homotopy extension property

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

## Idea

A [[morphism]] is said to have the *homotopy extension property* if [[homotopies]] out of its [[domain]] [[extension|extend]] to [[homotopies]] out of its [[codomain]].

More in detail, given a [[category]] with a concept of [[path space object]] and hence of [[right homotopy]], then a [[morphism]] $i \colon A \longrightarrow B$ is said to have the _right homotopy extension property_ with respect to an [[object]] $X$ if it has the [[left lifting property]] against one of the two path endpoint evaluation maps $p_0 \colon Path(X) \longrightarrow X$, i.e. if in every [[commuting square]] as below, a diagonal lift exists:

$$
  \array{
    A &\longrightarrow& Path(X)
    \\
    {}^{\mathllap{i}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p_0}}
    \\
    B &\longrightarrow& X
  }
  \,.
$$

Here the top morphism exhibits a [[right homotopy]] between two morphisms $A\to X$, and the diagonal filler is an [[extension]] of the top morphism along $i$ and exhibiting a compatible right homotopy between morphisms $B\to X$, whence the terminology.

The concept of homotopy extension property is the [[Eckmann-Hilton duality|Eckmann-Hilton dual]] of that of (left) _[[homotopy lifting property]]_, where instead one considers the presence of a [[cylinder object]], hence a notion of [[left homotopy]], and lifting in diagrams of the form

$$
  \array{
    A &\longrightarrow& X
    \\
    {}^{\iota_0}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{f}
    \\
    Cyl(A) &\longrightarrow& Y
  }
  \,.
$$

In situations where both [[path space objects]] as well as [[cylinder objects]] exist and are compatible, so that the concepts of left and right homotopy coincide, one may equivalently rephrase right homotopy extension in terms of left homotopy (and left homotopy lifting in terms of right homotopy).

The resulting definition is necessarily less transparent (def. \ref{LeftHomotopyExtensionPropertyInTop} below), but it happens to be more commonly used in the literature.
Specifically in the archetypical case that the ambient category is that of [[topological spaces]] and cylinders and path space objects are induced from the standard topological [[interval object]], a _[[Hurewicz cofibration]]_ (often just called _cofibration_ ) is a [[continuous function]] that satisfies the left homotopy extension property with respect to all topological spaces.

## Definition

### In topological spaces

\begin{definition}\label{LeftHomotopyExtensionPropertyInTop}
**(homotopy extension property for left homotopies in topological spaces)**
\linebreak
A [[continuous function]] $i \colon A \xrightarrow{\;} X$ of [[topological spaces]] is said to satisfy the (left) **homotopy extension property** (HEP) with respect to a space $Y$ if

* for any map $f \colon X \longrightarrow Y$ and any [[left homotopy]] $\eta \,\colon\,  A \times [0,1] \longrightarrow Y$ such that $\eta(-,0) = f \circ i$, there exists a [[left homotopy]] $\widehat{\eta} \,\colon\,  X \times [0,1] \longrightarrow Y$ such that $\widehat{\eta} \circ (i \times id) = \eta$;

equivalently:

* given a [[commuting diagram]] in [[Top|TopSp]] of solid arrows as shown below, there exists a dashed arrow $\widehat \eta$ making all sub-diagrams commute:

\begin{tikzcd}[column sep={between origins, 40pt}]
    A
    \ar[
      dd,
      "{i}"{left}
    ]
    \ar[
      rr,
      "{(\mathrm{id},\, 0)}"
    ]
    &&
    A \times [0,1]
    \ar[
      dd,
      "{ i \times \mathrm{id} }"{right}
    ]
    \ar[
      dl,
      "{\eta}"{left, yshift=3pt}
    ]
    \\
    &
    Y
    \\
    X
    \ar[
      rr,
      "{ (\mathrm{id},\, 0) }"{below}
    ]
    \ar[
      ur,
      "{f}"
    ]
    &&
    X \times [0,1]
    \ar[
      ul,
      dashed,
      "{ \widehat \eta }"{description}
    ]
\end{tikzcd}

The map $f$ is sometimes said to be the _initial condition_ of a _homotopy extension problem_ and $\widehat{\eta}$ is called the [[extension]] of the homotopy $\eta$ subject to that initial condition.

The map $i$ is called a **[[Hurewicz cofibration]]** if it satisfies this homotopy extension property with respect to all spaces $Y$.
\end{definition}

\begin{remark}
**(re-formulation in terms of right homotopies)**
\linebreak
In a [[convenient category of topological spaces|convenient]] [[cartesian closed category]] $TopSp$ of [[topological spaces]] (such as that of [[compactly generated topological spaces]]) the [[product]] $\dashv$ [[internal hom]]-[[adjunction]]

$$
  TopSp
    \underoverset
    {\underset{ (-)^{[0,1]} }{\longrightarrow}}
    {\overset{ (-) \times [0,1]  }{\longleftarrow}}
    {\;\;\;\;\;\bot\;\;\;\;\;}
  TopSp
$$

allows to pass to [[adjunct]] morphisms in Def. \ref{LeftHomotopyExtensionPropertyInTop}

$$
  \array{
    TopSp
    \big(
      A \times [0,1]
      ,\,
      Y
    \big)
    &\simeq&
    TopSp
    \big(
      A
      ,\,
      Y^{[0,1]}
    \big)
    \\
    \eta &\leftrightarrow& \eta'
  }
$$

and equivalently express the above diagram of [[left homotopies]] as the following (somewhat more transparent) diagram of [[right homotopies]], where $ev_0 \,\colon\, Y^{[0,1]}\to Y$ denotes [[evaluation]] at 0 $\gamma \mapsto \gamma(0)$:

\begin{tikzcd}
    A
    \ar[
      d,
      "{i}"{left}
    ]
    \ar[
      r,
      "{ \eta' }"{above}
    ]
    &
    Y^{[0,1]}
    \ar[
      d,
      "\mathrm{ev}_0"
    ]
    \\
    X
    \ar[
      r,
      "{f}"{below}
    ]
    \ar[
      ur,
      dashed,
      "\widehat \eta'"{description}
    ]
    &
    Y
\end{tikzcd}

In this equivalent formulation, the homotopy extension property is simply the [[right lifting property]] against the [[evaluation]] map $ev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y$ out of the [[path space]] of $Y$.
\end{remark}

(e.g. [May 1999, p. 43](#May99))

## Properties

### Closure

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

If a map $i\colon A\to X$ has the homotopy extension property with respect to a space $Y$, then for any map $g \colon A\to Z$, the [[pushout]] $g_*(i)=i\amalg_A Z :Z\to X\amalg_A Z$ has the homotopy extension property with respect to the space $Y$.

=--

This is a general statement about classes of morphisms defined by a [[left lifting property]], see at _[injective and projective morphisms -- closure properties](injective+or+projective+morphism#ClosureProperties)_

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

We would like to find $\tilde{F}$ to complete the commutative diagram

$$
  \array{
    A &\stackrel{g}\to&Z&\stackrel{f}\to& Y^I
    \\
    \downarrow^{i}&&\downarrow^{g_*(i)} &\nearrow {\tilde{F}}&  \downarrow^{ev_0}
    \\
    X&\stackrel{i_*(g)}\to&X\amalg_A Z&\stackrel{F}{\to}& Y
  }
  \,.
$$

Consider the external square obtained by composing the horizontal arrows:

$$
  \array{
    A &\stackrel{f\circ g}\to& Y^I
    \\
    \downarrow^{i} &\nearrow {\exists\tilde{G}}&  \downarrow^{ev_0}
    \\
    X&\stackrel{F\circ i_*(g)}{\to}& Y
  }
  \,.
$$

By the assumption on $i$, there is a $\tilde{G}$ as in the diagram, such that both triangles commute, i.e. $ev_0\circ\tilde{G}=F\circ i_*(g)$ and $\tilde{G}\circ i = \tilde{F}\circ g$.

If $i:A\to X$ is satisfying the HEP with respect to $Y$ then there is a diagonal in that external square which is some map $\tilde{G}:X\to Y^I$. This map together with $f:Z\to Y^I$, by the universal property of pushout, determines a unique map $\tilde{F}:X\amalg_A Z\to Y^I$ such that $\tilde{F}\circ i_*(g)=\tilde{G}$ and $\tilde{F}\circ g_*(i)=f$. We need to show only that
$ev_0\circ\tilde{F}=F$ as $\tilde{F}\circ g_*(i)=f$ holds by the construction of $\tilde{F}$ as stated.

By the definition of $\tilde{G}$ and the commutativity of the original double square diagram, $ev_0\circ \tilde{F}\circ i_*(g)=ev_0\circ\tilde{G}=F\circ i_*(g)$ and $ev_0\circ \tilde{F}\circ g_*(i)=ev_0\circ f=F\circ g_*(i)$. This is almost what we wanted except that we precompose the wanted identity with both maps into the pushout. Thus by the uniqueness part of the universal property of pushout it follows that $ev_0\circ\tilde{F}=F$.
=--

## Literature

Textbook accounts:

* {#May99} [[Peter May]], Section 6.1 of: *[[A concise course in algebraic topology]]*, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#Hatcher02} [[Allen Hatcher]], p. 14-17 in: *Algebraic Topology*, Cambridge University Press 2002 ([ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html))

See also:

* Wikipedia, *[Homotopy extension property](https://en.wikipedia.org/wiki/Homotopy_extension_property)*

The terminology "h-cofibration" is due to:

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], p. 16: _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))


homotopy factorization system

# Homotopy factorization systems

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## Idea

A **homotopy factorization system** in a [[model category]] is a presentation of an [[orthogonal factorization system in an (∞,1)-category|orthogonal factorization system]] in its [[simplicial localization|underlying]] [[(∞,1)-category]].

