Brown representability (online only)

Proof. Step 1: Constructing increasingly universal elements. We construct a sequence \[ X=:Z_0\xrightarrow {f_0}Z_1\xrightarrow {f_1}Z_2\longrightarrow \cdots \] and elements \(\xi _n\in F(Z_n)\) such that \(f_n^*(\xi _{n+1})=\xi _n\), the element \(\xi _n\) is \(n\)-universal for \(n\geq 1\), and \(T_{\xi _n}\) is surjective on \(S^k\) for every \(k\geq 1\).

Set \(\xi _0:=\eta \) and define \[ Z_1:=X\vee \bigvee _{m\geq 1}\ \bigvee _{\gamma \in F(S^m)}S^m. \] By the wedge axiom, there is a unique element \(\xi _1\in F(Z_1)\) whose restriction to \(X\) is \(\eta \) and whose restriction to the sphere indexed by \(\gamma \in F(S^m)\) is \(\gamma \). The inclusion of that sphere shows that \(T_{\xi _1}\) is surjective on \(S^m\) for every \(m\geq 1\). In particular, \(\xi _1\) is \(1\)-universal.

Suppose now that \(n\geq 1\) and that \((Z_n,\xi _n)\) has been constructed. By Remark 3.4.2, the map \[ T_{\xi _n}\colon [S^n,Z_n]_*\longrightarrow F(S^n) \] is a group homomorphism. Let \(K_n\) denote its kernel. Choose a representative \(g\colon S^n\to Z_n\) of every class \([g]\in K_n\), and form the pushout

The restrictions of \(\xi _n\) to all the attaching spheres are zero. The Mayer–Vietoris property therefore supplies an element \(\xi _{n+1}\in F(Z_{n+1})\) satisfying \(f_n^*(\xi _{n+1})=\xi _n\).

The map \(T_{\xi _{n+1}}\) remains surjective on every sphere, since \(T_{\xi _n}\) is surjective and \(\xi _n=f_n^*\xi _{n+1}\). Attaching \((n+1)\)-cells induces bijections \[ [S^k,Z_n]_*\xrightarrow {\cong }[S^k,Z_{n+1}]_* \] for \(k<n\), so \(\xi _{n+1}\) remains \(n\)-universal. Finally, every map \(S^n\to Z_{n+1}\) factors up to homotopy through \(Z_n\). If its class lies in the kernel of \(T_{\xi _{n+1}}\), a lift to \(Z_n\) lies in \(K_n\) and hence becomes null-homotopic after the corresponding cell is attached. Thus \(T_{\xi _{n+1}}\) is injective on \(S^n\), proving that \(\xi _{n+1}\) is \((n+1)\)-universal.

Step 2: Passing to the limit. Set \[ Z_X:=\colim _nZ_n. \] The mapping-telescope description from Lemma 2.4.29 gives a pushout square

On the summand \(Z_n\), the two maps are the identity and the transition map to the adjacent stage, assigned to the even and odd target wedges according to the parity of \(n\). The relations \(f_n^*(\xi _{n+1})=\xi _n\), together with the wedge axiom and the Mayer–Vietoris property, therefore produce an element \(\xi _X\in F(Z_X)\) restricting to \(\xi _n\) on every stage. In particular, the composite \(f_X\colon X=Z_0\to Z_X\) satisfies \(f_X^*(\xi _X)=\eta \).

For a fixed \(k\geq 1\), the map \(Z_n\to Z_X\) is obtained by attaching cells of dimension at least \(n+1\). Hence Remark 3.3.6 shows that \[ [S^k,Z_n]_*\longrightarrow [S^k,Z_X]_* \] is a bijection once \(n>k\). Since \(\xi _n\) is \(n\)-universal, \(T_{\xi _X}\) is consequently a bijection on \(S^k\). Thus \(\xi _X\) is universal. □

Proof of Theorem 3.4.3. If \(F\cong [-,Z]_*\) is representable, the wedge axiom follows from the universal property of coproducts. Mapping a pushout into \(Z\) gives a pullback of mapping animae. A pair of homotopy classes whose restrictions to the intersection agree can be represented together with a homotopy between those restrictions, and therefore lifts to a point of this pullback. This proves the Mayer–Vietoris property.

Conversely, suppose that \(F\) satisfies Brown’s conditions. Apply Proposition 3.4.5 to \(X=\pt \) and the unique element of \(F(\pt )\). We obtain a connected pointed anima \(Z\) and a universal element \(\xi \in F(Z)\). We claim that \[ T_\xi \colon [X,Z]_*\longrightarrow F(X) \] is a bijection for every connected pointed anima \(X\).

