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Foundational uniqueness and representability results on (co)homology theories in algebraic topology frequently make a point of assuming additivity, indicating at least some people think it's worthwhile considering theories not satisfying the wedge axiom despite the inapplicability of familiar results.

  1. What are some natural examples of such theories (and the failure of additivity)?
  2. What are some stupid examples?
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2 Answers 2

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Here's an example. (Rather than calling it a stupid example, I'll call it an example for which I know no applications.)

Let $A$ be a torsion-free abelian group and $B$ be an injective abelian group. Then we can define new versions of cohomology and homology: $$ \begin{align*} F^n(X,U) &= H^n(X,U) \otimes A \\ G_n(X,U) &= Hom(H^n(X,U), B) \end{align*} $$ Since $(-) \otimes A$ and $Hom(-,B)$ are functors that preserve exactness, all of the Eilenberg-Steenrod axioms except additivity automatically follow. You get a new "non-additive" cohomology or homology theory with $F^0(pt) = A$ and $G_0(pt) = B$.

For example, if $A$ is infinitely generated free abelian then the wedge axiom fails for $F^*$ because, for example, the first cohomology of $\bigvee S^1$ is $(\prod \Bbb Z) \otimes A$ instead of $\prod A$.

A more important example is typically furnished by $p$-adically completed homology (and similar theories, such as Morava $E$-theories). There, the homology theory should be viewed as taking values in a category of "complete" groups rather than the category of groups. A discussion of this appears in Appendix A of this paper of Barthel-Frankland.

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    $\begingroup$ Could there be a multiplicative example of this kind?? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 28, 2017 at 6:52
  • $\begingroup$ I second the request for multiplicative examples. Your first example is easier to understand, but the second one comes closer to the "naturality" request of my first question. But, given that I'm (sorry) completely new to these theories, why does one do this? $\endgroup$
    – jdc
    Commented May 28, 2017 at 22:57
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    $\begingroup$ If $A$ is a commutative ring then the first example is multiplicative, isn't it? For example, $A$ could be the ring of polynomials. $\endgroup$
    – Gregory Arone
    Commented May 29, 2017 at 6:40
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    $\begingroup$ One way to produce non-finitary homology theories is by the formula $h_*(X)=[E, X]_*$, where $E$ is an infinite CW-spectrum. This satisfies all the axioms for homology theory except the limit axiom. Another example is given by Tate homology, aka the homotopy cofiber of the norm map $(X^{\wedge 2})_{h\Sigma_2}\to ({X^{\wedge 2}})^{h\Sigma_2}$. $\endgroup$
    – Gregory Arone
    Commented May 29, 2017 at 6:44
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    $\begingroup$ @jdc One reason to do this is due to Sullivan. Sullivan studied spaces by decomposing them into rational information and p-complete information for each prime p, and then described a gluing procedure for putting this information back together (an "arithmetic square"). You really do need the p-completed version for this to work, because using homology with coefficients in the p-completed integers doesn't really give the right answer. $\endgroup$
    – Tyler Lawson
    Commented May 30, 2017 at 20:58
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The following example of a non-additive homology theory is due to James and Whitehead, in "Homology with zero Coeffients". It appears as an example in Rudyak's book "On Thom Spectra, Orientability, and Cobordism", pg 65.

Define

$\tilde{h}(X)=\frac{\prod_{n=0}^{\infty} \tilde{H}_n(X)}{\sum_{n=0}^{\infty} \tilde{H}_n(X)}$

where $\tilde{H_*}$ is (reduced) ordinary homology. In particular $\tilde{h}(S^n)=0$ for each $n$ but $\tilde{h}(\vee_{n=0}^\infty\; S^n)\neq 0$. James and Whitehead's paper also contains other examples which may be of interest.

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