## The enriched case

### Definition

Let $V$ be a [[monoidal model category]] (with cofibrant unit object) and $M$ a $V$-[[enriched model category]].  In the case $V =$ [[SSet]], the following definition is found in [Bousfield, section 6](#Bousfield).

\begin{definition}
  A **homotopy factorization system** in $M$ is a pair $(L,R)$ of classes of maps such that:

  1. Every map in $L$ is a cofibration, and every map in $R$ is a fibration.
  2. If $i:A\to B$ is in $L$ and $p:X\to Y$ is in $R$, then the induced [[pullback power]]
     $$ [i,p] : [B,X] \to [A,X] \times_{[A,Y]} [B,Y] $$
     is an acyclic fibration in $V$.
  3. Every morphism in $M$ factors as a map in $L$ followed by a map in $R$.
  4. $L$ and $R$ are closed under [[retracts]].

\end{definition}

### Remarks

It follows that $(L,R)$ is in fact a [[weak factorization system]].  For on the one hand; the underlying-set functor $V(I,-) : V\to Set$ takes acyclic fibrations to surjections since $I$ is cofibrant; thus $(L,R)$ have the lifting property.  And on the other hand, if $i$ has the left lifting property against $R$, then factoring it and applying the retract argument implies $i\in L$, and dually.

Note that $[i,p]$ is automatically a fibration, since $i$ is a cofibration and $p$ a fibration; thus the content of assertion (2) is that this map is a weak equivalence.  If $A$ (hence also $B$) is cofibrant and $Y$ (hence also $X$) is fibrant, then the pullback $[A,X] \times_{[A,Y]} [B,Y]$ is pullback of two fibrations between fibrant objects and thus a [[homotopy pullback]]; thus in this case the condition is equivalent to asking that the square
\begin{center}
  \begin{tikzcd}
    {}[B,X] \ar[r] \ar[d] & {}[A,X] \ar[d]\\
    {}[B,Y] \ar[r] & {}[A,Y]
  \end{tikzcd}
\end{center}
is a homotopy pullback square.  If $V$ is [[right proper]], then the condition for the pullback to be a homotopy pullback can be weakened to "$A$ (hence also $B$) is cofibrant OR $Y$ (hence also $X$) is fibrant", and thus becomes automatic if either all objects of $M$ are fibrant or all objects of $M$ are cofibrant.

## The unenriched case

### Definition

A hierarchy of notions of "homotopy factorization system" for unenriched model categories can be found in [Joyal, Appendix F](#Joyal).  Let $M$ be a model category and $(L,R)$ a pair of classes of maps.  Write $C$ for the class of cofibrations, $F$ for the class of fibrations, $M_{c}$ for the subcategory of cofibrant objects, $M_f$ for the subcategory of fibrant objects, $M_{c f}$ for the subcategory of fibrant and cofibrant objects, $C_{c f}$ for the class of cofibrations between fibrant and cofibrant objects, etc.

\begin{definition}

* $(L,R)$ is a **weak homotopy factorization system** if
  1. $L$ and $R$ are closed under weak equivalence in the [[arrow category]] $M^\to$, and
  2. $(L\cap C_{c f}, R\cap F_{c f})$ is a [[weak factorization system]] in $M_{c f}$.
* $(L,R)$ is a **homotopy factorization system** if it is a weak homotopy factorization system and in addition
  1. If $f\in L$ and $g f\in L$, then $g\in L$.
  2. If $g\in R$ and $g f\in R$, then $f\in R$.
* $(L,R)$ is a **strong homotopy factorization system** if it is a homotopy factorization system and in addition
  1. $(L\cap C, R\cap F)$ is a weak factorization system in $M$.

\end{definition}

## Relation between definitions

The relation between the enriched and unenriched notions is unclear to the author of this page, but here are some things that can be said.

\begin{proposition}
  Suppose either every object of $M$ is fibrant and cofibrant, or $V$ is right proper and either every object of $M$ is fibrant or every object of $M$ is cofibrant.  Then given an enriched hfs, by closing $L$ and $R$ under weak equivalence in $M^\to$ we obtain an unenriched weak hfs $(L',R')$.
\end{proposition}
\begin{proof}
  Since $(L,R)$ is a wfs, to show that $(L',R')$ is an unenriched weak hfs, it suffices to show that $L'\cap C_{c f} = L \cap C_{c f}$ and dually.  Note that any morphism in $C_{c f}$ is both cofibrant and fibrant in the Reedy model structure on $M^\to$; hence if two such morphisms are weakly equivalent in $M^\to$, there is a single weak equivalence relating them.  But the property of being a homotopy pullback square is preserved under weak equivalence; and under the given hypotheses, as remarked above, the homotopy lifting property can be expressed in terms of such a square, and is thus preserved by weak equivalences between cofibrations.  The proof for $R$ is dual (using the other Reedy model structure).
\end{proof}

It is unclear whether or under what conditions this weak hfs is a hfs or a strong hfs.

In the converse direction, the following are proven by [Joyal](#Joyal):

* An unenriched weak hfs $(L,R)$ is determined by $(L\cap C_{c f}, R\cap F_{c f})$ (called its **center**).
* If $(L,R)$ is an unenriched weak hfs, then $L\cap C_C$ has the left lifting property against $R\cap F_f$.
* If $(L,R)$ is an unenriched weak hfs, then every morphism from a cofibrant object to a fibrant one factors as a map in $L\cap C_c$ followed by one in $R\cap F_f$.

\begin{proposition}
  If $(L,R))$ is an unenriched weak hfs, the following are equivalent:

  1. If $f\in L$ and $g f\in L$, then $g\in L$.
  2. If $g\in R$ and $g f\in R$, then $f\in R$.
  3. The codiagonal of any map in $L\cap C_c$ belongs to $L$.
  4. The diagonal of any map in $R\cap F_f$ belongs to $R$.
  5. if $f\in L\cap C_{c f}$ and $g f \in L\cap C_{c f}$ and $g\in C$, then $g\in L$.
  6. if $g\in R\cap F_{c f}$ and $g f \in R\cap F_{c f}$ and $f\in F$, then $f\in R$.

\end{proposition}

The closure under diagonals and codiagonals suggests that some kind of homotopy orthogonality should exist, using [[simplicial resolutions]] rather than enrichment.

## References

* {#Bousfield} [[A. K. Bousfield]], *Constructions of factorization systems in categories*, Journal of Pure and Applied Algebra 9 (1977) 207-220, [pdf](http://web.math.rochester.edu/people/faculty/doug/otherpapers/Bousfield_Fact.pdf)

* {#Joyal} [[Andre Joyal]], *Notes on quasi-categories*, [pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf)

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[[!redirects homotopy factorization systems]]
[[!redirects weak homotopy factorization system]]
[[!redirects weak homotopy factorization systems]]
[[!redirects strong homotopy factorization system]]
[[!redirects strong homotopy factorization systems]]
[[!redirects homotopy factorization system in a model category]]
[[!redirects homotopy factorization systems in a model category]]
[[!redirects factorization system in a model category]]
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[[!redirects orthogonal factorization system in a model category]]
[[!redirects orthogonal factorization systems in a model category]]


homotopy fixed point

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

The refinement of the notion of [[fixed point]]/[[invariant]] to [[homotopy theory]]. Also _[[homotopy invariant]]_

See at

* [Invariants -- For infinity-group actions](invariant#ForInfinityGroupActions)

* [Infinity-action -- Invariants](infinity-action#Invariants)

## Properties

### Presentation

Generally, for $G$ an [[∞-group]] and $X$ equipped with an [[∞-action]], then the homotopy fixed points is the [[base change]] along/[[dependent product]] over the [[delooping]] $\mathbf{B}G$:

$$
  (-)^{h G}
  \;\colon\;
  G Act(\mathbf{H})
  \simeq
  \mathbf{H}_{/\mathbf{B}G}
  \overset
    {\;\;\;\prod_{\mathbf{B}G}\;\;\;}
    {\longrightarrow}
  \mathbf{H}
  \,.
$$

By the general discussion at _[[dependent product]]_ this is given by forming [[sections]], which by the discussion at [[∞-action]] here means forming sections of the $X$-[[associated ∞-bundle]] to the universal [[principal ∞-bundle]]

$$
  \array{
    X
      &\longrightarrow&
    X \sslash G
    \\
    && \big\downarrow
    \\
    && \mathbf{B}G
    \,.
  }
$$

Such sections

\begin{tikzcd}[column sep=6pt]
  \ast
  \ar[r]
  &
  \mathbf{B}G
  \ar[dr]
  \ar[rr, dashed]
  &&
  X /\!/ G
  \ar[dl]
  &
  X
  \ar[l]
  \\
  &
  &
  \mathbf{B}G
  &
\end{tikzcd}

are equivalently homotopy-[[equivariant function]] out of the point equipped with the trivial $G$-action:

$$
  X^{h G}
  \simeq
  G Act(\ast, X)
  \,.
$$

For [[geometrically discrete ∞-groupoid|geometrically discrete]] [[∞-group]] [[∞-actions]], hence for $\mathbf{H} = $ [[∞Grpd]], it is the [[Borel model structure]] which [[presentable (infinity,1)-category|presents]] the [[homotopy theory]]. By the discussion [there](Borel+model+structure#CofibrantReplacementAndHomotopyQuotientsFixedPoints), the [[derived hom space]] is computed as the hom-space out of plain [[actions]] of [[simplicial groups]] out of the total space of the simplicial [[universal principal bundle]] $\mathbf{E}G = W G$. Therefore in this case one finds

$$
  X^{h G}
  \simeq
  R Hom_G(\ast, X)
  \simeq
  Hom_G(E G, X)
  \,.
$$

This is the form in which homotopy fixed points are often defined in traditional literature (going back to [Thomason 83](#Thomason83) and as in the historical discussion of the [[Sullivan conjecture]]).

