It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some conditions, meaning the moduli space $Z$ of all such lifts is connected. This is mentioned in Theorem 1.1 of Lurie's "A Survey of Elliptic Cohomology" [Surv], for example.

In Remark 7.0.2 of Lurie's "Elliptic Cohomology II: Orientations" ([ECII]), Lurie says "...beware, however, that $Z$ is not contractible". In other words, $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is not the unique lift up to contractible choice (the gold standard of uniqueness in homotopy theory).

(Side note: in [ECII] and [Surv], Lurie is talking about the moduli stack of smooth elliptic curves, but the uniqueness up to homotopy of a derived stack $\overline{\mathsf{M}}_\mathrm{ell}^\mathrm{or}$ lifting the compactification of the moduli stack of smooth elliptic curves is also stated in the literature; for example, in Theorem 1.2 of Goerss' "Topological Modular Forms [after Hopkins, Miller, and Lurie]". I am interested in the compactified situation mostly, but both are related.)

Although I do not hope that $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ does possess this much stronger form of uniqueness, I would like to understand the reason for this failure:

Why is the moduli space $Z$ not contractible? and Does a similar statement apply in the compactified case?

To be a little more precise, let $\mathcal{O}^\mathrm{top}$ be the Goerss--Hopkins--Miller--Lurie sheaf of $\mathbf{E}_\infty$-rings on the small affine site of the moduli stack of elliptic curves $\mathsf{M}_\mathrm{ell}$. Denote this site by $\mathcal{U}$. The moduli space $Z$ can then be defined as the (homotopy) fibre product $$Z=\mathrm{Fun}(\mathcal{U}^{op}, \mathrm{CAlg})\underset{\mathrm{Fun}(\mathcal{U}^{op}, \mathrm{CAlg}(\mathrm{hSp}))}{\times}\{\mathrm{h}\mathcal{O}^\mathrm{top}\},$$ where $\mathrm{CAlg}$ is the $\infty$-category of $\mathbf{E}_\infty$-rings, and $\mathrm{CAlg(hSp)}$ is the 1-category of commutative monoid objects in the stable homotopy category. The presheaf $\mathrm{h}\mathcal{O}^\mathrm{top}$ can be defined using the Landweber exact functor theorem (at least on elliptic curves whose formal group admits a coordinate), and hence $Z$ can be seen as the moduli space of presheaves of $\mathbf{E}_\infty$-rings recognising the classical Landweber exact elliptic cohomology theories.

To prove uniqueness up to homotopy, I am aware one should use some arithmetic and chromatic fracture squares to break down the problem into rational, $p$-complete, $K(1)$-local, and $K(2)$-local parts. The $K(2)$-local part of $\mathcal{O}^\mathrm{top}$ is unique up to contractible choice by the Goerss--Hopkins--Miller theorems surrounding Lubin--Tate spectra (see Chapter 5 of [ECII] for a reference which you might already have open). The $K(1)$-local part also seems to be unique up to contratible choice, as all of the groups occuring in the Goerss--Hopkins obstruction theory vanish (this is discussed at length in Behrens' "The construction of $tmf$" chapter in the "TMF book" by Douglas et al). Similarly, the rational case also has vanishing obstruction groups; see ibid.

Edit: As pointed out by Tyler below, these claims about the function of Goerss--Hopkins obstruction theory above are wrong!

I'm then lead to believe that is something interesting (being a pseudonym for "I don't know what's") going on in the chromatic/arithmetic fracture squares gluing all this stuff together. Are their calculable obstructions/invariants to see this? Or otherwise known examples that contradict the contractibility of $Z$?

Any thoughts or suggestions are appreciated!


1 Answer 1


So the issue is with this:

all of the groups occuring in the Goerss--Hopkins obstruction theory vanish

In "generic terms", for the obstruction theory that you're running in either the $K(1)$-local case or the rational case you care about some bigraded obstruction groups $$ \mathfrak{E}xt^{s,t}(R;S) $$ where these are some nonabelian Ext-groups occuring in some algebraic category ($K(1)$-locally it is Andre-Quillen cohomology for derived $p$-complete $\psi$-$\theta$-algebras, and rationally it is Andre-Quillen cohomology for graded-commutative algebras over $\Bbb Q$, maybe relative to some fixed base algebra, maybe over some other algebra, maybe on a category of diagrams with some sheaf condition, etc etc etc).

The obstructions to existence of some object typically occur when $t-s = -1$, and to uniqueness (up to homotopy equivalence) occur when $t-s = 0$. These are the groups that vanish when constructing $Tmf$ by obstruction theory. Those two groups, however, tell you only about path components. In general, there is some obstruction-theoretic yoga that tells you $\mathfrak{E}xt^{s,t}(R;S)$ is part of some kind of fringed spectral sequence, computing $\pi_{t-s} \mathcal{M}$ of some moduli space of realizations. As a result, for contractibility of this moduli space we would need to know about what happens to $\mathfrak{E}xt^{s,t}$ for $t-s > 0$, and those groups don't vanish.

(Aside, the Goerss--Hopkins--Miller theorem about Lubin-Tate theories is so strong precisely because, in that situation, all of these higher groups vanish, and so these moduli spaces are homotopy discrete.)

The rational case makes it easiest to see this. For the rational case, roughly (ignoring the sheaf stuff), we have constructed $p$-complete $Tmf_p$ and we need to construct a map $H\Bbb Q[c_4, c_6] \to (Tmf_p)_{\Bbb Q}$ to the rationalization that has a particular effect on $\pi_*$. That map is unique up to homotopy. However, the space of maps $H\Bbb Q[c_4, c_6] \to (Tmf_p)_{\Bbb Q}$ does not have contractible components: $H\Bbb Q[c_4, c_6]$ is a free rational commutative algebra on generators in degrees $8$ and $12$, and so the space of maps to $(Tmf_p)_{\Bbb Q}$ is homotopy equivalent to $$ Map(S^8, (Tmf_p)_{\Bbb Q}) \times Map(S^{12},(Tmf_p)_{\Bbb Q}). $$ Its higher homotopy give many contributions from the rational higher homotopy of $Tmf_p$. In this case, freeness actually gives us vanishing of Ext groups when $s > 0$; it's $t$ that gives us trouble.

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    $\begingroup$ Thank you Tyler, for (a) being crystal clear, and (b) reminding me how the GHM theorem is so damn interesting! (I'll edit the question to not be so misleading about GH obstruction theory too) $\endgroup$ Commented Jun 7, 2021 at 16:09

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