Homology and cohomology theories

Convention 3.1.1 (Graded abelian groups). A graded abelian group is a sequence \(A_* = (A_n)_{n \in \Z }\) of abelian groups indexed by the integers. These form a 1-category \(\Ab ^{\Z }\) whose morphisms \(f_*\colon A_* \to B_*\) consist of homomorphisms \(f_n\colon A_n \to B_n\) for each \(n \in \Z \).

Definition 3.1.2 (Homology and cohomology theories). A reduced homology theory is a pair \((\widetilde E_*, \sigma )\) consisting of a functor \[ \widetilde E_*\colon \An _* \to \Ab ^{\Z } \] and a natural isomorphism \[ \sigma \colon \widetilde E_*(-) \iso \widetilde E_{*+1}(\Sigma (-)), \] called the suspension isomorphism, satisfying:

(1)

(Wedge axiom) For every collection \((X_i)_{i \in I}\) of pointed animae, the canonical map \[ \bigoplus _{i \in I}\widetilde E_*(X_i) \to \widetilde E_*\Big (\bigvee _{i \in I}X_i\Big ) \] is an isomorphism.

(2)

(Exactness) For every cofiber sequence \(X \xrightarrow {f} Y \xrightarrow {g} Z\) in \(\An _*\), the sequence of abelian groups \[ \widetilde E_*(X) \xrightarrow {f_*} \widetilde E_*(Y) \xrightarrow {g_*} \widetilde E_*(Z) \] is exact in each degree.

Dually, a reduced cohomology theory is a pair \((\widetilde E^*, \sigma )\) consisting of a functor \(\widetilde E^*\colon \An _*\catop \to \Ab ^{\Z }\) and a natural isomorphism \(\sigma \colon \widetilde E^*(-) \iso \widetilde E^{*+1}(\Sigma (-))\) satisfying:

(1)

(Wedge axiom) For every collection \((X_i)_{i \in I}\) of pointed animae, the canonical map \(\widetilde E^*\Big (\bigvee _{i \in I}X_i\Big ) \to \prod _{i \in I}\widetilde E^*(X_i)\) is an isomorphism.

(2)

(Exactness) For every cofiber sequence \(X \xrightarrow {f} Y \xrightarrow {g} Z\) in \(\An _*\), the sequence \(\widetilde E^*(Z) \xrightarrow {g^*} \widetilde E^*(Y) \xrightarrow {f^*} \widetilde E^*(X)\) is exact in each degree.

Exercise 3.1.3 (Long exact sequences of homology theories). Use the coPuppe sequence of Lemma 2.4.28 to show that every homology theory gives rise to a long exact sequence \[ \cdots \to \widetilde E_{n+1}(Z) \xrightarrow {\partial } \widetilde E_n(X) \xrightarrow {f_*} \widetilde E_n(Y) \xrightarrow {g_*} \widetilde E_n(Z) \xrightarrow {\partial } \widetilde E_{n-1}(X) \to \cdots . \] Formulate and prove the analogous statement for cohomology theories.

Exercise 3.1.4 (Long exact sequence of a pair). Let \(i\colon A \to X\) be a map of animae, and let \(j\colon X_+ \to X/A\) denote the cofiber map. Show that every homology theory gives rise to a natural long exact sequence \[ \cdots \to E_n(A) \xrightarrow {i_*} E_n(X) \xrightarrow {j_*} E_n(X,A) \xrightarrow {\partial } E_{n-1}(A) \xrightarrow {i_*} E_{n-1}(X) \to \cdots . \] Formulate and prove the analogous long exact sequence for cohomology: \[ \cdots \to E^n(X,A) \xrightarrow {j^*} E^n(X) \xrightarrow {i^*} E^n(A) \xrightarrow {\partial } E^{n+1}(X,A) \xrightarrow {j^*} E^{n+1}(X) \to \cdots . \]

Exercise 3.1.5. Show that precomposition with \(\Pi _{\infty }\colon \Top _* \to \An _*\) establishes a bijection between reduced cohomology theories in the sense of Definition 3.1.2 and pairs \((E^*,\sigma )\) consisting of a functor \(E^*\colon \Top _*\catop \to \Ab ^{\Z }\) and a natural isomorphism \(\sigma \colon E^*(-) \iso E^{*+1}(\Sigma (-))\) satisfying:

(1)

Weak homotopy equivalences induce isomorphisms on \(E^*\).

(2)

The wedge axiom holds for wedges of pointed CW-complexes whose basepoints are \(0\)-cells.

(3)

For every relative CW-pair \((X,A)\), the sequence \(E^*(X/A) \to E^*(X) \to E^*(A)\) is exact.

Formulate the analogous characterization of homology theories.

Hint: Use Theorem 2.3.4 to reduce to CW-complexes, and Proposition 2.4.12 to transport the relevant constructions.