### Relation to periodicity theorems

In the context of [[complex oriented cohomology theory]]

... ([Hill-Hopkins-Ravenel, theorem 8.4](#HillHopkinsRavenel))...

## Related concepts

* [[fixed point space]]

* [[crossed homomorphism]]

* [[Sullivan conjecture]]

* [[invariant theory]]

* [[fixed point spectrum]]

* [[geometric fixed point spectrum]]

* [[categorical fixed point spectrum]]

[[!include homotopy type representation theory -- table]]

## References

The original article:

* {#Thomason83} [[Robert Thomason]], *The homotopy limit problem*, pages 407-419 in: [[Haynes Miller]], [[Stewart Priddy]] (eds.) *Proceedings of the Northwestern Homotopy Theory Conference*, Contemporary Mathematics **19**, AMS 1983 ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019), [[ThomasonHomotopyLimit.pdf:file]])

Further discussion, for [[groupoids]] and [[stacks]]:

* R. Virk, *On fixed point stacks* ([arXiv:2112.06871](https://arxiv.org/abs/2112.06871))

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

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], Theorem 8.4 in: _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_ ([[HHRKervaire.pdf:file]])

[[!redirects homotopy fixed points]]

[[!redirects homotopy fixed point space]]
[[!redirects homotopy fixed point spaces]]


homotopy group

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### Context
#### Homotopy theory
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#### Algebraic topology
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[[!include algebraic topology - contents]]
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=--

# Contents
* table of contents
{: toc}

## Idea

The _homotopy groups_ $\pi_n(X,x)$ of a [[pointed object|pointed]]
[[topological space]] $(X,x)$ are a sequence of [[groups]] that generalise the [[fundamental group]] $\pi_1(X,x)$ to higher [[homotopy (as a transformation)|homotopies]].

The $n$th homotopy group $\pi_n(X,x)$ has as elements [[equivalence classes]] of [[spheres]] $\gamma : S_*^n \to X_*$ in $X$ where two such are regarded as equivalent if there is a [[left homotopy]] $\gamma \Rightarrow \gamma'$ between them, fixing the base point. The group operation is given by gluing of two spheres at their basepoint.

In degree 0, $\pi_0(X,x)$ is not a group but merely a [[pointed set]].  In degree $n \geq 2$ all homotopy groups are [[abelian groups]].  Only $\pi_1(X,x)$ may be an arbitrary group.  In general, $\pi_n(X,x)$ is an $n$-[[k-tuply groupal n-groupoid|tuply groupal]] set.

Lower homotopy groups [[action|act]] on higher homotopy groups; the nonabelian group cohomology of this gives the Postnikov invariants of the space.  All of this data put together allows one to reconstruct the original space, at least up to weak [[homotopy type]], through its [[Postnikov system]].

These definitions only depend on the [[homotopy type]] of $X$, by definition: a [[weak homotopy equivalence]] between topological spaces is a [[continuous map]] that induces an [[isomorphism]] on all homotopy groups.

Accordingly, homotopy groups are defined for all other models of [[homotopy types]], notably for [[simplicial sets]]. See at _[[simplicial homotopy group]]_ for more.

There are generalizations of the concept to [[stable homotopy groups]] of [[spectra]] and to [[equivariant homotopy groups]] for [[topological G-spaces]] and [[equivariant spectra]].

## Definition
 {#Definition}

### For topological spaces
 {#ForTopologicalSpaces}

Let $X$ be a [[topological space]], let $x : * \to X$ be a point, to be called the _base point_.

For $n \in \mathbb{N}$, let $S^n$ be the pointed $n$-[[sphere]].

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

The underlying set of $\pi_n(X,x)$ is the set of
[[equivalence classes]] of basepoint-preserving [[continuous functions]]

$$
  \gamma : S^n \to X
$$

where two such are regarded as equivalent if there is a basepoint-preserving [[left homotopy]] between them.

=--

Now we will put some structure on that set.

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

There are $n$ independent equators through the basepoint of $S^n$.  Given two maps $f, g: S^n \to (X,x)$, form their [[coproduct|copairing]] in the category of pointed spaces to get a map

$$
  S^n \vee S^n \to (X,x)
$$

(where $\vee$ indicates the [[wedge sum]]); then combine this with a map $S^n \to S^n \vee S^n$ that maps the $i$th equator to the basepoint and each hemisphere to one copy of the sphere.  The result is a map $S^n \to (X,x)$, called the $i$th __concatenation__ of $f$ and $g$:

$$
  S^n \to_i S^n \vee S^n \stackrel{[f,g]}{\to} (X,x)
  \,.
$$

One can check that each of these operations respects homotopy equivalence and hence equips $\pi_n(X,x)$ with the structure of a [[magma]].

=--

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

These magmas are in fact [[groups]]; in particular:

* the [[constant function]] that maps all of $S^n$ to $x$ represents the __null element__ of $\pi_n(X,x)$, which is an identity for every concatenation.

=--

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

This is closely related to the statement that the [[positive dimension spheres are H-cogroup objects]] (see there) in the [[homotopy category]] of [[pointed topological spaces]].

=--

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

This may seem like quite a complicated kind of structure, but it is actually quite simple up to homotopy.  First of all, all $n$ concatenations of given maps $f$ and $g$ are homotopic, so we speak of simply a single __concatenation__ for $n \geq 1$ (and none for $n = 0$).  By the [[Eckmann-Hilton argument]], this concatenation will be commutative up to homotopy for $n \geq 2$.  In any case, it is [[associativity|associative]] and [[inverse|invertible]] up to homotopy, and the null element is an identity up to homotopy.

=--

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

The result is that the set $\pi_n(X,x)$ of equivalence classes is an abelian group for $n \geq 2$, a group for $n = 1$, and a pointed set for $n = 0$ (when the null element is the only structure).

=--

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

If $X$ is [[connected space|path-connected]], then all of the $\pi_n(X,a)$ are [[isomorphism|isomorphic]].  Accordingly, it\'s traditional to just write $\pi_n(X)$ in that case.  (This is why we must use $\Pi_n(X)$ for the homotopy $n$-groupoid.)  However, there may be many different isomorphisms between $\pi_n(X,a)$ and $\pi_n(X,b)$ (given by $\pi_{n+1}(X)$), so a more careful treatment requires keeping track of the basepoint even in the connected case.

=--

### For simplicial sets

See [[simplicial homotopy group]].

### For Lie groupoids

* [[homotopy groups of a Lie groupoid]]

### For objects in a general $\infty$-stack $(\infty,1)$-topos

[[Top]] is the archetypical [[(∞,1)-topos]].
The definition of homotopy groups for objects in [[Top]]
is just a special case of a general definition of
homotopy groups of objects of [[∞-stack]] [[(∞,1)-topos]]es.

This is described in detail at

* [[homotopy groups in an (∞,1)-topos]].

## Truncated and connected spaces {#truncconn}

Often it is useful to talk about spaces whose homotopy groups are all trivial above or below a certain degree, for instance in the context of [[Postnikov towers]] and [[Whitehead towers]].

For $n = -1, 0, 1, 2, \ldots, \infty$:

* a space is **$n$-[[homotopy n-type|truncated]]** if all homotopy groups _above_ degree $n$ are trivial.

* a space is **$n$-[[n-connected space|connected]]** (or **$n$-simply connected**) if it is [[inhabited space|inhabited]] and all homotopy groups _at or below_ degree $n$ are trivial.

Vacuously, every space is $\infty$-truncated, and precisely the inhabited spaces are $(-1)$-connected.  On the other end, precisely the weakly [[contractible spaces]] are $\infty$-connected, and a space is $(-1)$-truncated iff it is weakly contractible if inhabited.  (So [[classical mathematics|classically]], using [[excluded middle]], a space is $(-1)$-truncated iff it is either [[empty space|empty]] or weakly contractible.)

To extend one step further in [[negative thinking]], every space (even the empty space) is $(-2)$-connected, and precisely a weakly contractible space (but not the empty space) is $(-2)$-truncated.

* A [[pointed space]] is a **degree-$n$ [[Eilenberg-Mac Lane space]]** if it is both $n$-truncated as well as $(n-1)$-connected, i.e. if its only nontrivial homotopy group is in degree $n$.

A weakly contractible space is an Eilenberg–MacLane space in every degree, and these are the only Eilenberg–MacLane spaces in degree $\infty$ or $-1$.  In degree $0$, they are the pointed [[discrete space | discrete spaces]] (and those weakly homotopy equivalent to such).  In degree $1$, they are (up to weak homotopy equivalence) precisely the [[classifying spaces]] of [[groups]].  And so on.

