TITLE: Is there the Whitehead theorem for cohomology theory?
QUESTION [12 upvotes]: We know that there are Whitehead theorem for homotopy and homology theory.
Is there the Whitehead theorem for cohomology theory for 1-connected CW complexes?

REPLY [18 votes]: Conceptually, the following two theorems (both due to Whitehead) are Eckmann-Hilton duals.
Theorem.  A weak homotopy equivalence between CW complexes is a homotopy equivalence.
Theorem.  A homology isomorphism between simple spaces is a weak homotopy equivalence.
They don't look dual, but they are.  See
J.P. May. 
The dual Whitehead Theorems.
London Math. Soc.  Lecture Note Series 
Vol. 86(1983), 46--54.
The point is that the second statement is really about cohomology, and the standard
cellular proof of the first statement dualizes word-for-word to a ``cocellular'' proof
of the second.  Cocellular constructions are what appear in Postnikov towers, and they
can be used more systematically than can be found in the literature.  Yet another plug:
they are central in the upcoming book "More Concise Algebraic Topology'' by Kate Ponto
and myself.