Spectra and Stable Homotopy Groups — Examples

For every $n\ge 1$, [F1] gives $\mathrm{deg}:{\pi }_n(S^n)\stackrel{\cong }{\to }\mathbb{Z}$, with the identity sent to $1$.

Suspension preserves degree by [F1]. Consequently the square

By [F2], its colimit is the zero stem and hence is $\mathbb{Z}$. The identity at any stage represents the compatible element $1$, so its stable class maps to $1$. $\mathrm{\square }$

The homotopy class $[f]\in {\pi }_{n+k}(S^n)$ is a legal representative in the colimit defining ${\pi }_k^s$.

By definition of that colimit's bonding map, $(n,[f])\sim (n+1,[\mathrm{\Sigma }f])$. Iterating gives $(n,[f])\sim (n+r,[{\mathrm{\Sigma }}^{r}f])$ for every $r\ge 0$.

A based homotopy $f\simeq f^{\mathrm{\prime }}$ gives the same stage class by [F2], and its suspensions do likewise, so it gives the same stable class. This is consistent with strict-map functoriality for suspension prespectra. Nothing here says that the original unstable class can be recovered from its stable image. $\mathrm{\square }$

At level $n$, the canonical smash associator gives [F1] $({\mathrm{\Sigma }}^{\mathrm{\infty }}{S}^r{)}_n=S^n\wedge S^r\cong S^{n+r}={\mathbb{S}}_{n+r}=({\mathrm{sh}}^r\mathbb{S}{)}_n\mathrm{.}$

Both structure maps add a leading $S^1$ coordinate. Smash coherence says the level homeomorphisms commute with these structure maps, so they form a strict prespectrum isomorphism.

Applying [F2] $r$ times gives ${\pi }_k({\mathrm{sh}}^r\mathbb{S})\cong {\pi }_{k-r}(\mathbb{S})={\pi }_{k-r}^s$. The opposite sign in the Step-1 scaffold is incompatible with the direct reindexing and is not used. $\mathrm{\square }$

By [F1], the loop $[a,b]$ corresponds to the word $ab{a}^{-1}{b}^{-1}$ in the free group on $a,b$. No two adjacent letters in this four-letter word are formal inverses, so it is already reduced and is not the empty word. The reduced-word uniqueness in [F2] therefore proves that $[a,b]\mathrm{\ne }1$. Thus it is a nonzero initial-stage element of

Since $X$ is connected, [F5] with connectivity parameter $1$ and $i=1$ gives the suspension homomorphism $E:{\pi }_1(X)\to {\pi }_2(\mathrm{\Sigma }X)$. This is its surjective endpoint; no injectivity is promised. The target is abelian by [F3]. Every homomorphism from $F(a,b)$ to an abelian group kills the commutator, so $E([a,b])=0$.

Freudenthal uses the two-cone suspension. Collapsing its basepoint meridian gives the reduced suspension $S^1\wedge X$ used by ${\mathrm{\Sigma }}^{\mathrm{\infty }}X$. The meridian is a nonempty contractible CW subcomplex, so [F4] makes the collapse an isomorphism on ${\pi }_2$; the quotient formula is the reduced-suspension formula in the prespectrum definition. Thus the image found in step 1.2 is the prespectrum bonding image.

In the stable colimit, $[0,[a,b]]=[1,E([a,b])]=[1,0]=0$. Thus a genuinely nontrivial unstable class dies before stabilization. No value of ${\pi }_4(S^3)$ or positive stable stem is assumed. $\mathrm{\square }$