## Examples

### In low dimensions

The $0$th homotopy 'group' $\pi_0(X,a)$ can be identified with the set of all [[path component | path components]] of $X$, with the component containing $a$ as the basepoint.  Similarly, the [[fundamental infinity-groupoid|fundamental 0-groupoid]] $\Pi_0(X)$ is the set of all path components without a chosen basepoint.  Note that $\Pi_0(X)$ is traditionally written $\pi_0(X)$, even without a basepoint.

The $1$st homotopy group $\pi_1(X,a)$ is precisely the [[fundamental group]] of $X$ at $a$.  This is the original example from which all others derived.  It was once written simply $\pi(X,a)$ with the $\pi$ standing for Poincaré, who invented it.

+--{.query}
At least, that\'s where I *think* that it comes from ...  ---Toby
=--

### Of the circle

The first homotopy group of the [[circle]] $S^1$ is the group of [[integer | integers]].

$$
  \pi_1(S^1) \simeq \mathbb{Z}
  \,.
$$

A formalization of a proof of this in [[homotopy type theory]] is in ([Shulman](#ShulmanPi1S1)).

### Of spheres

See [[homotopy groups of spheres]].

## History

In the early years of the 20th century it was known that the nonabelian fundamental group $\pi_1(X,a)$ of a space $X$ with base point $a$ was useful in geometry and complex analysis. It was also known that the abelian [[homology group | homology groups]] $H_n(X)$ existed for all $n \geq 0$ and that if $X$ is connected then $H_1(X)$ is isomorphic to the abelianisation of any $\pi_1(X,a)$.

Consequently it was hoped to generalise the fundamental group to higher dimensions, producing nonabelian groups whose abelianisations would be the homology groups.

In 1932, E. Čech proposed a definition of higher homotopy groups using maps of spheres, but the paper was rejected for the Zurich ICM since it was found that these groups $\pi_n(X,a)$ were abelian for $n \geq 2$, and so do not generalise the fundamental group in the way that was originally desired.  Nonetheless, they have proved to be extremely important in homotopy theory, although more difficult to compute in general than homology groups.  See [[weak homotopy equivalence]].

## Properties

It was early realised that the fundamental groupoid $\Pi_1(X)$ operates on the family of groups $\{\pi_n(X,a) | a \in X\}$ which should thus together be regarded as a module over $\pi_1(X,a)$.

A key property of homotopy groups is the _Whitehead theorem_: if $f:X \to Y$ is a map of connected [[m-cofibrant spaces]] (spaces each of the homotopy type of a [[CW complex]]), and $f$ induces isomorphisms $\pi_n(X,a) \to \pi_n(Y,f(a))$ for some $a$ and all $n \geq 1$, then $f$ is a [[homotopy equivalence]].

However, the homotopy groups by themselves, even considering the operations of $\pi_1$, do not characterise homotopy types. See also [[algebraic homotopy|algebraic homotopy theory]].

See also the _[[Freudenthal suspension theorem]]_.

## Some general nonsense

Using the [[Eckmann-Hilton duality]] between
[[cohomology]] and [[homotopy (as an operation)]] one may discuss homotopy groups along the same lines as the discussion of [[cohomology groups]]  (see there).

From that perspective we might say that:

for $B, X$ any two objects in an [[(∞,1)-topos]] $\mathbf{H}$, the "homotopy of $X$ with co-coefficients $B$" is the [[hom-set]]
  $$
    \pi_H(X,B) := H(B,X) := \pi_0 \mathbf{H}(B,X)
    \,,
  $$
  where $\pi_H$ denotes the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathbf{H}$.

For the special case that the object $B$ here is a [[co-group object]], this _homotopy set_ $\pi(X,B)$ naturally inherits the structure of a [[group]].

The standard example is that where $B = S^n$ is the $n$-[[sphere]]. This naturally comes with an co-group structure up to homotopy, which is precisely the structure underlying the co-category structure of the [[interval object]] and more generally that underlying the mechanism of the [[Trimble n-category]].

As opposed to [[cohomology]] where people are used to talking about [[generalized cohomology]], "homotopy" usually just means this ordinary homotopy for $B = S^n$.

## Related concepts

* [[homotopy group of a spectrum]]

* [[weak homotopy equivalence]]

* [[p-localization]]

* [[Goodwillie calculus]]

* [[homotopy groups in an (infinity,1)-topos]]

* [[finite homotopy type]], [[nilpotent homotopy type]], [[p-local homotopy type]], [[p-complete homotopy type]]

* [[equivariant homotopy group]]

## References

### General

On the early history of the notion of the higher homotopy groups (beyond the [[fundamental group]]):

* [[Peter Hilton]], p. 289 (9 of 11) in: *Subjective History of Homology and Homotopy Theory*, Mathematics Magazine **61** 5 (1988) 282-291 [[doi:10.2307/2689545](https://doi.org/10.2307/2689545)]

> "At a meeting in Vienna in 1931 [[Eduard Čech|Čech]] gave a paper in which he described certain groups from the homotopy point of view. He had no applications of these groups. Moreover, he had only one theorem, that they were [[abelian group|commutative]]. And he was persuaded by people, and we know that [[Pavel Aleksandrov|Alexandroff]] played a role here, that they could not be interesting, because it was thought that any information that could be obtained from [[abelian groups|Abelian groups]] must come from the [[ordinary homology|homology]]. [[Witold Hurewicz|Hurewicz]] redefined the [[homotopy groups]] and immediately gave important applications in a series of four notes which were intended as preliminary publications. In that series of four papers he showed the significance of what we now call [[obstruction theory]]."

Modern accounts (for more see at *[[homotopy theory]]*):

* [[Paul Goerss]], [[J. F. Jardine]], section I.7 of: _[[Simplicial homotopy theory]]_,  Progress in Mathematics, Birkhäuser (1996)

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 3.4 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* {#Hutchings} [[Michael Hutchings]], _Introduction to higher homotopy groups and obstruction theory_ ([pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf))

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 3 of _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

### Formalization in HoTT

Homotopy groups and their properties can naturally be formalized in [[homotopy type theory]]. In this context a proof that $\pi_1(S^1)\simeq \mathbb{Z}$ is in

* {#ShulmanPi1S1} [[Mike Shulman]], _[P1S1.v](https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v)_

* [[Dan Licata]], [[Mike Shulman]], _Calculating the fundamental group of the circle in homotopy type theory_ ([pdf](http://www.cs.cmu.edu/~drl/pubs/ls13pi1s1/ls13pi1s1.pdf))

and a proof that

$\pi_k(S^n) \simeq \left\{ \array{ 0 & for \;k \lt n \\ \mathbb{Z} & for\; k = n} \right.$

is in

* [[Dan Licata]], [[Guillaume Brunerie]], _[PiNSN.agda](https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiNSN.agda)_

[[!redirects homotopy groups]]

[[!redirects homotopy n-tuply groupal set]]
[[!redirects homotopy n-tuply groupal sets]]

[[!redirects homotopy groupal set]]
[[!redirects homotopy groupal sets]]

[[!redirects homotopy pointed set]]
[[!redirects homotopy pointed sets]]

[[!redirects homotopy abelian group]]
[[!redirects homotopy abelian groups]]


homotopy group of a spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

Stable homotopy groups are [[homotopy groups]] as seen in [[stable homotopy theory]].

A [[morphism]] inducing an [[isomorphism]] on all stable homotopy groups is called a [[stable weak homotopy equivalence]].

## Definition

### For pointed topological spaces

Given a [[pointed topological space]] $X$, its _stable homotopy groups_ are the [[colimit]] of ordinary [[homotopy groups]] of its [[reduced suspensions]]

$$
  \pi_n^S(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{n+k}(\Sigma^k X)
  \,.
$$

### For sequential spectra

\begin{definition}
  \label{StableHomotopyGroupsOfSequentialSpectrum}

Given a  [[sequential spectrum]] $E$, in the form of a sequence of component spaces $E_n$ with structure maps $\Sigma E_n \to E_{n+1}$, then for $k \in \mathbb{Z}$ the $n$th _homotopy group_ of $E$ is the [[colimit]]

$$
  \begin{aligned}
  \pi_n(E)
    & \coloneqq
  \underset{\longrightarrow}{\lim}_k
  \pi_{n+k}(E_k)
  \\
  & \coloneqq
  \underset{\longrightarrow}{\lim}
  \left(
    \cdots
      \to
    \pi_{n+k}(E_k)
       \stackrel{\Sigma}{\longrightarrow}
    \pi_{n+k+1}(\Sigma E_k)
       \stackrel{\pi_{n+k+1}(\sigma_k^E)}{\longrightarrow}
    \pi_{n+k+1}(E_{k+1})
       \to
    \cdots
  \right)
  \end{aligned}
$$

over the [[homotopy groups]] of the component spaces.

\end{definition}

For sequential spectra in [[simplicial sets]], the same formula applies for the [[geometric realization]] of the component simplicial sets.

