# Anderson duality

If $A$ is an abelian group, consider the functor from spectra to graded abelian groups

$\mathrm{Sp} \to \mathrm{Ab}_{*}, \quad X \mapsto \mathrm{Hom}_{\mathrm{Ab}_*} (\pi_{*}(X) ,\ A),$ where $A$ is considered to be in degree zero.

This is generally not a cohomology theory, because it does not satisfy exactness. If $A$ is injective, then this does satisfy exactness, and in fact defines a cohomology theory. By Brown representability it is represented by a spectrum $I_A$. If $X$ is any spectrum, we write $I_A X$ for the function spectrum $F(X, I_A)$. By definition of the spectrum $I_A$, the homotopy groups of $I_A X$ are simply $\mathop{\pi_*} I_A X = \mathop{\pi_*} F(X,\ I_A ) = (I_A)^{-*}(X) = \mathrm{Hom}_{\mathrm{Ab}_*} (\pi_{-*}(X) ,\ A).$

Examples of injective groups are $\mathbf{Q}$ and $\mathbf{Q}/\mathbf{Z}$, so we get spectra $I_{\mathbf{Q}}$ and $I_{\mathbf{Q}/\mathbf{Z}}$. Note that we also get a map $I_{\mathbf{Q}} \to I_{\mathbf{Q}/\mathbf{Z}}.$ The group $\mathbf{Z}$ is not injective, so we cannot apply the above construction to yield $I_{\mathbf{Z}}$. But we can instead define $I_{\mathbf{Z}}$ as the (homotopy) fibre of this map: $I_{\mathbf{Z}} := \mathrm{fib}(I_{\mathbf{Q}} \to I_{\mathbf{Q}/\mathbf{Z}}),$ essentially mirroring the algebraic short exact sequence $0\to\mathbf{Z} \to \mathbf{Q}\to\mathbf{Q}/\mathbf{Z}\to0.$

If $X$ is now a general spectrum, then the Anderson dual of $X$ is the function spectrum

$I_{\mathbf{Z}}X := F(X, I_{\mathbf{Z}}).$ This sits in a fibre sequence $I_{\mathbf{Z}}X \to I_{\mathbf{Q}}X \to I_{\mathbf{Q}/\mathbf{Z}}X.$

Remark. If $X$ is a spectrum, then $I_{\mathbf{Q}/\mathbf{Z}}X$ is called the Brown-Comenetz dual of $X$. Its $n$-th homotopy group is the Pontryagin dual of $\pi_{-n} X$.

## The short exact sequence

The above fibre sequence induces a long exact sequence on homotopy groups. Part of this sequence reads $\mathrm{Hom} (\pi_{-n-1}(X) ,\ \mathbf{Q}) \to \mathrm{Hom} (\pi_{-n-1}(X) ,\ \mathbf{Q}/\mathbf{Z}) \to \mathop{\pi_n} I_{\mathbf{Z}} X \to \mathrm{Hom} (\pi_{-n}(X) ,\ \mathbf{Q} ) \to \mathrm{Hom} (\pi_{-n}(X) ,\ \mathbf{Q}/\mathbf{Z} ).$ We can truncate this into a short exact sequence by working out what the cokernel of the left-most map, and the kernel of the right-most map is.

Let us begin with the kernel of the map on the right. We have a short exact sequence $0 \to \mathbf{Z} \to \mathbf{Q} \to \mathbf{Q}/\mathbf{Z} \to 0,$ and $\mathrm{Hom}(A,-)$ is left-exact for any abelian group $A$, so we get an exact sequence $0 \to \mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Z}) \to \mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}) \to \mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}/\mathbf{Z}).$ In other words, $\mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Z})$ is the kernel we were looking for.

For the map on the left, we use a similar argument. Namely, $\mathrm{Hom}(A,-)$ is not right-exact, but we do have an exact sequence $\mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}) \to \mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}/\mathbf{Z}) \to \mathrm{Ext}(\pi_{-n}(X),\ \mathbf{Z}) \to \mathrm{Ext}(\pi_{-n}(X),\ \mathbf{Q}).$ As $\mathbf{Q}$ is injective, this last Ext-group vanishes, leaving us with an exact sequence $\mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}) \to \mathrm{Hom}(\pi_{-n}(X),\ \mathbf{Q}/\mathbf{Z}) \to \mathrm{Ext}(\pi_{-n}(X),\ \mathbf{Z}) \to 0,$ i.e., the cokernel we were after is $\mathrm{Ext}(\pi_{-n}(X),\ \mathbf{Z})$.