For surjectivity, let \(\eta \in F(X)\) and let \(\widetilde \eta \in F(X\vee Z)\) correspond under the wedge axiom to \((\eta ,\xi )\). Applying Proposition 3.4.5 to \((X\vee Z,\widetilde \eta )\) produces a universal element \(\widetilde \xi \in F(\widetilde Z)\) and a map \(X\vee Z\to \widetilde Z\) carrying \(\widetilde \xi \) to \(\widetilde \eta \). Its restriction \(g\colon Z\to \widetilde Z\) induces isomorphisms on all positive homotopy groups because both \(\xi \) and \(\widetilde \xi \) are universal. Since both animae are connected, \(g\) is an isomorphism by Proposition 2.4.22. Composing the restriction \(X\to \widetilde Z\) with an inverse of \(g\) gives a class in \([X,Z]_*\) which maps to \(\eta \).

For injectivity, suppose that \(f,g\colon X\to Z\) satisfy \(f^*\xi =g^*\xi =: \eta \). Form the pushout

where \(\nabla \) is the fold map. The Mayer–Vietoris property gives an element \(\widetilde \eta \in F(\widetilde X)\) restricting to \(\xi \) on \(Z\) and to \(\eta \) on \(X\). Apply Proposition 3.4.5 to \((\widetilde X,\widetilde \eta )\). As in the surjectivity argument, the resulting composite \(Z\to \widetilde X\to \widetilde Z\) is an isomorphism because it carries one universal element to another. The pushout square then shows, after composing with its inverse, that both \(f\) and \(g\) are homotopic to the same map \(X\to Z\). Hence \([f]=[g]\). □

Proof. Consider a pushout square of connected pointed animae

The pushout property gives an isomorphism \(\cofib (k)\iso \cofib (j)\). The long exact cohomology sequences of Exercise 3.1.3 therefore fit into a commutative diagram

Suppose that \(\alpha \in h^n(A)\) and \(\beta \in h^n(B)\) satisfy \(k^*\alpha =l^*\beta \). The boundary of \(\beta \) vanishes, since its image under the right-hand isomorphism is the boundary of \(l^*\beta =k^*\alpha \). Hence \(\beta \) lifts to some \(\gamma _0\in h^n(X)\). The element \(\alpha -i^*\gamma _0\) lies in the kernel of \(k^*\), so it comes from \(h^n(\cofib (k))\). Transporting a preimage across the left-hand isomorphism and then mapping it to \(h^n(X)\) gives an element \(\gamma _1\) whose restriction to \(B\) vanishes and whose restriction to \(A\) is \(\alpha -i^*\gamma _0\). Thus \(\gamma _0+\gamma _1\) restricts to \((\alpha ,\beta )\), proving the required surjectivity. □

Proof of Theorem 3.2.6. Let \((h^*,\sigma )\) be a reduced cohomology theory. For each \(m\geq 0\), the restriction of \(h^m\) to connected pointed animae satisfies the wedge axiom by definition and the Mayer–Vietoris property by Lemma 3.4.6. The abstract Brown representability theorem, Theorem 3.4.3, gives a connected pointed anima \(Y_m\) and a natural isomorphism \[ h^m(X)\cong [X,Y_m]_* \] on connected pointed animae.

Set \[ E_m:=\Omega Y_{m+1}\qquad (m\geq 0). \] Since \(\Sigma X\) is connected for every pointed anima \(X\), the suspension isomorphism gives natural isomorphisms \[ h^m(X)\cong h^{m+1}(\Sigma X)\cong [\Sigma X,Y_{m+1}]_*\cong [X,E_m]_* \] for every pointed anima \(X\). Thus \(E_m\) represents \(h^m\) on all of \(\An _*\). The same calculation shows that \(\Omega E_{m+1}\) also represents \(h^m\). The Yoneda lemma therefore supplies an isomorphism \[ E_m\iso \Omega E_{m+1} \] compatible with the given suspension isomorphism. These isomorphisms make \((E_m)_{m\geq 0}\) into a spectrum \(E\). The representing isomorphisms identify \(h^k\) with \(\widetilde E^k\) for \(k\geq 0\), and for \(k<0\) we obtain \[ \widetilde E^k(X) =[X,\Omega ^{-k}E_0]_* \cong h^0(\Sigma ^{-k}X) \cong h^k(X). \] This proves the first assertion.

Now let \(\alpha \colon \widetilde E^*\to \widetilde F^*\) be a natural transformation commuting with suspension. For each \(m\geq 0\), the degree-\(m\) component is a natural transformation \[ [-,E_m]_*\longrightarrow [-,F_m]_*, \] so Chapterexercise 3.1 identifies it with postcomposition by a unique homotopy class of pointed maps \(f_m\colon E_m\to F_m\). Compatibility with suspension says precisely that the squares

commute in \(\Ho (\An _*)\). Choosing representatives and the corresponding homotopies gives a morphism of spectra \(f\colon E\to F\) in the sense of Definition 3.2.2. Its induced natural transformation is \(\alpha \) in nonnegative degrees, and hence in every degree by compatibility with suspension. □