(For details see [this definition](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyGroups).)

\begin{remark}
  \label{StableHomotopyGroupsOfOmegaSpectrum}

If a [[sequential spectrum]] $X$ is an [[Omega-spectrum]], then its colimiting stable homotopy groups according to Def. \ref{StableHomotopyGroupsOfSequentialSpectrum} reduce to the actual [[homotopy groups]] of the component spaces $X_n \coloneqq \Omega^\infty \Sigma^n X$, in that:

$$
  X \; \text{is Omega-spectrum}
  \;\;\;\;\;
   \Rightarrow
  \;\;\;\;\;
  \pi_k(X)
    \simeq
  \left\{
    \array{
      \pi_{k+n}\big( X_n \big)
      &\vert&
      k + n \geq 0
      \\
      \pi_k\big(X_0\big)
      &\vert&
      k \geq 0
      \\
      \pi_0 X_{\vert k \vert}
      &\vert&
      k \lt 0
      \\
    }
  \right.
  \,.
$$

\end{remark}

(For details see [this example](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyGroupsOfOmegaSpectrum).)

## Properties

### Suspension isomorphism

Let

$$
  \Sigma
  \;\colon\;
  SeqSpec
  \longrightarrow
  SeqSpec
$$

be the operation of forming degreewise the [[smash product]] with the circle, the (un-derived, [[reduced suspension|reduced]]) [[suspension]] of $X$.

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

For $X$ a [[sequential spectrum]], smashing with $S^1$ constitutes [[natural isomorphisms]] of stable homotopy groups of $X$ with the stable homotopy groups in one degree higher of the suspension spectrum of $X$

$$
  S^1 \wedge (-)
    \;\colon\;
  \pi_\bullet(X)
    \stackrel{\simeq}{\longrightarrow}
  \pi_{\bullet+1}(\Sigma X)
  \,.
$$

=--

(e.g. [Schwede 12, part I, prop. 2.6](#Schwede12))

### For suspension spectra

For $E = \Sigma^\infty X$ the [[suspension spectrum]] of a [[pointed topological space]], we have

$$
  \pi_n^S(X) \simeq \pi_n(\Sigma^\infty X)
  \,.
$$

### As a homology theory

The assignment of [[stable homotopy groups]] to [[topological spaces]] $X$ ([[CW-complexes]])

$$
  X \mapsto \pi_\bullet^{st}(X) \coloneqq \pi_\bullet(\Sigma^\infty X)
$$

satisfies the [[axioms]] of a [[generalized homology theory]]. As such this is also called _[[stable homotopy homology theory]]_.

## Examples

* [[stable homotopy groups of spheres]]

* [[Thom's theorem]]

## Related concepts

* [[homotopy group]]

* [[free infinite loop space]]

* [[equivariant homotopy group]]

## Examples

* The homotopy groups of a [[suspension spectrum]] $\Sigma^\infty X$ of a [[pointed topological space]] $X$ are the _[[stable homotopy groups]]_ of $X$.

* In particular the homotopy groups of the [[sphere spectrum]] are the _[[stable homotopy groups of spheres]]_.

* [[Thom's theorem]] says that the homotopy groups of the [[Thom spectrum]] $M O$ form the (unoriented) [[cobordism ring]].

## References

* [[Frank Adams]], part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Schwede12} [[Stefan Schwede]], _[[Symmetric spectra]]_ (2012)

[[!redirects homotopy groups of a spectrum]]

[[!redirects homotopy groups of spectra]]

[[!redirects stable homotopy group]]
[[!redirects stable homotopy groups]]


homotopy group with coefficients

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

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of _[[homotopy groups]]_.

## Definition

Given a [[pointed space]] $X$ and [[finitely generated|finitely generated]] [[abelian group]] $A$ and $n\ge2$,
we set
$$
  \pi_n(X,A)
    \;\coloneqq\;
  [P^n(A),X]
  \,,
$$
where
$P^n(A)$ is the $n$th [[Peterson space]] of $A$
and $[-,-]$ denotes [[pointed homotopy classes]].

## Related concepts

* [[homotopy groups]]

* [[Peterson space]]

## References

* [[Franklin P. Peterson]], _Generalized Cohomotopy Groups_.  American Journal of Mathematics 78:2 (1956), 259–281.  [doi:10.2307/2372515](http://dx.doi.org/10.2307/2372515)

* [[Joseph A. Neisendorfer]], _Homotopy groups with coefficients_,
Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338.  [doi:10.1007/s11784-010-0020-1](http://dx.doi.org/10.1007/s11784-010-0020-1).

[[!redirects homotopy groups with coefficients]]


homotopy groups in an (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

In the [[(∞,1)-topos]] [[Top]] to every object -- every [[topological space]] -- $X$ is associated
the set $\pi_0(X)$ of connected components and the [[homotopy group]]s $\pi_n(X,x)$ for $x \in X$ and $n \in \mathbb{N}$, $n\gt 0$.

By the general logic of [[space]], we may think of the objects in an arbitrary [[∞-stack]] [[(∞,1)-topos]] as generalized spaces of sorts. Accordingly, there is a notion of **homotopy groups of an $\infty$-stack** .

But care has to be taken. It turns out that there are actually _two_ different notions of homotopy groups in an arbitrary $(\infty,1)$-topos, two notions that accidentally coincide for [[Top]]:

* there is a notion of **categorical homotopy group**:

  every $(\infty,1)$-topos $\mathbf{H}$ is [[power]]ed over [[∞Grpd]] [[model structure on simplicial sets|usually modeled]] as [[SSet]], hence for every object $X \in \mathbf{H}$ there is the **categorical $n$-sphere object** $X^{S^n_c}$, where $S^n_c = \Delta^n/\partial \Delta^n$.

* there should be a notion of **geometric homotopy group**, induced from the [[monodromy]] of [[locally constant ∞-stack]]s on objects $X \in \mathbf{H}$.

For instance let $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ be the $(\infty,1)$-topos of [[Lie ∞-groupoid]]s. An ordinary smooth [[manifold]] $X$ is represented in $\mathbf{H}$ by a [[sheaf]] of [[set]]s on [[Diff]]. This has no higher nontrivial categorical homotopy groups -- $\pi_{n \gt 0}^{cat}(X) = 0$ -- reflecting the fact regarded as a smooth [[∞-groupoid]], $X$ is a categorically [[discrete groupoid]].

But of course the manifold $X$ may have nontrivial [[homotopy group]]s in terms of its underlying [[topological space]]. For instance if $X = S^1$ is the circle, then the _geometric_ first homotopy group is nontrivial, $\pi_1^{geom}(X) = \mathbb{Z}$.

We discuss below both cases. The case of categorical homotopy groups is fully understood, for the case of geometric homotopy groups at the moment only a few aspects are in the literature, more is in the making. Some authors of this page ([[Urs Schreiber|U.S.]]) thank [[Richard Williamson]] for pointing this out.

## Categorical homotopy groups

See [[categorical homotopy groups in an (∞,1)-topos]].

## Geometric homotopy groups

See [[geometric homotopy groups in an (∞,1)-topos]].

[[!redirects homotopy group of an infinity-stack]]
[[!redirects homotopy group of an ∞-stack]]
[[!redirects homotopy groups of an infinity-stack]]
[[!redirects homotopy groups of an ∞-stack]]
[[!redirects homotopy groups of infinity-stacks]]
[[!redirects homotopy groups of ∞-stacks]]
[[!redirects homotopy group (of an infinity-stack)]]
[[!redirects homotopy group (of an ∞-stack)]]
[[!redirects homotopy groups (of an infinity-stack)]]
[[!redirects homotopy groups (of an ∞-stack)]]
[[!redirects homotopy groups (of infinity-stacks)]]
[[!redirects homotopy groups (of ∞-stacks)]]

[[!redirects fundamental group of an infinity-stack]]
[[!redirects fundamental group of an ∞-stack]]
[[!redirects fundamental groups of infinity-stacks]]
[[!redirects fundamental groups of ∞-stacks]]

[[!redirects homotopy group of an infinity-stack]]

[[!redirects homotopy groups in an (∞,1)-topos]]
[[!redirects homotopy group in an (∞,1)-topos]]
[[!redirects homotopy group in an (infinity,1)-topos]]


homotopy groups of a cubical Kan complex

# Contents
* table of contents
{: toc}

## A relation on the $n$-cubes of a cubical set with trivial boundary

We make use of the notation established at [[cubical set]] and [[category of cubes]].

Throughout this page, we shall let $X$ be a [[cubical set]], let $x : \square^{0} \rightarrow X$ be a $0$-cube of $X$, and let $n \geq 0$ be an integer.

