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A cell spectrum is obtained by attaching stable cells, with successive cofibers of spectra wedges of suspensions of spectra of the sphere spectrum. Cellular models provide the cofibrant objects needed to compute derived maps of topological spectra. Stable cells may occur in arbitrary integer degrees.

A topological spectrum can be described sequentially by based spaces $E_j$, $j\ge 0$, and structure mapsIterating these maps gives compatible maps $S^{k-j}\wedge E_j\to E_k$. Equivalently, an indexed prespectrum on a universe for spectra $U$ has spaces $E(V)$ and unital associative structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for finite-dimensional inclusions $V\subset W\subset U$. Here $S^A$ denotes the one-point compactification of the real inner product space $A$.

A map of topological spectra $u:D\to E$ is a family of based maps $u_j:D_j\to E_j$ satisfyingThe indexed version requires the same compatibility for every inclusion of indexing subspaces. These are point-set maps, before passing to spectrum homotopy or the stable homotopy category.

An Omega-spectrum is a topological spectrum whose adjoint structure mapsare weak homotopy equivalences. Consequently ${\pi }_{k}E={\pi }_{k+j}{E}_j$ whenever $k+j\ge 0$, using the structure identifications. In the genuine indexed point-set convention, the word spectrum already includes the stronger condition that $E(V)\to {\Omega }^{W\ominus V}E(W)$ is a homeomorphism; a general system without that condition is called an indexed prespectrum. Its associated spectrum is obtained by spectrification. The sequential and indexed conventions present the same stable theory.

A cell spectrum is built from the zero topological spectrum by successive stable cell attachments. Concretely, attach a free spectrum on the inclusion $S^{m-1}\hookrightarrow D^m$ at some level $j$ by a pushout along its boundary map; its quotient is the stable cell sphere ${\Sigma }^{m-j}\mathbb{S}$. Iterating such attachments and taking the colimit gives a cell spectrum, with successive quotients wedges of integer suspensions of spectra of the sphere spectrum. Negative cell degrees are allowed by choosing $j>m$. In the genuine indexed model apply spectrification to this cellular prespectrum. The compatibility of the structure maps is part of both the objects and their maps.

In the stable homotopy category, a ring spectrum is a unital associative monoid object for the smash product of spectra. Thus it consists of a topological spectrum $E$ with multiplication and unitwhere $\mathbb{S}$ is the sphere spectrum, and the identities areThe canonical associativity and unit identifications of the symmetric monoidal category are understood in these formulas. All identities are identities of maps in the stable homotopy category.

A commutative ring spectrum also satisfieswhere $\tau :E\wedge E\to E\wedge E$ interchanges the two factors. This is the ring-object convention; a coherently structured ring spectrum has additional point-set or higher-homotopy data. In particular homotopy-category commutativity alone is not an assertion of such a coherent enhancement.

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 derived smash product of spectra is the symmetric monoidal product on the stable homotopy category. It has unit sphere spectrum, preserves cofiber sequences of spectra and homotopy colimits of spectra in each variable, and satisfies ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

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.