Holmstrom Generalized cohomology

Generalized cohomology

Kono and Tamaki: Generalized cohomology.

Generalized cohomology

A generalized cohomology theory is a sequence of functors satisfying axioms I to VI below.

Generalized cohomology

Any cohomology theory satisfying axioms I - VI above has the following three exact sequences.

We say that $(X; A, B)$ is a triple if $(X,A)$ and $(X,B)$ are CW pairs. We say that $(X;A;B)$ is a triad if $(X,A)$ and $(A, B)$ are CW pairs.

Exact sequence for triple:

$\ldots \to h^{n-1}(A,B) {}_{\to}^{\ \delta^{n-1}} h^n(X,A) {}_{\to}^{\ j^*} h^n(X,B) {}_{\to}^{\ i^*} h^n(A,B) \to \ldots$

Exact sequence for a triad:

$\ldots \to h^{n-1}(A,A \cap B) {}_{\to}^{\ \Delta} h^n(X,A \cup B) {}_{\to}^{\ j^*} h^n(X,B) {}_{\to}^{\ i^*} h^n(A, A \cap B) \to \ldots$

where $\Delta$ is the excision isomorphism followed by $\delta^{n-1}$ .

Mayer-Vietoris exact sequence for a triad:

$\ldots \to h^{n-1}(A \cap B) {}_{\to}^{\ \Delta} h^n(A \cup B) {}_{\to}^{\ \alpha} h^n(A) \oplus h^n(B) {}_{\to}^{\ \beta} h^n(A \cap B) \to \ldots$

where $\beta$ is the difference map, and where $\Delta: h^{n-1}(A \cap B) \to h^n(B, A \cap B) \to h^n(A \cup B, A) \to h^n(A \cup B)$ , where the middle map is excision.

category: Standard theorems

Generalized cohomology

We present here the Eilenberg-Steenrod axioms, following Kono-Tamaki. Here we consider CW-complexes only, it would probably be better to consider compactly generated spaces.

We consider the category $CWp$ of CW pairs, and the category $finCWp$ of finite CW pairs. A CW pair is a pair $(X,A)$ consisting of a CW complex $X$ and a subcomplex $A$ , which can be empty.

There is a covariant endofunctor $\rho$ on $CWp$ sending $(X, A)$ to $(A, \emptyset)$ .

We consider a sequence of contravariant functors $h^n : CWp \to Ab$ together with natural transformations $\delta^n: h^n \circ \rho \to h^{n+1}$ for $n \in \mathbb{Z}$ .

Axioms:

I, II, III: Functoriality of $h^n$ and naturality of $\delta^n$
IV (Exactness): For any $(X,A) \in CWp$ , the following sequence is exact:

 $\ldots \to h^{n-1}(A) \to h^n(X,A) \to h^n(X) \to h^n(A) \to h^{n+1}(X,A) \to \ldots$

V (Homotopy): Homotopic maps in $CWp$ give same map on cohomology
VI (Excision): Let $(X,A), (X,B) \in CWp$ . The inclusion $(A, A \cap B) \to (A \cup B, B)$ induces an isomorphism on each cohomology group.
VII (Dimension): $h^n(pt) = 0$ for all nonzero $n$ .
VIII (Additivity): “The cohomology of a disjoint union of spaces is the product of the cohomology of each space”

category: Properties

Generalized cohomology

The phrase “generalized cohomology” is usually used to refer to a cohomology theory for topological spaces which satisfies the (generalized) Eilenberg-Steenrod axioms. The first such theory to appear, except for “ordinary” (singular) cohomology was topological K-theory.