Spectra and Stable Homotopy Groups

On this page a based space is a compactly generated weak Hausdorff space $X$ equipped with a chosen point ${\ast }_X\in X$. Products are the compactly generated products from the published CGWH convention, and every quotient is kified after taking the ordinary quotient.

A based space $X$ is well-pointed when the inclusion

$\{{\ast }_X\}\hookrightarrow X$

is an unbased cofibration, in the sense of the published homotopy-extension property. Thus well-pointedness is an additional hypothesis: it is not being asserted for every based CGWH space. The structure spaces of every sequential prespectrum on this page are required to be well-pointed. No choice principle is used in these conventions.

Let $X$ and $Y$ be based CGWH spaces. Their wedge inside the k-product is

$X\vee Y=(X\times \{{\ast }_Y\})\cup (\{{\ast }_X\}\times Y)\subseteq X{\times }_{k}Y\mathrm{.}$

The smash product is the based, kified quotient

$X\wedge Y=k((X{\times }_{k}Y)\mathrm{/}(X\vee Y)),$

based at the collapsed wedge. If $q:X{\times }_{k}Y\to X\wedge Y$ is the quotient map, then the quotient universal property says that a based map $X\wedge Y\to Z$ is equivalently a continuous map $X{\times }_{k}Y\to Z$ that is constant with value ${\ast }_Z$ on $X\vee Y$. The kification does not change this test against a CGWH target.

We write $x\wedge y$ for $q(x,y)$. No claim that $X\wedge Y$ is well-pointed is made without further hypotheses.

For based CGWH spaces $X,Y,Z$, the formulas

$(x\wedge y)\wedge z⟼x\wedge (y\wedge z),x\wedge y⟼y\wedge x,$

and $S^0\wedge X\to X$, $1\wedge x\mapsto x$, induce natural based homeomorphisms

$(X\wedge Y)\wedge Z\cong X\wedge (Y\wedge Z),X\wedge Y\cong Y\wedge X,S^0\wedge X\cong X\mathrm{.}$

These maps satisfy the usual pentagon, triangle, and symmetry coherence identities.

A sequential prespectrum $E$ consists of well-pointed based CGWH spaces $E_n$, indexed by $n\ge 0$, and based structure maps

${\sigma }_n:S^1\wedge E_n⟶E_{n+1}\mathrm{.}$

The sphere coordinate is always written first. Its adjoint structure map is

${\tilde{\sigma }}_n:E_n⟶\mathrm{\Omega }{E}_{n+1},{\tilde{\sigma }}_n(x)(t)={\sigma }_n(t\wedge x)\mathrm{.}$

The loop space carries the compactly generated compact-open topology. The loop--suspension adjunction identifies ${\sigma }_n$ and ${\tilde{\sigma }}_n$.

On this page a spectrum means an Omega-prespectrum: every ${\tilde{\sigma }}_n$ is a weak homotopy equivalence. This is terminology in the strict sequential setting only; no stable model structure or replacement functor is part of the definition.

For a well-pointed based CGWH space $X$, its suspension prespectrum is

$({\mathrm{\Sigma }}^{\mathrm{\infty }}X{)}_n=S^n\wedge X,$

where $S^n=(S^1{)}^{\wedge n}$ and $S^0$ is the smash unit. Its structure map

$S^1\wedge (S^n\wedge X)⟶S^{n+1}\wedge X$

is the canonical associativity homeomorphism, with the new $S^1$ coordinate placed first. Coherence makes the iterated construction unambiguous.

The sphere prespectrum is

$\mathbb{S}={\mathrm{\Sigma }}^{\mathrm{\infty }}{S}^0,{\mathbb{S}}_n=S^n\mathrm{.}$

These are prespectra. The definition does not assert that an arbitrary suspension prespectrum is an Omega-spectrum.

Let $E=(E_n,{\sigma }_n)$ and $F=(F_n,{\tau }_n)$ be sequential prespectra. A strict map of prespectra $f:E\to F$ is a family of based maps $f_n:E_n\to F_n$ such that, for every $n\ge 0$,

$f_{n+1}\circ {\sigma }_n={\tau }_n\circ (1_{S^1}\wedge f_n):S^1\wedge E_n⟶F_{n+1}\mathrm{.}$

A structure-compatible homotopy $H:f\simeq g$ is a family of based homotopies $H_n:E_n\times I\to F_n$ satisfying

$H_{n,0}=f_n,H_{n,1}=g_n,$

and for which every time slice $H_{n,t}$ is strict:

$H_{n+1,t}\circ {\sigma }_n={\tau }_n\circ (1_{S^1}\wedge H_{n,t})(t\in I)\mathrm{.}$

Thus this page uses the strict level category and this specified notion of homotopy; it does not identify arbitrary zigzags or introduce stable maps.

