144
6 Operads
The labeling ◦ uv is given as the composition Vert(◦ uv Γ ) = Vert(Γ )
→ N. Notice
that
g(◦ uv Γ ) = g(Γ ) + s.
Definition 6.22 We call the labeled graph ◦ uv Γ the contraction of Γ .
For a labeled graph Γ and a modular module E we consider an analog
E(Γ ) :=
v∈Vert(Γ )
E
Leg(v); (v)
(6.66)
of the vector space (6.33). Each graph isomorphism φ
φ : Γ 0
∼ =
−→ Γ 1 of labeled
graphs clearly induces an isomorphism
E(φ) : E(Γ 0 )
∼ =
−→ E(Γ 1 )
of the spaces (6.66). Mimicking (6.34), we define for a finite set S and a genus g ∈ A
F(E)(S; g) :=
G E(Γ )
∼
(6.67)
with the sum taken over all labeled graphs Γ = (Γ, ,) having Leg(Γ ) = S and
g(Γ ) = g. The relation ∼ identifies x ∈ E(Γ 0 ) with its image E(φ φ)(x) ∈ E(Γ 1 )
for any isomorphism φ
φ : Γ 0 → Γ 1 of labeled graphs that induces the identity map
of the set of the legs.
Proposition 6.9 The modular module F(E) =
F(E)(S; g) | S ∈ Cor, g ∈ A
is
a modular operad.
Proof. The actions (6.48) and compositions (6.51) are defined as in the proof of
Proposition 6.1; we leave the details for the reader. To define the ◦ uv -operations,
we recall the contraction of Definition 6.22 and notice the canonical isomorphism
E(Γ ) ∼ = E(◦ uv Γ ) which induces isomorphisms of the quotients
◦ uv : F(E)
S {u, v}; g
∼ = F(E)
S; g + s).
The axioms of modular operads are easy to verify.
It is simple to see that, if A = N and s = 1, the modular operad F(E) is
stable if and only if E(S; g) = 0 implies (S, g) ∈ S. The following analog of
Proposition 6.2 holds.