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For isometry families $f_x:U\to U\prime$ and $g_y:U\prime \to U\prime \prime$, the complement of $g_y{f}_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same Thom space as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y{⋉}_g(X{⋉}_{f}E)\cong (Y\times X){⋉}_{g\circ f}E$ in the stable homotopy category.

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 finite spectrum is a topological spectrum equivalent to a cell spectrum with finitely many cells. Such an object is compact in the stable homotopy category and has a Spanier-Whitehead dual. Compactness means maps out of it commute with filtered homotopy colimits of spectra.

A map of topological spectra is a compatible family of based maps of the level spaces: $u_{j+1}{\sigma }_j^D={\sigma }_j^E(1\wedge u_j)$. It is distinct from its class in the stable homotopy category.

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.

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.

Let ${\mathrm{Map}}_{\mathrm{sp}}(D,E)$ be the space of point-set maps of topological spectra. Its underlying set is the first set in the comparison. A member is a compatible collection $D_j\to E_j$, rather than independently chosen maps at the levels.

Because the source is cellular and the target is an Omega-spectrum, taking spectrum homotopy classes gives all maps in the stable homotopy category:Thus the first set maps onto the second. Two point-set maps have the same stable class precisely when they are spectrum-homotopic, using the cellular cylinder and fibrant target. The homotopy must commute with the structure maps throughout; unrelated levelwise homotopies are not the definition.

Write $f_x$ for the linear isometry specified at $x$. Choose a cofinal sequence $V_i$ from the indexing family $I$, and write the target part of the supplied subordinate flag for a family of linear isometries as $W_i$. After a cofinal enlargement it hasBoth flags exhaust their universe for spectra. Compactness of the parameter space ensures that a finite target stage can contain all images of each fixed source stage.

Let ${\xi }_i$ be the vector bundle on $X$ with fiberHere $\ominus$ is the orthogonal complement in the indicated ambient subspace. Its rank is $\dim W_i-\dim V_i$. Let $\mathrm{Th}({\xi }_i)$ be its Thom space, the disk bundle modulo its sphere bundle, so that every fiber is compactified and all its points at infinity become the single basepoint. For the zero bundle this convention gives $\mathrm{Th}(0_X)=X_{+}$. Define a target indexed prespectrum on the flag byThe smash product here is of based spaces, not yet the derived smash product of spectra.

To specify all structure maps, for $i\le j$ use the continuous fiberwise orthogonal isomorphismBoth sides identify with the complement of $f_x(V_i)$ in $W_j$. In particular this is the isometry obtained by orthogonally projecting onto $f_x(V_j\ominus V_i)$ and its complement; it is not an assertion that the displayed summands coincide separately. Pass to fiberwise one-point compactifications, identify the last summand with $V_j\ominus V_i$ using $f_x^{-1}$, and apply the source structure map. This definesMore explicitly, decompose $z+w\in W_j\ominus f_x(V_i)$, where $z\in W_j\ominus W_i$ and $w\in ({\xi }_i{)}_x$, as $w^{\prime }+f_x(v)$ with $w^{\prime }\in ({\xi }_j{)}_x$ and $v\in V_j\ominus V_i$. Send its pair with $e\in E(V_i)$ to the pair $(w^{\prime },{\sigma }_{V_i,V_j}(v,e))$ over $x$. The notation for finite vectors extends to the compactifications, sending infinities and basepoints to the basepoint. Orthogonal decomposition and associativity of the source structure maps prove the prespectrum identities.

Extend the indexed prespectrum from the cofinal flag to the target indexing family and apply spectrification $L$. The point-set definition isOn a map of topological spectra $u:E\to E^{\prime }$ it is induced at stage $i$ by $1_{\mathrm{Th}({\xi }_i)}\wedge u(V_i)$. Thus the prescribed isometry family, indexing family and flag have all been used. A common cofinal refinement identifies the resulting objects in the stable homotopy category; the point-set model retains the chosen flag. If $f$ is a constant standard inclusion, the complement bundles are trivial and this recovers the usual reindexed $X_{+}\wedge E$.

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.

A ring spectrum is a unital associative monoid object of the stable homotopy category under the smash product of spectra. It has $\mu :E\wedge E\to E$ and $\eta :\mathbb{S}\to E$, satisfying the associative and two unit identities. Structured ring spectra additionally specify coherent point-set or higher homotopies; a ring object alone does not assert those enhancements.

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 ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

A sequential model of a topological spectrum consists of based spaces $E_j$ and structure maps $S^1\wedge E_j\to E_{j+1}$. Indexed models use finite-dimensional subspaces of a universe for spectra and structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$. Different standard models present the same stable homotopy category.

The stable homotopy category is obtained from topological spectra by inverting stable equivalences of spectra. Its morphisms are abelian groups; for a cellular source and fibrant target they are spectrum homotopy classes of point-set maps of topological spectra.

Stable homotopy theory studies topological spectra, which retain the information that persists under repeated reduced suspension. Its stable homotopy category admits invertible suspension of spectra and exact cofiber sequences of spectra.

Suspension of spectra shifts the stable degree. Unlike reduced suspension of based spaces, it is invertible in the stable homotopy category, with inverse denoted ${\Sigma }^{-1}$.