+-- {: .num_defn}
###### Notation

We denote by $Z_{n}(X,x)$ the set of $n$-cubes $\sigma : \square^{n} \rightarrow X$ of $X$ with the property that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes for every integer $1 \leq i \leq n$ and every integer $0 \leq \epsilon \leq 1$.

$$
   \array{
      \square^{n-1} & \overset{p}{\rightarrow}        & \square^{0} \\
  \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n-i})} \downarrow &                                   & \downarrow \mathrlap{x} \\
   \square^{n} & \underset{\sigma}{\rightarrow} & X
   }
$$

=--

+-- {: .num_defn #NotationEquivalenceRelation}
###### Notation

Let $\sim$ be the relation on $Z_{n}(X,x)$ given by identifying $\sigma_{0}$ and $\sigma_{1}$ if there is an $(n+1)$-cube $h : \square^{n+1} \rightarrow X$ of $X$ such that the following diagrams in $\mathsf{Set}^{\square^{op}}$ commute

$$
   \array{
      \square^{n}  & \overset{\square^{n} \otimes i_{0}}{\rightarrow}    & \square^{n+1} \\
             & \underset{\sigma_{0}}{\searrow} & \downarrow h \\
             &                         & X
   }
$$

$$
   \array{
      \square^{n}  & \overset{\square^{n} \otimes i_{1}}{\rightarrow}    & \square^{n+1} \\
             & \underset{\sigma_{1}}{\searrow} & \downarrow h \\
             &                         & X
   }
$$

and such that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes for every integer $1 \leq i \leq n$ and every integer $0 \leq \epsilon \leq 1$.

$$
   \array{
      \square^{n} & \overset{p}{\rightarrow}        & \square^{0} \\
  \mathllap{y(I^{i-1} \otimes i_{\epsilon} \otimes I^{n+1-i})} \downarrow &                                   & \downarrow \mathrlap{x} \\
  \square^{n+1} & \underset{\sigma}{\rightarrow} & X
   }
$$

=--

+-- {: .num_defn}
###### Remark

The commutativity of the first two diagrams in Notation \ref{NotationEquivalenceRelation} asserts that $h$, viewed as an arrow $\square^{n} \otimes \square^{1} \rightarrow X$ of $\mathsf{Set}^{\square^{op}}$, defines a [[cubical homotopy | homotopy]] from $\sigma_{0} : \square^{n} \rightarrow X$ to $\sigma_{1} : \square^{n} \rightarrow X$.

=--

## The relation $\sim$ is in fact an equivalence relation

+-- {: .num_defn}
###### Proposition

Let $\sigma$ be an $n$-cube of $X$ which belongs to $Z_{n}(X,x)$. Then $\sigma \sim \sigma$.

=--

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

We take $h$ to be the arrow $\sigma \circ \big( \square^{n} \otimes y(p) \big) : \square^{n+1} \rightarrow \square^{n} \rightarrow X$ of $\mathsf{Set}^{\square^{op}}$.

=--

+-- {: .num_defn}
###### Proposition

Let $X$ be equipped with the structure of a cubical Kan complex. Let $\sigma_{0}$ and $\sigma_{1}$ be $n$-cubes of $X$ which belong to $Z_{n}(X,x)$. Suppose that $\sigma_{0} \sim \sigma_{1}$. Then $\sigma_{1} \sim \sigma_{0}$.

=--

+-- {: .proof}

=--

## Homotopy groups of a cubical Kan complex

+-- {: .num_defn}
###### Notation

We denote by $\pi_{n}(X,x)$ the set $Z_{n}(X,x) / \sim$.

=--


homotopy groups of a Lie groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _geometric [[homotopy group]]s_ of a [[Lie groupoid]] $X$ are those of its [[nLab:geometric realization]] $|X|$ when regarded as a [[simplicial manifold]]. Equivalently, regarding $X$ as an object in the [[(∞,1)-topos]] [[?LieGrpd]], its homotopy groups are those of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi(X) \in $ [[∞Grpd]].

## Definition

For $X = (X_1 \stackrel{\to}{\to} X_0)$ a [[Lie groupoid]] and $x : * \to X$ a [[point]], let

$$
  X_\bullet =
  \left(
    \cdots
    X_1 \times_{X_0} X_1 \times_{X_0} X_1
   \stackrel{\to}{\stackrel{\to}{\to}}X_1 \times_{X_0} X_1 \stackrel{\to}{\to} X_0
  \right)
$$

be its [[nerve]] regarded as a [[simplicial manifold]].

+-- {: .un_remark}
###### Remark

When regarding each manifold $X_n$ as a [[diffeological space]], hence a [[sheaf]] on the [[site]] [[CartSp]] then $X_\bullet in PSh(CartSp)^{\Delta^{op}} \simeq [CartSp^{op},sSet]$ is the [[simplicial presheaf]] on [[CartSp]] that presents $X$ as an object in the [[(∞,1)-topos]] [[?LieGrpd]] of [[∞-Lie groupoid]]s.

=--

+-- {: .un_defn}
###### Definition

Regard $X_\bullet$ as a [[simplicial topological space]] by forgetting the [[smooth structure]]. Write $|X_\bullet| \in $ [[Top]] for its [[geometric realization of simplicial topological spaces|geometric realization as a simplicial topological space]].

The **geometric homotopy groups** of $X$ are defined to be the ordinary [[homotopy group]]s of the [[topological space]] $|X_\bullet|$:

$$
  \pi_n(X,x) := \pi_n(|X_\bullet|,x)
  \,.
$$

=--

In this form the definition originates in ([Segal](#Segal)).

## Properties

Regard $X$ as an [[∞-Lie groupoid]] under the natural embedding $LieGrpd \hookrightarrow \infty LieGrpd$. By the discussion at [[?LieGrpd]] this is a [[locally ∞-connected (∞,1)-topos]], which means that its terminal [[geometric morphism]] comes with a further [[left adjoint]] $\Pi$

$$
  (\Pi \dashv \Delta \dashv \Gamma) : \infty LieGrpd \to \infty Grpd
  \,.
$$

We say that $\Pi(X) \in \infty Grpd \simeq Top$ is the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] of $X$.

+-- {: .un_prop}
###### Observation

The geometric homotopy groups of $X$ are those of $\Pi(X) \in Top$.

=--

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

By the discussion at [[∞-Lie groupoid]] we have precisely that $\Pi(X)$ is presented by the [[geometric realization]] of the [[simplicial topological space]] underlying the [[nerve]] of $X$.

=--

## References

The definition of the homotopy groups of a Lie groupoid as those of its geometric realization appearently goes back to

* [[Graeme Segal]], _Classifying spaces and spectral sequences_ , IHES Publ. Math. 34 (1968) 105–112.
{#Segal}

An equivalent definition is in

* A. Haefliger, Groupoïdes d'holonomie et espaces classiants , Astérisque 116 (1984), 70-97

reproduced in section 3 of

* [[Graeme Segal]], _Classifying spaces related to foliations_ , Topology 17 (1978), 367-382.

[[!redirects fundamental group of a Lie groupoid]]

[[!redirects homotopy groups of Lie groupoids]]


homotopy groups of spheres

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

# Homotopy groups of spheres
* table of contents
{: toc}

***

> My initial inclination was to call this book _[[The Music of the Spheres]]_, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. ([D. Ravenel 86, preface](#Ravenel86))

> With all due respect to anyone interested in them, the stable homotopy groups of spheres are a mess. ([[Stable homotopy and generalised homology|J. F. Adams 74, p 204]])

***

## Idea

The [[homotopy groups]] of [[spheres]] $\pi_{n+k}(S^n)$ are the [[homotopy classes]] of maps $S^{n+k} \longrightarrow S^n$

$$
  \pi_{n+k}(S^n) \coloneqq [S^{n+k}, S^n]
  \,.
$$

For fixed $k$, the [[colimit]] over $n$ with respect to the [[suspension]] homomorphism

$$
  \pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})
$$

over all $\pi_{n+k}(S^n)$ (called the $k$-[[stem]]) is called the _[[stable homotopy groups]] of spheres_ (also: the "stable $k$-[[stem]]")

$$
  \pi_k^S = \coloneqq \underset{\longrightarrow}{\lim}_n \pi_{n+k}(S^n)
 \,.
$$

In fact, by the [[Freudenthal suspension theorem]], the value of the $\pi_{n+k}(S^n)$ stabilizes for $n \gt k+1$ (depend only on $k$ in this range), whence the name.

The [[stable homotopy groups]] of sphere are equivalently the [[homotopy groups of a spectrum]] for the [[sphere spectrum]] $\mathbb{S}$

$$
  \pi_k^S = \pi_k(\mathbb{S})
  \,.
$$

The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of [[algebraic topology]] and [[stable homotopy theory]] were devised to compute more and more of the stable stems. This notably include the [[Adams spectral sequence]], the [[Adams-Novikov spectral sequence]].

## Tables
 {#Tables}

The first few  stable homotopy groups of the [[sphere spectrum]] $\mathbb{S}$ are

| $k =$ | 0 | 1 | 2 | 3 | 4  | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S}) = $ | $\mathbb{Z}$  | $\mathbb{Z}/2$  |  $\mathbb{Z}/2$ |  $\mathbb{Z}/{24}$ |  $0$ |  $0$ | $\mathbb{Z}/2$ |   $\mathbb{Z}/{240}$ | $(\mathbb{Z}/2)^2$ |  $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/6$ | $\mathbb{Z}/{504}$ | $0$ | $\mathbb{Z}/3$ | $(\mathbb{Z}/2)^2$ | $\mathbb{Z}/{480} \oplus \mathbb{Z}/2$ | $\cdots$ |

The following tables show the [[p-primary group|p-primary decomposition]] of these and the following stable homotopy groups.

The horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. The bottom rows in each case are given by the [[image of the J-homomorphism]].

See at _[fundamental theorem of finitely generated abelian grouops -- Graphical representation](fundamental+theorem+of+finitely+generated+abelian+groups#GraphicalRepresentation)_ for details on the notation used in these table, and  see example \ref{InterpretTable} below for illustration.

(the following graphics are taken from [Hatcher](#Hatcher), based on  ([Ravenel 86](#Ravenel86))

**$p = 2$-primary component**

**$p = 3$-primary component**

**$p = 5$-primary component**

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

The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

=--

## Examples

### 0th stem

See at _[[Hopf degree theorem]]_.

