The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.


However, could you help me to appreciate the mathematical meanings of the following:


1. What is the significance of the meanings of "Anderson dual" to the sphere spectrum? (For example, I understand the significant meanings of Pontryagin dual and torsion subgroup. Can "Anderson dual" be explained as simple as that of Pontryagin dual?)
2. What is the significance of the meanings of (a shift of) $I\mathbb{C}^{\times}$, a “character dual” to the sphere spectrum?
3. (A shift of) the Anderson dual $I\mathbb{Z}\_1$ to the sphere spectrum?


Please feel free to correct my statements and to manifest the meaning behind my question. Many thanks!



Sphere spectrum, Character dual and Anderson dual
The Anderson dualizing spectrum $I\_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi\_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded abelian groups. Since $\mathbf{Q/Z}$ is an injective $\mathbf{Z}$-module, this functor is representable by a spectrum $I\_\mathbf{Q/Z}$, called the Brown-Comenetz dualizing spectrum. After $p$-completion, this spectrum is equivalent to the spectrum you denoted $I\mathbf{C}^\times$. If you do the same construction with $\mathbf{Q/Z}$ replaced by $\mathbf{Q}$, you obtain the Eilenberg-Maclane spectrum $\mathbf{Q}$. There is a canonical map $\mathbf{Q}\to I\_\mathbf{Q/Z}$, and the fiber of this map is defined to be $I\_\mathbf{Z}$. This has the property that it is $(-1)$-coconnective, $\pi\_0 I\_\mathbf{Z} \cong \mathbf{Z}$, $\pi\_{-1} I\_\mathbf{Z} = 0$, and $\pi\_{-n} I\_\mathbf{Z} = \mathrm{Hom}(\pi\_{1-n} S, \mathbf{Q/Z})$.


There are several reasons why the Anderson dualizing spectrum is significant. I'll try to explain some of them, in the order in which I understand them the best.


* It is the dualizing sheaf of the affine derived scheme $\mathrm{Spec}(S)$. In order for this to make sense, I have to tell you what "dualizing sheaf" means. Let $A$ be a *connective* $\mathbf{E}\_\infty$-ring (this definition fails drastically in the nonconnective setting). Then, an $A$-module $\omega\_A$ is said to be a dualizing sheaf if:


	1. $\omega\_A$ is coconnective,
	2. $\pi\_n \omega\_A$ is a finitely generated $\pi\_0 A$-module for every $n$,
	3. the map $A\to \mathrm{Map}\_A(\omega\_A, \omega\_A)$ is an equivalence (this is the dualizing property), and
	4. $\omega\_A$ has finite injective dimension as an $A$-module, i.e., there is some integer $n$ such that for every discrete $A$-module $M$, the groups $\pi\_k\mathrm{Map}\_A(M,\omega\_A)$ vanish for $k>n$.You can check that $I\_\mathbf{Z}$ satisfies all of these properties when $A = S$. Moreover, one can prove (see Proposition 6.6.2.1 of Lurie's SAG) that dualizing sheaves are unique up to elements of $\mathrm{Pic}(A)$; when $A = S$, this group is just $\mathbf{Z}$, generated by $S^1$. It follows that any other dualizing sheaf for $S$ is equivalent to a suspension of $I\_\mathbf{Z}$.


There is an analogue of Serre duality in this spectral setting. Let me first explain the even periodic case, and then discuss what happens in the $K(n)$-local setting. In the even periodic setting, one might consider objects like $KU$, $KO$, and $\mathrm{TMF}$ and its variants. All of these can be obtained as the global sections of sheaves of $\mathbf{E}\_\infty$-rings on Deligne-Mumford stacks ($\mathrm{Spec}(\mathbf{Z})$, $\mathrm{Spec}/\!\!/C\_2$, and the moduli stack $\mathcal{M}\_{\mathrm{ell}}$ of elliptic curves, respectively). Each of these Deligne-Mumford stacks are --- in the classical world --- smooth. In particular, they are [Gorenstein](https://stacks.math.columbia.edu/tag/0AWV). 


One can actually show that the associated derived stacks are Gorenstein in the spectral world, too. (The cases mentioned above are due to Heard-Stojanoska, and Stojanoska, respectively.) One finds the extremely interesting statement, e.g., that $I\_\mathbf{Z} \mathrm{TMF} := \mathrm{Map}(\mathrm{TMF}, I\_\mathbf{Z})$ is in the Picard group of $\mathrm{TMF}$ (which is known to be cyclic, by work of Mathew-Stojanoska). This can be generalized: one can prove that most reasonable derived locally even periodic Deligne-Mumford stacks $\mathfrak{X}$ satisfy the property that $\mathrm{Map}(\Gamma(\mathfrak{X}, \mathcal{O}\_\mathfrak{X}), I\_\mathbf{Z})$ is in the Picard group of $\Gamma(\mathfrak{X}, \mathcal{O}\_\mathfrak{X})$. I have a proof of the latter fact [here](https://arxiv.org/abs/1711.04806), as Theorem 3.14; I'm sure experts already know of this result.


