6.4 Modular Operads
143
which can easily be converted into a diagram analogous to the one in Proposition 6.2.
Free modular operads admit a description parallel to free cyclic operads given in
Sect. 6.1. Let us firstly introduce some necessary notions.
Definition 6.21 A labeled graph is a couple Γ = (Γ, ,) consisting of a graph Γ as
in Definition 6.8 and a labeling : Vert(Γ ) → A. The genus g(Γ ) ