \to K$ and $e \circ !: K \times K \to 1 \to K$, add the axiom that $j = \pi_1 \circ i: U \to K \times K \to K$ (provably monic in finite limit logic) is a [[regular monomorphism]]: the [[equalizer]] of its [[cokernel pair]];

1. Add the axiom that $(0, j): 1 + U \to K$ is a [[monomorphism|monic]] [[epimorphism|epic]].

Some commentary might be in order. Clearly $U$ plays the role of the [[group of units]] of $K$, realized as a [[subobject]] by $j: U \to K$. Axiom 3. says that $0$ and $U$ exhaust all of $K$, but without going so far to say that $(0, j): 1 + U \to K$ is an [[isomorphism]], an inappropriately strong condition in the case of $Top$ (as it would force the point $0: 1 \to K$ to be open, making $K$ a [[discrete space]]).

Axiom 2. is more subtle: a mono $k: A \to B$ in $Top$ is regular iff $A$ has the subspace topology inherited from $B$ via $k$. So Axiom 2. interpreted in $Top$ says that the subspace topology on $U$ coming from its inclusion into $K$ coincides with the topology it has by definition, viz. the subspace topology coming from its embedding $i$ in $K \times K$. Notice that inversion on $U$ is continuous if we use the definitional topology, since inversion is effected by permuting the two factors of $K \times K$. Thus Axiom 2. is a sneaky way of forcing inversion on $U$ with the subspace topology from $K$ to be continuous (and in fact it is equivalent to continuity of inversion).

## Examples

* The [[real numbers]] with their [[metric topology]] as a [[Euclidean space]];

* The [[complex numbers]], similarly,

etc.

[[!redirects topological field]]
[[!redirects topological fields]]