cohomology

# Contents

## Definition

###### Definition

A reduced cohomology theory is a functor

$\tilde E^\bullet \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}$

from the opposite of pointed topological spaces (CW-complexes) to $\mathbb{Z}$-graded abelian groups (“cohomology groups”), in components

$\tilde E \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X)) \,,$

and equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form

$\sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)$

such that:

1. (homotopy invariance) If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopy $f_1 \simeq f_2$ between them, then the induced homomorphisms of abelian groups are equal

$f_1^\ast = f_2^\ast \,.$
2. (exactness) For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced mapping cone, then this gives an exact sequence of graded abelian groups

$\tilde E^\bullet(Cone(i)) \overset{j^\ast}{\longrightarrow} \tilde E^\bullet(X) \overset{i^\ast}{\longrightarrow} \tilde E^\bullet(A) \,.$

We say $\tilde E^\bullet$ is additive if in addition

• (wedge axiom) For $\{X_i\}_{i \in I}$ any set of pointed CW-complexes, then the canonical comparison morphism

$\tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i)$

is an isomorphism, from the functor applied to their wedge sum, example \ref{WedgeSumAsCoproduct}, to the product of its values on the wedge summands, .

We say $\tilde E^\bullet$ is ordinary if its value on the 0-sphere $S^0$ is concentrated in degree 0:

• (Dimension) $\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

A homomorphism of reduced cohomology theories

$\eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet$

is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute

$\array{ \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F^\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E^{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F^{\bullet + 1}(\Sigma X) } \,.$

(e.g. AGP 02, def. 12.1.4)

## Properties

### Brown representability

The Brown representability theorem says that for any reduced cohomology theory $\tilde E^\bullet$ there is an Omega-spectrum $E$ which represents $\tilde E^\bullet$ on pointed connected CW-complex $X$, in that

$\tilde E^n(X) \simeq [X,E_n]_\ast \,.$

### Relation to unreduced cohomology

For an unreduced cohomology theory $E^\bullet$ the induced reduced cohomology is

$\tilde E^k(X,x_0) \coloneqq E^k(X,\{x_0\}) = ker(H^k(X)\to H^k(\{x_0\}))$

For more see at generalized cohomology – Relation btween reduced and unreduced.

## References

See the references at generalized (Eilenberg-Steenrod) cohomology.

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

Revised on April 21, 2016 05:47:54 by Urs Schreiber (131.220.184.222)