CALCULATION OF THE RATIONAL HOMOTOPY OF π_{∗}(Q(2))

MARK BEHRENS

In this section we derive one of the motivating properties of Q(2) (and probably Q(N) for other N). Namely, the the rational homotopy of Q(2) is concentrated in degrees −2, −1, and 0. This should be contrasted to the spectrum T MF and the fiber of the map

T MF ⟶^{ϕ_{q}^{∗}−ϕ_{f}^{∗}} T MF (2)

both of which have rational homotopy in infinitely many stems. The idea is that Q(2) is not only a large chunk of L_{K(2)}S^{0}, but also is similar to it in the sense that its homotopy is almost all torsion. An appropriate connective version of Q(2) ought to have the property that it is all torsion except in degree 0, like the sphere itself.

The rational story simplifies the situation in that, rationally, the stacks M^{1} and M(2)^{1} have cohomology concentrated in degree zero isomorphic to rational rings of (weakly) modular forms.

H^{0}(M_{ℂ}^{1}; O) = MF_{ℂ} = ℂ[c_{4}, c_{6}, Δ^{−1}]
H^{0}(M(2)_{ℂ}^{1}; O) = MF (2)_{ℂ} = ℂ[q_{2}, q_{4}, Δ^{−1}]

Thus, computing the rational homotopy of Q(2) amounts to computing the cohomology of the following chain complex C_{ℂ}(Q(2)):

MF_{ℂ} ⟶^{(ϕ_{f}^{∗}−ϕ_{q}^{∗})⊕(ψ_{[2]}^{∗}−1)} MF (2)_{ℂ} ⊕ MF_{ℂ} ⟶^{(ψ_{d}^{∗}+1)⊕−ϕ_{f}^{∗}} MF (2)_{ℂ}.

Our approach will be to filter this complex with a simple two-stage filtration. The cohomology of the associated graded will be computed in terms of modular functions. We will then compute the differentials between the two layers of the associated graded.

1. The rings of modular functions

Our computations will be made easier once we identify the weight 0 component of these rings. These are the rings of modular functions, and they are well understood. In MF, the element Δ of weight 12 is defined to be (c_{4}^{3} − c_{6}^{2})/1728. The numerator is irreducible, and thus one finds

MF_{0}(ℂ) = ℂ[j]

where the element j (the j-invariant) is given by

j = c_{4}^{3}/Δ.

On the other hand, in MF (2), the discriminant is given by Δ = q_{4}^{2}(16q_{2}^{2} − 64q_{4}). Thus inverting Δ is the same as inverting q_{4} and the element μ defined to be

μ = 16q_{2}^{2} − 64q_{4} = −64(ϵ + δ^{2}).

Then the level 2 modular functions are given by

MF (2)_{0} = ℂ[j_{2}, j_{2}^{−1}]

where the element j_{2} is given by

j_{2} = q_{4}/μ.

The appearance of Laurent polynomials reflects the fact that M(2) has two cusps, so it rationally looks like a twice punctured ℙ^{1}.

2. A filtration of the complex

We shall define a two stage filtration on C_{ℂ}^{∗}(Q(2)) and compute the cohomology of the associated graded complex E_{0}C_{ℂ}^{∗}(Q(2)). Define a filtration via:

F_{0}C^{∗} = C^{∗}
F_{1}C^{∗} = (MF_{ℂ} → MF_{ℂ} → 0 → ⋯)
F_{2}C^{∗} = 0

Thus the associated graded is given by

E_{0}^{∗,0}C^{∗} = (0 → MF (2)_{ℂ} ⟶^{ψ_{d}^{∗}+1} MF (2)_{ℂ} → 0 ⋯)
E_{0}^{∗,1}C^{∗} = (MF_{ℂ} ⟶^{ψ_{[2]}^{∗}−1} MF_{ℂ} → 0 → ⋯)
E_{0}^{∗,k}C^{∗} = 0 for k > 1

We first compute the cohomology of E_{0}^{∗,1}C^{∗}. If f is a weight k modular form, then the map ψ_{[2]}^{∗} − 1 is given by

ψ_{[2]}^{∗}f − f = (2^{k} − 1)f.

Thus rationally, ψ_{[2]}^{∗} − 1 is an isomorphism in non-zero weights. Thus the cohomology consists precisely of the weight 0 (weakly) modular forms, otherwise known as the modular functions. We therefore have

H^{0}(E_{0}^{∗,1}C^{∗}) = ℂ[j]
H^{1}(E_{0}^{∗,1}C^{∗}) = ℂ[j]
H^{s}(E_{0}^{∗,1}C^{∗}) = 0 for s > 1.

Computing H^{∗}(E_{0}^{∗,0}C^{∗}) is slightly more complicated. Recall that we have

MF (2)_{ℂ} = ℂ[q_{2}, q_{4}, q_{4}^{−1}, μ^{−1}]

where μ = 16q_{2}^{2} − 64q_{4}. Thus an additive basis of MF (2)_{ℂ} is given by

{q_{4}^{i}μ^{j}q_{2}^{ϵ} : i, j ∈ ℤ, ϵ = 0, 1}.

In Section ??, we computed ψ_{d}^{∗}, which gives us

(ψ_{d}^{∗} + 1)(q_{4}^{i}μ^{j}q_{2}^{ϵ}) = 2^{8i−4j} · (−2)^{ϵ}q_{4}^{j}μ^{i}q_{2}^{ϵ} + q_{4}^{i}μ^{j}q_{2}^{ϵ}.

