Section 15.4 Spectra Revisited

Brown’s theorem produces, for each generalized cohomology theory \(E^*\text{,}\) a sequence of representing spaces; we now show that these spaces assemble into a spectrum.

Recall (Definition 11.17) that a spectrum \(X_\bullet\) is a sequence of pointed spaces \(\{X_n\}_{n \geq 0}\) together with structure maps \(\Sigma X_n \to X_{n+1}\text{.}\)

For this construction we need the cohomology version of Brown’s theorem (Hatcher, Theorem 4E.1): a reduced cohomology theory satisfying the wedge axiom is representable on all based CW complexes, allowing disconnected representing spaces. The connected-space statement above alone does not give this conclusion. For example, \(\widetilde H^0(-;G)\) is represented by the discrete based space \(G\text{.}\)

Thus, given \(E^*\text{,}\) there are based CW spaces \(\{X_n\}_{n \geq 0}\) with natural isomorphisms

\begin{equation*} [Y, X_n]_* \;\cong\; E^n(Y). \end{equation*}

To produce structure maps \(\Sigma X_n \to X_{n+1}\) we use the suspension axiom for \(E^*\) and the Yoneda lemma.

Theorem 15.12. Yoneda lemma.

For any functor \(F \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}\) and any object \(X \in \mathcal{C}\text{,}\) natural transformations \(\mathrm{Hom}_{\mathcal{C}}(-, X) \to F\) are in bijection with the elements of \(F(X)\text{:}\)

\begin{equation*} \mathrm{Nat}\bigl(\mathrm{Hom}_{\mathcal{C}}(-, X),\, F\bigr) \;\cong\; F(X). \end{equation*}

The bijection is concrete: a natural transformation \(\eta\) corresponds to \(\eta_X(\mathrm{id}_X) \in F(X)\text{;}\) an element \(c \in F(X)\) corresponds to the natural transformation defined on \(g \colon Y \to X\) by \(g \mapsto g^*(c)\text{.}\)

Corollary 15.13.

Two objects \(X,Y\in\mathcal{C}\) are isomorphic if and only if there is a natural isomorphism

\begin{equation*} \mathrm{Hom}_{\mathcal{C}}(-,X)\cong\mathrm{Hom}_{\mathcal{C}}(-,Y). \end{equation*}

Indeed, the two natural isomorphisms correspond to maps \(X\to Y\) and \(Y\to X\text{,}\) and their composites correspond to the identity transformations. In the homotopy category, this says that representing CW spaces are unique up to based homotopy equivalence.

Building the structure maps.

By the suspension axiom \(E^{n+1}(\Sigma Y) \cong E^n(Y)\) together with \([Y, X_n]_* \cong E^n(Y)\text{,}\) we have

\begin{equation*} [\Sigma Y, X_{n+1}]_* \;\cong\; [Y, X_n]_*. \end{equation*}

The suspension–loop adjunction \([\Sigma Y, Z]_* \cong [Y, \Omega Z]_*\) gives

\begin{equation*} [Y, \Omega X_{n+1}]_* \;\cong\; [Y, X_n]_* \quad \text{for every } Y. \end{equation*}

Choose the direction of this natural isomorphism from \([Y,X_n]_*\) to \([Y,\Omega X_{n+1}]_*\text{.}\) By Yoneda it is induced by a based map

\begin{equation*} \epsilon_n\colon X_n\longrightarrow\Omega X_{n+1}, \end{equation*}

obtained by applying the isomorphism to \([\mathrm{id}_{X_n}]\text{.}\) Here \(Y\) ranges over based CW complexes. To justify the conclusion about spaces, choose a CW approximation \(Z\to\Omega X_{n+1}\text{.}\) It induces bijections on based homotopy classes of maps from CW complexes. Thus \(Z\) and \(X_n\) represent the same functor, so Yoneda gives a homotopy equivalence \(X_n\simeq Z\text{.}\) Consequently \(\epsilon_n\) is a weak equivalence. Its adjoint \(\Sigma X_n\to X_{n+1}\) is the required structure map.

Definition 15.14.

A spectrum \(\{X_n\}\) is an \(\Omega\)-spectrum if the maps \(X_n \to \Omega X_{n+1}\) adjoint to the structure maps are weak equivalences.

Corollary 15.15.

Every generalized cohomology theory \(E^*\) gives rise to an \(\Omega\)-spectrum \(E\text{.}\)

Example 15.16.

Applied to reduced ordinary cohomology \(\widetilde H^*(-; G)\) this recovers the Eilenberg–MacLane spectrum \(HG = \{K(G, n)\}_{n \geq 0}\) from Example 11.20, where \(K(G,0)\) is the discrete group \(G\text{,}\) based at \(0\text{.}\)

Remark 15.17.

The converse holds as well: an \(\Omega\)-spectrum \(E=\{E_n\}_{n\geq0}\) defines a reduced generalized cohomology theory. In nonnegative degrees it is given by \(h^n(Y)=[Y,E_n]_*\text{;}\) in any integer degree \(k\text{,}\) use

\begin{equation*} h^k(Y)=[\Sigma^rY,E_{k+r}]_*,\qquad r\geq\max\{0,-k\}. \end{equation*}

The adjoint structure maps and the suspension–loop adjunction identify the groups for successive choices of \(r\text{.}\) See Hatcher, Theorem 4.58, for the cohomology axioms. Inverting weak equivalences in a suitable category of spectra gives the stable homotopy category.

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