Thus we have proved the following.

Proposition 1. If $X$ is a spectrum and $n \in \mathbf{Z}$, then there is a short exact sequence of the form $0 \to \mathrm{Ext} (\pi_{-n-1}(X) ,\ \mathbf{Z}) \to \mathop{\pi_n} I_{\mathbf{Z}} X \to \mathrm{Hom} (\pi_{-n}(X) ,\ \mathbf{Z} ) \to 0,$ and this sequence is natural in $X$.

Note that if $\pi_{-n} (X)$ is finitely generated, then the Hom-group on the right is a free abelian group (in particular, the sequence is split). If $\pi_{-n-1}(X)$ is finitely generated, then the Ext-group on the left is a torsion abelian group.

## Anderson dual of chains

Let $E$ and $X$ be spectra. Using the adjunction between the smash product and the mapping spectrum, we can compute the Anderson dual of the $E$-chains $E\otimes X$ on $X$: \begin{align*} I_\mathbf{Z} (E\otimes X) &= F(E\otimes X,\ I_\mathbf{Z})\\
&\simeq F(X, \ F(E, I_\mathbf{Z}))\\
&= F(X, \ I_\mathbf{Z}E). \end{align*} In particular, we find that the homotopy groups of $I_\mathbf{Z}(E\otimes X)$ are the $I_\mathbf{Z}E$-cohomology groups of $X$: $\pi_* (I_\mathbf{Z}(E\otimes X)) \cong (I_\mathbf{Z}E)^{-*}(X).$ Combining this with the short exact sequence from Proposition 1, we get a ‘universal coefficient theorem’ as follows.

Proposition 2. If $X$ and $E$ are spectra and $n\in \mathbf{Z}$, then there is a short exact sequence of the form $0 \to \mathrm{Ext} (E_{n-1}(X) ,\ \mathbf{Z}) \to (I_\mathbf{Z}E)^{n}(X) \to \mathrm{Hom} (E_{n}(X) ,\ \mathbf{Z} ) \to 0,$ and this sequence is natural in both $X$ and $E$.

For example, if $E=H\mathbf{Z}$, this recovers the usual universal coefficient theorem for $\mathbf{Z}$-cohomology. Indeed, Proposition 1 shows that the homotopy groups of $I_{\mathbf{Z}}H\mathbf{Z}$ are given by $\mathbf{Z}$ in degree 0 and vanish elsewhere, so the Anderson dual of $H\mathbf{Z}$ is (equivalent to) $H\mathbf{Z}$.

## Applications to K-theory

The Anderson duals of real and complex K-theory are also known.

Theorem. We have an equivalence $I_{\mathbf{Z}}KU \simeq KU.$

The proof is relatively simple, as outlined in [HS14, §2]. The version for real K-theory is much harder.

Theorem ([HS14], Thm. 8.1). We have an equivalence $I_{\mathbf{Z}}KO \simeq \Sigma^4 KO.$

Combined with Proposition 2, we get a universal coefficient theorem for real K-theory: $0 \to \mathrm{Ext} (KO_{n-1}(X) ,\ \mathbf{Z}) \to KO^{n-4}(X) \to \mathrm{Hom} (KO_{n}(X) ,\ \mathbf{Z} ) \to 0.$

Compare this to the usual universal coefficient spectral sequence for $KO$: $\operatorname{Ext}_{KO_*}^{*,*}( KO_*(X),\ KO^* ) \Rightarrow KO^*(X).$ The homotopy groups $KO_*$ have infinite homological dimension (see, e.g., page 8 of [Mei12] for a sketch), which makes this spectral sequence difficult to work with. By contrast, the universal coefficient theorem coming from Anderson duality is much more manageable.

Stojanoska also discusses the Anderson self-duality of $\mathrm{Tmf}$: see [Sto12].

## References

[HS14] Drew Heard and Vesna Stojanoska. K-theory, Reality, and Duality. May 2014. arXiv:1401.2581.

[Sto12] Vesna Stojanoska. Duality for Topological Modular Forms. June 2012. arXiv:1105.3968.

[Mei12] Lennart Meier. United Elliptic Homology. June 2012. PhD thesis. Available online.