Define rational subspaces V_{i,j,ϵ} < MF (2) for i, j ∈ ℤ and ϵ = 0, 1 by

V_{i,j,ϵ} = ℂ{q_{4}^{j}μ^{j}q_{2}^{ϵ}, q_{4}^{j}μ^{i}q_{2}^{ϵ}}.

Note that these subspaces are invariant under ψ_{d}^{∗} − 1.

CALCULATION OF THE RATIONAL HOMOTOPY OF π_{∗}(Q(2)) 3

Fixing i ≠ j and ϵ = 0 and restricting ψ_{d}^{∗} − 1 to V_{i,j,0}, we may write the matrix representation

ψ_{d}^{∗} − 1 = [ \begin{matrix}1 & 2^{8j−4i} \\ 2^{8i−4j} & 1 \end{matrix} ] .

This matrix is singular if and only if i = −j, in which case its kernel and cokernel both have dimension 1. The kernel is spanned by j_{2}^{−i} − 2^{12i}j_{2}^{i}. The cokernel is the subspace of MF (2)_{ℂ} spanned by {j_{2}^{i}, j_{2}^{−i}} subject to the linear relation j_{2}^{−i} ≡ 2^{12i}j_{2}^{i}. Thus it is spanned by \overline{j_{2}^{i}}, the image of j_{2}^{i} in the quotient.

If i ≠ j, and ϵ = 1, then the matrix representation of ψ_{d}^{∗} + 1 restricted to V_{i,j,1} is given by

ψ_{d}^{∗} − 1 = [ \begin{matrix}1 & −2^{8j−4i+1} \\ −2^{8i−4j+1} & 1 \end{matrix} ] .

This matrix is always non-singular.

If i = j, then V_{i,i,ϵ} is just 1-dimensional. The effect of ψ_{d}^{∗} + 1 on V_{i,i,ϵ} is multiplication by 2^{4i} · (−2)^{ϵ} + 1. This is never 0.

We conclude that the cohomology of the associated graded is given by

H^{0}(E_{0}^{∗,0}C^{∗}) = 0
H^{1}(E_{0}^{∗,0}C^{∗}) = ℂ{j_{2}^{−i} − 2^{12i}j_{2}^{i} : i > 0}
H^{2}(E_{0}^{∗,0}C^{∗}) = ℂ\{\overline{j_{2}^{i}} : i > 0\}
H^{s}(E_{0}^{∗,0}C^{∗}) = 0 for s > 2.

3. The spectral sequence

To compute the rational homotopy of Q(2), we are left computing the differentials in the spectral sequence

H^{s}(E_{0}^{k,t}C^{∗}) ⇒ π_{2k−s}(Q(2)_{ℂ}).

This is actually just a long exact sequence, since the spectral sequence has only two non-trivial rows. The differentials are given by

ϕ_{q}^{∗} − ϕ_{f}^{∗} :H^{0}(E_{0}^{∗,1}C^{∗}) → H^{1}(E_{0}^{∗,0}C^{∗})
−ϕ_{f}^{∗} :H^{1}(E_{0}^{∗,1}C^{∗}) → H^{2}(E_{0}^{∗,0}C^{∗})

Using the formulas derived in Sections ?? and ??, we compute

ϕ_{q}^{∗}(j^{k}) = (j_{2}^{−1} + 3 · 2^{8} + 3 · 2^{16}j_{2} + 2^{24}j_{2}^{2})^{k}
ϕ_{f}^{∗}(j^{k}) = (j_{2}^{−2} + 3 · 2^{4}j_{2}^{−1} + 3 · 2^{8} + 2^{12}j_{2})^{k}

Thus we have

(ϕ_{q}^{∗} − ϕ_{f}^{∗})(j^{k}) = j_{2}^{−2k} + 2^{12k}j_{2}^{2k} + A

where A is an element of the span of terms j_{2}^{−i} + 2^{12i}j_{2}^{i} for i < k. We conclude that the kernel of ϕ_{q}^{∗} − ϕ_{f}^{∗} is spanned by 1_{MF}, and the cokernel is spanned by the images of the basis elements

{j_{2}^{−i} + 2^{12i}j_{2}^{i} : i > 0, i odd}.

Our formula for ϕ_{f}^{∗} tells us that the other differential −ϕ_{f}^{∗} also has a kernel spanned by 1_{MF}, and the cokernel is spanned by the images of the elements

\{\overline{j_{2}^{i}} : i > 0, i odd\}.

4 MARK BEHRENS

Thus we have shown that the rational homotopy of Q(2) is given by

π_{−2}(Q(2)_{ℂ}) = ℂ\{\overline{j_{2}^{i}} : i > 0, i odd\}
π_{−1}(Q(2)_{ℂ}) = ℂ\{1_{MF} \} ⊕ ℂ\{\overline{j_{2}^{−i} + 2^{12i}j_{2}^{i}} : i > 0, i odd\}
π_{0}(Q(2)_{ℂ}) = ℂ{1_{MF} }
π_{k}(Q(2)_{ℂ}) = 0 for all other k

The subspaces in the 0 and −1 stems spanned by 1_{MF} correspond to the torsion-free elements 1 and ζ in π_{0}(L_{K(2)}S^{0}) and π_{−1}(L_{K(2)}S^{0}), respectively.