Fix $k\in \mathbb{Z}$ and choose $n_0\ge 0$ such that $n+k\ge 1$ whenever $n\ge n_0$. For a sequential prespectrum $E$, the bonding homomorphism is

$\begin{array}{rl}{\displaystyle b_n:{\pi }_{n+k}(E_n)}& {\displaystyle ⟶{\pi }_{n+k+1}(E_{n+1}),}\\ {\displaystyle [f]}& {\displaystyle ⟼[{\sigma }_n\circ (1_{S^1}\wedge f)\circ \chi ],}\end{array}$

where $\chi :S^{n+k+1}\cong S^1\wedge S^{n+k}$ is the canonical sphere-coordinate homeomorphism. Equivalently, $b_n$ is suspension followed by $({\sigma }_n{)}_{\ast }$.

The $k$th stable homotopy group of $E$ is

${\pi }_k(E)={\mathrm{colim}}_{n\ge n_0}({\pi }_{n+k}(E_n),b_n)\mathrm{.}$

Concretely this is the disjoint union of the stage groups modulo $(n,x)\sim (n+1,b_{n}x)$. Two representatives are added after advancing both to a common stage. This is well defined because the $b_n$ are homomorphisms. After increasing $n_0$ if necessary, all degrees $n+k$ are at least two, so all stage groups and the resulting colimit are abelian. The next lemma proves that the displayed group does not depend on the chosen finite initial cutoff.

Let $n_0\ge 0$, and let $A_{n_0}\to A_{n_0+1}\to A_{n_0+2}\to \cdots$ be the sequential group system used to define a stable homotopy group. For every $N\ge n_0$, inclusion of the tail induces a canonical group isomorphism

${\mathrm{colim}}_{n\ge N}{A}_n\stackrel{\cong }{\to }{\mathrm{colim}}_{n\ge n_0}{A}_n\mathrm{.}$

In particular, deleting finitely many legal terms from the system defining ${\pi }_k(E)$ does not change the resulting group.

A strict map $f:E\to F$ of sequential prespectra induces, for every $k\in \mathbb{Z}$, a homomorphism

${\pi }_k(f):{\pi }_k(E)⟶{\pi }_k(F)\mathrm{.}$

These maps preserve identities and composition. Structure-compatible homotopic strict maps induce the same homomorphism.

In this strict sequential-prespectrum setting, a stable weak equivalence is, by definition, a strict map $f$ for which ${\pi }_k(f)$ is an isomorphism for every $k\in \mathbb{Z}$.

For a sequential prespectrum $E$, its shift is

$(\mathrm{sh}E{)}_n=E_{n+1},$

with structure map ${\sigma }_{n+1}:S^1\wedge E_{n+1}\to E_{n+2}$. Its sequential suspension (the right-shifted object) is

$(sE{)}_0=\ast ,(sE{)}_{n+1}=E_n,$

with the unique structure map out of $S^1\wedge \ast$ at level zero and ${\sigma }_{n-1}:S^1\wedge E_{n-1}\to E_n$ at level $n\ge 1$.

With the grading convention ${\pi }_k(E)={\mathrm{colim}}_n{\pi }_{n+k}(E_n)$, there are canonical isomorphisms

$\boxed{{\displaystyle {\pi }_k(\mathrm{sh}E)\cong {\pi }_{k-1}(E),{\pi }_k(sE)\cong {\pi }_{k+1}(E)\mathrm{.}}}$

For the shift, put $m=n+1$ and delete the missing finite initial term:

${\mathrm{colim}}_n{\pi }_{n+k}(E_{n+1})={\mathrm{colim}}_m{\pi }_{m+(k-1)}(E_m)={\pi }_{k-1}(E)\mathrm{.}$

For $sE$, discard its zero level and put $m=n-1$:

${\mathrm{colim}}_{n\ge 1}{\pi }_{n+k}(E_{n-1})={\mathrm{colim}}_{m\ge 0}{\pi }_{m+(k+1)}(E_m)={\pi }_{k+1}(E)\mathrm{.}$

The cofinal-tail lemma makes both reindexings canonical. These signs correct the reversed formulas in the Step-1 scaffold.

