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A cofiber sequence of spectra $A\to B\to C\to \Sigma A$ induces exact sequences after either mapping into or out of a fixed topological spectrum, and on stable homotopy groups. In particular the second map is zero precisely when the first map admits a right inverse in the stable homotopy category.

A generalized homology theory obeys the homotopy, exactness, suspension and wedge axioms, without imposing the ordinary dimension axiom. A represented homology theory is obtained by taking stable homotopy groups after a smash product of spectra.

Use derived smash products of spectra and replace a based space by a based CW complex when necessary. The reduced represented homology theory of $E$ isThus represented homology theory is covariant in $X$: a based map induces a map of suspension spectra, and hence a map on the stable homotopy groups of the displayed smash product of spectra. In a well-based sequential model the same definition iswith $j$ sufficiently large and transition maps given by suspension followed by the structure map.

The reduced represented cohomology theory isThe brackets denote maps in the stable homotopy category. This is contravariant in $X$, by precomposition. For the Omega-spectrum $E$ it is also the group of based homotopy classes $[X,E_q{]}_{\ast }$, where for negative $q$ we take $E_q={\Omega }^{-q}{E}_0$. More generally, for $j\ge \max (q,0)$ the Omega-spectrum identifications give ${\tilde{E}}^q(X)=[{\Sigma }^{j-q}X,E_j{]}_{\ast }$; the shifts and loops supply its abelian group structure. In particularAll these definitions are reduced: the basepoint itself contributes zero.

Choose a small skeleton $\mathcal{F}$ of the finite spectra, including every integer suspension. There is such a set of representatives: a finite list of stable cells and their attaching maps specifies a finite cell model, and the possible attaching maps form sets. For every $F\in \mathcal{F}$ and every stable map $v:F\to D$, take one copy of $F$ and formDefine the evaluation map $e:X\to D$ by taking its restriction to the $(F,v)$ summand to be $v$. Complete it to the cofiber sequence of spectraThis is the universal evaluation phantom map construction.

First, $p$ is nonzero. If $p=0$, applying $[D,-{]}_{\mathrm{st}}$ to the cofiber sequence of spectra gives the exact portionSince $p_{\ast }(1_D)=p=0$, exactness supplies $s:D\to X$ with $es=1_D$. Thus $D$ is a retract of $X$. This is also a wedge summand: complete $s$ to a cofiber sequence of spectra $D\to X\to C^{\prime }\to \Sigma D$. The left inverse $e$ makes this triangle split, giving $X\cong D\vee C^{\prime }$ in the stable homotopy category. One can see the splitting directly from exactness: the connecting map is zero because $e$ extends $1_D$ across $s$, and consequently the quotient map $X\to C^{\prime }$ admits a section. This contradicts the assumption on $D$. Hence $p\ne 0$.

For any finite spectrum $F$ and any $v:F\to D$, its chosen representative has a summand inclusion ${\iota }_v:F\to X$ with $e{\iota }_v=v$, after transferring along the representative isomorphism. Consecutive maps of a cofiber sequence of spectra compose to zero, soThus $p$ is a phantom map of spectra. To establish the requested conclusion on all spaces, rather than just coefficient groups, let $K$ first be a finite based CW complex. Put $A={\Sigma }^{\infty }K$. The Spanier-Whitehead dual $A^{\vee }=F(A,\mathbb{S})$ is a finite spectrum. The evaluation and coevaluation adjunction of Spanier-Whitehead duality identifiesUnder this identification $p_{\ast }$ is postcomposition with $p$. Its source is a finite spectrum, including when $k$ is negative, so the calculation $pv=0$ proves that $p_{\ast }$ is zero on ${\tilde{D}}_k(K)$ for every $k$.

Now let $Y$ be an arbitrary based CW complex. It is the filtered union, or filtered homotopy colimit of spaces, of its finite based subcomplexes $K$. The suspension spectrum functor and smash product of spectra preserve that homotopy colimit. The shifted sphere spectrum is compact, so taking stable homotopy groups givesEvery element of the first group therefore comes from a finite $K$, where $p_{\ast }$ has just been proved zero. Consequently $p_{\ast }$ is zero on $Y$ as well. A based CW complex replacement extends the conclusion to arbitrary based spaces in the homotopy category. We have constructedThis uses continuity of represented homology theory, not a continuity assertion for represented cohomology theory; the latter need not hold.

The sphere spectrum is $\mathbb{S}={\Sigma }^{\infty }{S}^0$. It is the unit of the smash product of spectra. Its integer suspensions of spectra represent stable homotopy groups.

A map of topological spectra inducing isomorphisms on every stable homotopy group is a stable equivalence of spectra.

The $k$th stable homotopy group of $E$ is ${\pi }_{k}E=[{\Sigma }^k\mathbb{S},E{]}_{\mathrm{st}}$, equivalently ${\mathrm{colim}}_j{\pi }_{k+j}{E}_j$ in a sequential model.