### 1st stem

The [[first stable homotopy group of spheres]] (the first [[stable stem]]) is the [[cyclic group of order 2]]:

\[
  \label{ComplexHopfFibrationGeneratingpi1}
  \array{
    \pi_1^s &\simeq& \mathbb{Z}/2
    \\
    [h_{\mathbb{C}}] &\leftrightarrow& [1]
  }
\]

where the generator $[1] \in \mathbb{Z}/2$ is represented by the [[complex Hopf fibration]] $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$.

\begin{imagefromfile}
        "file_name": "PuncturedSphereBordismBetweenTwoCircles.jpg",
        "web": "nlab",
        "width": 200,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), the generator (eq:ComplexHopfFibrationGeneratingpi1) is represented by the [[1-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[U(1)]])

$$
  \array{
    \pi_1^s & \simeq & \Omega_1^{fr}
    \\
    [h_{\mathbb{C}}] & \leftrightarrow & [S^1_{fr=1}]
    \,.
  }
$$

Moreover, the relation $2 \cdot [S^1_{Lie}] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 2 [[open balls]] inside [[generalized the|the]] [[2-sphere]].

### 2nd stem

The [[second stable homotopy group of spheres]]...

### 3rd stem

The [[third stable homotopy group of spheres]] (the third [[stable stem]]) is the [[cyclic group]] of [[order of a group|order]] 24:

\[
  \label{QuaternionicHopfFibrationGeneratingpi3}
  \array{
    \pi_3^s &\simeq& \mathbb{Z}/24
    \\
    [h_{\mathbb{H}}] &\leftrightarrow& [1]
  }
\]

where the generator $[1] \in \mathbb{Z}/24$ is represented by the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

\begin{imagefromfile}
        "file_name": "K3CobordismBetween24ThreeSpheres.jpg",
        "web": "nlab",
        "width": 260,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), the generator (eq:QuaternionicHopfFibrationGeneratingpi3) is represented by the [[3-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[SU(2)]] $\simeq$ [[Sp(1)]] )

$$
  \array{
    \pi_3^s & \simeq & \Omega_3^{fr}
    \\
    [h_{\mathbb{H}}] & \leftrightarrow & [S^3_{fr=1}]
    \,.
  }
$$

Moreover, the relation $24 \cdot [S^3_{Lie}] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 24 [[open balls]] inside [[generalized the|the]] [[K3]]-manifold (e.g. [Wang-Xu 10, Sec. 2.6](third+stable+homotopy+group+of+spheres#WangXu10), [Bauer 10](third+stable+homotopy+group+of+spheres#Bauer10), [SP 17](third+stable+homotopy+group+of+spheres#SP17)).

Equivalently, the elements of $\pi_3^s \,\simeq\, \Omega^{fr}_3$ are detected by half the [[Todd classes]] of [[coboundary|cobounding]] manifolds with [[special unitary group]]-[[tangential structure]] on their [[stable tangent bundle]] (elements of the [[MSUFr]]-[[bordism ring]]):

We have the following [[short exact sequence]] of the [[MSU]]-, [[MSUFr]]- and [[MFr]]-[[bordism rings]] ([Conner-Floyd 66, p. 104](third+stable+homotopy+group+of+spheres#ConnerFloyd66))

\[
  \label{HalfToddClassesOnShortExactSequenceOfSUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^{SU}_{8\bullet+4}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{SU,fr}_{8\bullet+4}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_{8\bullet + 3}
  &
  \simeq
  &
  \pi^s_{8\bullet+3}
  \\
  &
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e_{\mathbb{R}}}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
\]

which produces from half the [[Todd class]] of cobounding $(SU,fr)$-manifolds the [[KO]]-theoretic [[Adams e-invariant]] $e_{\mathbb{R}}$ ([Adams 66, p. 39](e-invariant#Adams66)) of the boundary manifold in $\Omega^{fr}_{8k + 3} \simeq \pi^s_{8k+3}$. For $k = 0$ this detects the third stable homotopy group of spheres, by the following:

+-- {: .num_prop #AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres}
###### Proposition
**([Adams 66, Example 7.17  and p. 46](e-invariant#Adams66))**

In degree 3, the [[KO]]-theoretic [[e-invariant]] $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$  and hence reflects the full [[third stable homotopy group of spheres]]:

$$
  \array{
    \pi^s_3
      &
      \underoverset{
        \simeq
      }{
        e_{\mathbb{R}}
      }{
        \;\;\longrightarrow\;\;
      }
      &
    \mathbb{Z}/24
    &
    \subset
    &
    \mathbb{Q}/\mathbb{Z}
    \\
    [h_{\mathbb{H}}]
    &&\mapsto&&
    \left[\tfrac{1}{24}\right]
  }
$$

while $e_{\mathbb{C}}$ sees only "half" of it (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

=--

## Properties

### Serre finiteness theorem

+-- {: .num_theorem }
###### Theorem
**([[Serre finiteness theorem]])**

The [[homotopy group]] $\pi_{n+k}(S^k)$ is a [[finite group]] except

1. for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$;

1. $k = 2m$ and $n = 2m -1$ in which case

   $$
     \pi_{4m - 1}(S^{2m}) \simeq \mathbb{Z} \oplus F_m
   $$

   for $F_m$ a [[finite group]].

=--

([Serre 53](#Serre53), see [Ravenel 86, Chapter I, Lemma 1.1.8](#Ravenel86))

### Nishida nilpotence theorem

The [[Nishida nilpotence theorem]]...

### Relation to the framed bordism ring
 {#RelationToFramedBordismRing}

By [[Thom's theorem]], for any [[(B,f)-structure]] $\mathcal{B}$, there is an [[isomorphism]] (of [[commutative rings]])

$$
  \Omega^{\mathcal{B}}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(M\mathcal{B})
$$

from the [[cobordism ring]] of manifolds with stable normal $\mathcal{B}$-structure to the [[homotopy groups of a spectrum|homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].

Now for $\mathcal{B} = Fr$ [[framing]] structure, then

$$
  M Fr \simeq \mathbb{S}
$$

is equivalently the [[sphere spectrum]]. Hence in this case [[Thom's theorem]] states that there is an isomorphism

$$
  \Omega^{fr}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(\mathbb{S})
$$

between the framed cobordism ring and the stable homotopy groups of spheres.

For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance ([Wang-Xu 10, section 2](#WangXu10)) and ([Putnam](#Putnam)).

For instance

* $\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;

* $\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.

### J-homomorphism and Adams e-invariant

The following characterizes the [[image]] of the [[J-homomorphism]]

$$
  J \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
$$

from the [[homotopy groups]] of the [[stable orthogonal group]] to the stable homotopy groups of spheres. This was first conjectured in ([Adams 66](#Adams66)) (since called the _Adams conjecture_) and then proven in ([Quillen 71](#Quillen71)).

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

By the discussion at _[orthogonal group -- homotopy groups](orthogonal%20group#HomotopyGroups)_ we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

=--

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

The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]] of the
[[J-homomorphism]], and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

* for $n = 0 \;mod \;$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism is [[injection|injective]], hence its image is $\mathbb{Z}_2$,

* for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$, where $B_{2k}$ is the [[Bernoulli number]]

* for all other cases the image is necessarily zero.

=--

[[!include image of J -- table]]

## Related concepts

* [[formalization of the homotopy groups of spheres in homotopy type theory]]

* [[EHP spectral sequence]]

* [[Adams–Novikov spectral sequence]]

* [[Music of the Spheres]]

## References

### General

Introductions and surveys include

* Alex Wright, _Homotopy groups of spheres: A very basic introduction_ ([pdf](http://www-personal.umich.edu/~alexmw/HomotopyGroupsOfSoheres.pdf))

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([pdf](http://math.mit.edu/~guozhen/homotopy%20groups.pdf), [[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

* {#Putnam} [[Andrew Putman]], _Homotopy groups of spheres and low-dimensional topology_ ([pdf](http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf), [[PutmanHomotopyGroupsOfSpheres.pdf:file]])

* {#Hatcher} [[Allen Hatcher]], _Pictures of stable homotopy groups of spheres_ ([html](http://www.math.cornell.edu/~hatcher/stemfigs/stems.html))

* {#MahowaldRavenel87} [[Mark Mahowald]], [[Doug Ravenel]], _Towards  a Global Understanding of the Homotopy Groups of Spheres_, in: [[Samuel Gitler]] (ed.): _The Lefschetz Centennial Conference: Proceedings on Algebraic Topology II_, Contemporary Mathematics volume 58, AMS 1987  ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf), [[MahowaldRavenelHomotopyGroupsOfSpheres.pdf:file]], [ISBN:978-0-8218-5063-3](https://bookstore.ams.org/conm-58-2))

  > ([[chromatic homotopy theory]])

* [[Haynes Miller]], [[Doug Ravenel]], _Mark Mahowald's work on the homotopy groups of spheres_ ([pdf](http://www-math.mit.edu/~hrm/ksem/miller-ravenel.pdf))

* {#Ravenel86} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, since 1986 ([web](http://www.math.rochester.edu/people/faculty/doug/mu.html))

* [[Stanley Kochmann]], _Stable Homotopy Groups of Spheres -- A Computer-Assisted Approach_, Lecture Notes in Mathematics, 1990

