Holmstrom Classical Brown representability

http://golem.ph.utexas.edu/category/2012/08/brown_representability.html

http://mathoverflow.net/questions/104866/brown-representability-for-non-connected-spaces

http://mathoverflow.net/questions/1346/representablity-of-cohomology-ring

http://mathoverflow.net/questions/1438/why-is-homology-not-corepresentable

We follow Kono and Tamaki.

We say that a functor $CW_* \to Set$ is a homotopy functor if $H(f)=H(g)$ whenever $f$ is homotopic to $g$. I guess this is the same as a functor on the homotopy category.

A contravariant representable homotopy functor must satisfy the Wedge axiom and a Mayer-Vietoris property: For two subspaces $A, B$ of a space $X$, and two elements $a \in H(A), b \in H(B)$, there exists an element $c \in H(A \cup B)$ which “restricts” to $a$ and to $b$.

A homotopy functor satisfying these two properties is called a Brown functor. (We also require $H(pt) \neq \emptyset$)

Thm: A Brown functor takes values in $Set_*$. Moreover, $H(\Sigma X)$ is a group and $H(\Sigma^k X)$ is an abelian group for $k \geq 2$.

Thm: Any Brown functor is representable. Similar statement for finite CW complexes, under some additional hypothesis.

nLab page on Classical Brown representability