Cohomology

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1.2 Cohomology

Definition 1.19. [Rud98] A reduced generalized cohomology is a contravariant functor \(\HH ^*: \Spc _*^{op}\to \mathrm {grAb}\) from the homotopy category of spaces to the category of \(\Z \)-graded abelian groups, such that

(Additive) For every set of pointed spaces \(X_\alpha \) for \(\alpha \in A\), the canonical homomorphism \(\HH ^*(\bigvee _{\alpha \in A}X_\alpha )\to \prod _{\alpha \in A} \HH ^*(X_\alpha )\) is an isomorphism.
(Mayer-Vietoris) For every homotopy cofiber sequence in pointed space \(X\to Y\to Z\), there is long exact sequence \[ ...\to \HH ^{n-1}(X)\xrightarrow {\delta _{n-1}} \HH ^n(Z)\to \HH ^n(Y) \to \HH ^n(X)\xrightarrow {\delta _n} HH^{n+1}(Z)\to ... \]

A reduced generalized homology is a covariant functor \(\HH _*: \Spc _*\to \mathrm {grAb}\) from the homotopy category of spaces to the category of \(\Z \)-graded abelian groups, such that

(Additive) For every set of pointed spaces \(X_\alpha \) for \(\alpha \in A\), the canonical homomorphism \(\coprod _{\alpha \in A} \HH _*(X_\alpha )\to \HH _*(\bigvee _{\alpha \in A}X_\alpha )\) is an isomorphism.
(Mayer-Vietoris) For every homotopy cofiber sequence in pointed space \(X\to Y\to Z\), there is long exact sequence \[ ...\to \HH ^{n+1}(Z)\xrightarrow {\delta _{n}} \HH _n(X)\to \HH _n(Y) \to \HH _n(Z)\xrightarrow {\delta _{n-1}} HH_{n-1}(X)\to ... \]

Remark 1.20. Given a spectrum \(E\), the functor \(E^n(X):=[\Sigma ^{-n}\Sigma ^\infty X, E]\) (where \(\Sigma ^{-n}=\Omega ^n\) for \(n>0\)), is a cohomology.

The additive property follows from \[ [\Sigma ^{-n}\Sigma ^\infty (\bigvee _{\alpha \in A}X_\alpha ), E]\cong [\bigvee _{\alpha \in A}\Sigma ^{-n}\Sigma ^\infty (X_\alpha ), E]\cong \prod _{\alpha \in A}[\Sigma ^{-n}\Sigma ^\infty (X_\alpha ), E]] \]

The Mayer-Vietoris property follows from that a homotopy cofiber sequence is taken to homotopy cofiber sequence \[ ...\to \Sigma ^{-(n+1)}\Sigma ^infty Z\to \Sigma ^{-n}\Sigma ^infty X\to \Sigma ^{-n}\Sigma ^infty Y\to \Sigma ^{-n}\Sigma ^infty Z\to \Sigma ^{-(n-1)}\Sigma ^infty X\to ... \] Then there is long exact sequence \[ ...\to E^{n-1}(X)=[\Sigma ^{-(n-1)}\Sigma ^infty X, E]\to E^n(Z)=[\Sigma ^{-n}\Sigma ^infty Z, E]\to E^n(Y)\to E^n(Z)\to E^{n-1}(X)\to ... \]

Equivalently, the functor \(E_n(X):=[\Sigma ^n, \Sigma ^\infty X \wedge E]\) is a generalized homology.