Holmstrom Reduced cohomology

Reduced cohomology

The category $CW_*$ of pointed CW complexes.

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Reduced cohomology

See also Generalized cohomology

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Reduced cohomology

See Generalized cohomology for some notation.

A reduced cohomology theory is a sequence of functors $\tilde{h}^n: CW_* \to Ab$ together with natural transformations $\sigma^n: \tilde{h}^n \to \tilde{h}^{n+1} \circ \Sigma$, satisfying the following axioms:

• IV’ (Exactness): For any inclusion $A \to X$ in $CW_*$, the sequence: $\tilde{h}^n(X/A) \to \tilde{h}^n(X) \to \tilde{h}^n(A)$ is exact
• V’ (Homotopy): Homotopic morphisms in $CW_*$ induce the same map on cohomology
• VI’ (Suspension): For $X \in CW_*$, the map $\sigma^n(X)$ is an isomorphism for all $n$.

We can also consider an additivity axiom, saying that “the cohomology of a wedge product is the product of the individual cohomology groups”.

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Reduced cohomology

The reduced cohomology theories are in bijection with the generalized cohomology theories under composition with the functor from CW pairs to $CW_*$ sending $(X,A)$ to $(X/A, *)$.

If a map of pointed CW complexes induces an isomorphism on ordinary homology, then it also induces an isomorphism on any reduced cohomology theory. If the complexes involved are finite, the same holds for ordinary cohomology.

(Milnor exact sequence): For a theory satisfying additivity, we have, for any nested union $X = \bigcup X_i$, an exact sequence $0 \to \lim^1 \tilde{h}^{n-1}(X_i) \to \tilde{h}^{n}(X) \to \lim \tilde{h}^{n}(X_i) \to 0$

Roughly, the following is true: A cohomology theory on finite CW complexes is determined by its Coefficient group, and the same is true for a theory on CW complexes, provided it satisfies additivity.

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Reduced cohomology

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Reduced cohomology

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