One practical consequence of this is the following. For a spectrum $A$, define $I\_\mathbf{Z} A = \mathrm{Map}(A,I\_\mathbf{Z})$. Then there is a short exact sequence
$$0\to \mathrm{Ext}^1\_\mathbf{Z}(\pi\_{-d-1} X, \mathbf{Z}) \to \pi\_d I\_\mathbf{Z} X \to \mathrm{Hom}\_\mathbf{Z}(\pi\_{-d} X, \mathbf{Z}) \to 0.$$
Now, suppose that $E$ is a spectrum such that $I\_\mathbf{Z} E \simeq \Sigma^k E$ for some $k$ (so that $I\_\mathbf{Z} E \in \mathrm{Pic}(E)$). Suppose we want to understand the $E$-theory of a spectrum $X$. Apply the short exact sequence above to the spectrum $E\wedge X$; this gives a short exact sequence
$$0\to \mathrm{Ext}^1\_\mathbf{Z}(E\_{-d-1} X, \mathbf{Z}) \to (I\_\mathbf{Z} E)^d X \to \mathrm{Hom}\_\mathbf{Z}(E\_{-d} X, \mathbf{Z}) \to 0.$$
But the middle term is, by the "Anderson self-duality" of $E$, equivalent to $E^{d+k} X$. This gives an interesting and useful "universal coefficients" exact sequence.
* Another reason one might care about the spectrum $I\_\mathbf{Z}$ is because of its appearance in the Picard group of the $K(n)$-local category, which goes by the name Gross-Hopkins duality. Let $n>0$. After $K(n)$-localizing, the difference between $I\_\mathbf{Q/Z}$ and $I\_\mathbf{Z}$ vanishes: $L\_{K(n)} I\_\mathbf{Q/Z} \simeq \Sigma L\_{K(n)} I\_\mathbf{Z}$. Suppose $X$ is an $E\_n$-local spectrum. We define another spectrum $I\_n X = \Sigma L\_{K(n)} I\_\mathbf{Z} X \simeq L\_{K(n)} \mathrm{Map}(L\_n X, I\_{\mathbf{Q/Z}})$, and let $I\_n = I\_n L\_{K(n)} S$. Then, Gross and Hopkins proved two remarkable statements: first, the spectrum $I\_n$ is invertible in the $K(n)$-local category (this is not very surprising), and second, there is an isomorphism $(E\_n)^\vee\_\ast I\_n \simeq \Sigma^{n^2-n} (E\_n)\_\ast[\det]$ of $(E\_n)\_\ast[\![\Gamma\_n]\!]$-modules, where $\det:\Gamma\_n \to \mathbf{Z}\_p^\times$ is the determinant map, and $\Gamma\_n$ is the Morava stabilizer group at height $n$. (I've implicitly been fixing the Honda formal group over $\mathbf{F}\_{p^n}$.)


The proof of this result is really cool; it passes through an important object in rigid analytic geometry (the Gross-Hopkins period map). In any case, if $p\gg n$, then one can actually identify $I\_n$ itself with a $K(n)$-local object $S[\det]$, called the determinantal sphere. This object is incredibly interesting, and plays an important role in the chromatic story at height $n$.
* The Anderson dualizing spectrum also plays an important (but unpublished) role in orientation theory. The following simple question is an open problem: let $\kappa$ be a perfect field of characteristic $p>0$, and let $H$ be a formal group of finite height over $\kappa$. Denote by $E(\kappa, H)$ the associated Morava $E$-theory. Then, is there an $\mathbf{E}\_\infty$-orientation $MU \to E(\kappa,H)$? One can reduce this to understanding the spectrum of units $gl\_1 E(\kappa,H)$. In unpublished work, Hopkins and Lurie have shown that the fiber of the map $gl\_1 E(\kappa, H) \to L\_n gl\_1 E(\kappa, H)$ (which is known as the discrepancy spectrum) is equivalent to $\Sigma^n I\_\mathbf{Q/Z}$ in nonnegative degrees (they also did more). This allows one to construct a map $\Sigma^{n+1} H\mathbf{Z} \to gl\_1 E(\kappa, H)$, the very existence of which is enough to obstruct the existence of certain $\mathbf{E}\_\infty$-orientations of $E(\kappa, H)$.
* I have heard that there is some relationship with TQFTs, but I don't know how that goes. There are people here more qualified than me who can say more about this.