For $k\in \mathbb{Z}$, the $k$th stable stem of the sphere is

${\pi }_k^s:={\pi }_k(\mathbb{S})={\mathrm{colim}}_{n\ge n_0}{\pi }_{n+k}(S^n),$

where $n_0$ is any index for which $n+k\ge 1$ thereafter. The bonding map is the suspension homomorphism determined by the sphere-prespectrum structure homeomorphism $S^1\wedge S^n\cong S^{n+1}$. Tail independence makes the notation independent of $n_0$.

For integers $k\ge 0$ and $n\ge 1$, the sphere-system bonding map

$E:{\pi }_{n+k}(S^n)⟶{\pi }_{n+k+1}(S^{n+1})$

is an isomorphism when $n>k+1$ and is a surjection when $n=k+1$.

For every $k\ge 0$, ${\pi }_k(\mathbb{S})$ is canonically isomorphic to the eventual value of

${\pi }_{n+k}(S^n)\stackrel{E}{\to }{\pi }_{n+k+1}(S^{n+1})\stackrel{E}{\to }\cdots \mathrm{.}$

This eventual group is the classical $k$th stable homotopy group of spheres.

Let $E,F,G$ be sequential prespectra, with structure maps denoted ${\sigma }^E,{\sigma }^F,{\sigma }^G$. A pairing $E\wedge F\to G$ is a family of based maps

${\mu }_{m,n}:E_m\wedge F_n⟶G_{m+n}$

together with based homotopies making it compatible with each structure coordinate. Suppressing only the canonical associators, the two required comparisons are

${\mu }_{m+1,n}({\sigma }_m^E\wedge 1)\simeq {\sigma }_{m+n}^G(1_{S^1}\wedge {\mu }_{m,n})$

on $S^1\wedge E_m\wedge F_n$, and

${\mu }_{m,n+1}(1\wedge {\sigma }_n^F)\simeq {\sigma }_{m+n}^G(1_{S^1}\wedge {\mu }_{m,n})({\tau }_{E_m,S^1}\wedge 1)$

on $E_m\wedge S^1\wedge F_n$. Here ${\tau }_{E_m,S^1}$ moves the new suspension coordinate past $E_m$; this twist is part of the formula, not an implicit sign.

A unital multiplication on $E$ is such a pairing $E\wedge E\to E$, a based unit $\eta :S^0\to E_0$, and specified compatible homotopies from the two unit composites to the identity and between $\mu (\mu \wedge 1)$ and $\mu (1\wedge \mu )$, with the canonical associators inserted. A commutativity datum additionally gives compatible comparison homotopies between ${\mu }_{m,n}$ and ${\mu }_{n,m}{\tau }_{E_m,E_n}$, including the permutation of the $m$ and $n$ suspension-coordinate blocks. That block permutation has degree $(-1{)}^{mn}$ on $S^{m+n}$. All homotopies are required to respect the two displayed structure comparisons.

Let $E$ be a sequential prespectrum with a unital multiplication. If $x\in {\pi }_p(E)$ and $y\in {\pi }_q(E)$ are represented at stages $m,n$ by

$f:S^{m+p}\to E_m,g:S^{n+q}\to E_n,$

then the smash $f\wedge g$, followed by ${\mu }_{m,n}$ and the canonical sphere coordinate rearrangement, defines a product

${\pi }_p(E)\times {\pi }_q(E)⟶{\pi }_{p+q}(E)\mathrm{.}$

It is well defined, associative, and unital. If the multiplication has the commutativity datum of the preceding definition, then

$xy=(-1{)}^{pq}yx\mathrm{.}$

This page proves no value of a positive stable stem. It also does not prove Brown representability, a stable model structure, a fibrant or cofibrant replacement theorem, localization at stable weak equivalences, or the existence of a stable homotopy category.

Here stable weak equivalence means only the definition already given for a strict map: isomorphism on ${\pi }_k$ for every $k\in \mathbb{Z}$. That condition must not be silently replaced by levelwise homotopy equivalence, strict levelwise homotopy, or invertibility in an unconstructed category.

Later pages may use the sequential-prespectrum definitions, their colimit groups, the corrected shift bookkeeping, and products arising from explicit pairings. Any use of replacement, localization, Brown representability, or a calculated positive sphere stem needs a separate supplier.