* [[Doug Ravenel]], _Nilpotence and Periodicity in Stable Homotopy Theory_, Annals of Mathematics Studies **128**, Princeton University Press (1992) [[ISBN:9780691025728](https://press.princeton.edu/books/paperback/9780691025728/nilpotence-and-periodicity-in-stable-homotopy-theory-am-128-volume), [pdf](https://people.math.rochester.edu/faculty/doug/mybooks/nilpb.pdf), [webpage](https://people.math.rochester.edu/faculty/doug/nilp.html)]

* {#Kochmann96} [[Stanley Kochmann]], section 5 of  of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _Stable homotopy groups of spheres_, PNAS October 6, 2020 117 (40) 24757-2476 ([arXiv:2001.04247](https://arxiv.org/abs/2001.04247), [doi:10.1073/pnas.2012335117](https://doi.org/10.1073/pnas.2012335117))

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _More stable stems_ ([arXiv:2001.04511](https://arxiv.org/abs/2001.04511))

A tabulation of stable homotopy groups of spheres is in

* [[Doug Ravenel]], Appendix 3 of _[[Complex cobordism and stable homotopy groups of spheres]]_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA3.pdf))

Original articles on basic properties:

* {#Serre53} [[Jean-Pierre Serre]], _Groupes d'homotopie et classes de groupes abelien_, Ann. of Math. 58 (1953), 258–294 ([jstor:1969789](https://www.jstor.org/stable/1969789))

Early computation of unstable homotopy groups of spheres $\pi_{n+k}(S^k)$ up to $n\leq 19$:

* [[Hirosi Toda]], _Composition Methods in Homotopy Groups of Spheres_, Annals of Mathematics Studies Vol. 49, Princeton University Press (1962) ([jstor:j.ctt1bgzb5t](https://www.jstor.org/stable/j.ctt1bgzb5t))

See also

* Wikipedia, _[Homotopy groups of spheres](http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres)_

* [[eom]], _[Spheres, homotopy groups of the](http://www.encyclopediaofmath.org/index.php/Spheres,_homotopy_groups_of_the)_

* MO, _[Computational complexity of computing homotopy groups of spheres](http://mathoverflow.net/questions/31004/computational-complexity-of-computing-homotopy-groups-of-spheres)_

### Image of the J-homomorphism

Discussion of the [[image]] of the [[J-homomorphism]] is due to

* {#Adams66} [[John Adams]], _On the groups $J(X)$ IV_, Topology 5: 21,(1966)  _Correction_, Topology 7 (3): 331 (1968)

* {#Quillen71} [[Daniel Quillen]], _The Adams conjecture_, Topology. an International Journal of Mathematics 10: 67–80 (1971)

### Formalization in homotopy type theory

For formalization in [[homotopy type theory]] see

* [[formalization of the homotopy groups of spheres in homotopy type theory]]

* [[UF-IAS-2012]], _[HomotopyGroupsOfSpheres](http://uf-ias-2012.wikispaces.com/HomotopyGroupsOfSpheres)_

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

[[!redirects homotopy group of a sphere]]
[[!redirects homotopy groups of a sphere]]
[[!redirects homotopy groups of spheres]]

[[!redirects homotopy group of spheres]]

[[!redirects stable homotopy group of spheres]]
[[!redirects stable homotopy groups of spheres]]


homotopy groups of spheres in HoTT

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

=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

This entry is about [[homotopy groups of spheres]] formalized in [[homotopy type theory]] (HoTT).

The [[homotopy groups of spheres]] are a fundamental concept in [[algebraic topology]] and [[homotopy theory]]. They are the [[homotopy classes]] of [[maps]] between [[n-spheres|$n$-spheres]], as $n$ varies. These may equivalently be understood as collection of different ways to [[space attachment|attach]] spheres to each other. For instance, the [[homotopy type]] of a [[CW complex]] is completely determined by the homotopy types of the attaching maps.

The following is a quick reference for the state of the art on formalizing and computing [[homotopy groups of spheres]] in [[homotopy type theory]].

## Table

      n/k
      0
      1
      2
      3
      4

      0
      π0(S0)
      π0(S1)
      π0(S2)
      π0(S3)
      π0(S4)

      1
      π1(S0)
      π1(S1)
      π1(S2)
      π1(S3)
      π1(S4)

      2
      π2(S0)
      π2(S1)
      π2(S2)
      π2(S3)
      π2(S4)

      3
      π3(S0)
      π3(S1)
      π3(S2)
      π3(S3)
      π3(S4)

      4
      π4(S0)
      π4(S1)
      π4(S2)
      π4(S3)
      π4(S4)

## Formalized proofs

The following is a list of results for which at least one [[proof]] has been formalized in HoTT.

### Calculation of $\pi_4(S^3)$ {#pi4s3}

* [Brunerie 2016](#Brunerie16) has proved that there exists an $n$ such that $\pi_4(S^3)$ is $\mathbb{Z}_n$.  Given a computational interpretation, we could run this proof and check that $n$ is 2. Added June 2016: Brunerie now has a proof that $n=2$, using cohomology calculations and a Gysin sequence argument.

See also at *[[first stable homotopy group of spheres]]*.

### Calculation of $\pi_3(S^2)$ {#pi3s2}

* [[Peter LeFanu Lumsdaine]] has constructed the Hopf fibration as a dependent type. Lots of people around know the construction, but I don't know anywhere it's written up. Here's some [Agda code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Hopf.agda) with it in it.

* [Brunerie 2016](#Brunerie16), proof that the total space of the Hopf fibration is $S^3$, together with $\pi_n(S^n)$, imply this by a long-exact-sequence argument.
* This was [formalized in Lean](https://github.com/leanprover/lean2/blob/master/hott/homotopy/sphere2.hlean) in 2016.

### $\pi_n(S^2)=\pi_n(S^3)$ for $n\geq 3$  {#hopff}

* This follows from the Hopf fibration and long exact sequence of homotopy groups.
* It was [formalized in Lean](https://github.com/leanprover/lean2/blob/master/hott/homotopy/sphere2.hlean) in 2016.

See also at *[[second stable homotopy group of spheres]]*.

### Freudenthal Suspension Theorem  {#fsusp}

Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \le 2n - 2$

* [[Peter LeFanu Lumsdaine]]'s encode/decode-style proof, [formalized](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Freudenthal.agda) by [[Dan Licata]], using a clever lemma about maps out of two n-connected types.

### Calculation of $\pi_n(S^n)$ {#pinsn}

* [[Dan Licata]] and [[Guillaume Brunerie]]'s [encode/decode-style proof using iterated loop spaces](https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiNSN.agda) (for single-loop presentation).
* [[Guillaume Brunerie]]'s proof (for suspension definition).
* [[Dan Licata]]'s [proof from Freudenthal suspension theorem](https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiSnSusp.agda) (for suspension definition).

See also at **[[Hopf degree theorem]]**.

### Calculation of $\pi_k(S^n)$ for $k \lt n$ {#piksn}

* [Brunerie 2016](#Brunerie16) (see the [[HoTT book]]).
* [[Dan Licata]]'s encode/decode-style proof [for pi_1(S^2) only](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Pi1S2.agda).
* [[Dan Licata]]'s encode/decode-style proof [for all k < n](https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiKSNLess.agda) (for single-loop presentation).
* [[Dan Licata]]'s [proof from Freudenthal suspension theorem](https://github.com/dlicata335/hott-agda/blob/master/homotopy/PiSnSusp.agda) (for suspension definition).

### Calculation of $\pi_2(S^2)$ {#pi2s2}

* [Brunerie 2016](#Brunerie16), using the total space the Hopf fibration.
* [[Dan Licata]]'s encode/decode-style [proof](https://github.com/dlicata335/hott-agda/tree/master/homotopy/pi2s2).

### Calculation of $\pi_1(S^1)$ {#pi1s1}

* [[Mike Shulman]]'s proof by contractibility of total space of universal cover ([HoTT blog](http://uf-ias-2012.wikispaces.com/ofpost)).
* [[Dan Licata]]'s encode/decode-style proof ([HoTT blog](http://homotopytypetheory.org/2012/06/07/a-simpler-proof-that-%CF%80%E2%82%81s%C2%B9-is-z/)). A [paper](http://arxiv.org/abs/1301.3443) mostly about the encode/decode-style proof, but also describing the relationship between the two.

* [[Guillaume Brunerie]]'s proof using the flattening lemma.

See also at *[[circle type]]* for more.

## Related entries

* [[first stable homotopy group of spheres]]

* [[second stable homotopy group of spheres]]

* [[third stable homotopy group of spheres]]

* [[synthetic homotopy theory]]

* [[Hopf construction in homotopy type theory]]

* [[cohomology in homotopy type theory]]

## References

* {#Brunerie16} [[Guillaume Brunerie]], *[[On the homotopy groups of spheres in homotopy type theory]]* $[$[arxiv:1606.05916](https://arxiv.org/abs/1606.05916)$]$

* {#Ljungstroem22} [[Axel Ljungström]], *The Brunerie Number Is -2* (June 2022) $[$[blog entry](https://homotopytypetheory.org/2022/06/09/the-brunerie-number-is-2/)$]$

[[!redirects homotopy groups of spheres in homotopy type theory]]

[[!redirects formalization of the homotopy groups of spheres in homotopy type theory]]

category: homotopy theory