An example of an homology theory which fails to send weak equivalences to isomorphism

Question

As well known, singular homology sends weak homotpy equivalences into isomorphisms in homology. The lecture notes which I am reading mention that this fact is peculiar of singular homology and does not hold true in general for any homology theory (in the sense of Eilenberg and Steenrod). So I was wondering, which is an example of an homology theory which fails to satisfy this property?

Answer 1

If one is explicitly mentioning E.S. axioms, I would find it hard to argue that you can interpret the question so there are any examples. When one mentions the E.S. axioms one almost always either restricts to CW complexes, in which case weak homotopy equivalences are equivalent to homotopy equivalences, or one explicitly requires that weak homotopy equivalences are sent to isomorphisms.

Now there are certainly collections of groups that you can assign to a topological space that many would be happy to call a homology theory, but they do not satisfy all the E.S. axioms. In these cases, they might not send weak equivalences to isomorphisms. Switching to cohomology for the sake of an example: Cech cohomology does not send weak equivalences to equivalences, for example, the topologist's sine curve has nontrivial first cohomology when it has the weak homotopy type of two points.

Answer 2

The answer is not at all trivial because it requires a deeper knowledge about the many existing homology theories.

The Warsaw circle $C$ has the property that $p : C \to P$ = one-point space is a weak homotopy equivalence (since all $\pi_n(C) = 0$. Cech homology with integer coefficients has the property that $\check H_1(C) \approx \mathbb Z$, thus $p_* : \check H_1(C) \to \check H_1(P)$ is not an isomorphism. Unfortunately Cech homology with integer coefficients is not exact, thus it does not satisfy all Eilenberg-Steenrod-axioms.

However, there are variants of Cech homology with arbitrary coefficients, and if we take coefficients in a field $F$, then Cech homology becomes exact and you get an example (since $\check H_1(C) \approx F$).

If you prefer integer coefficients, you can take Steenrod homology.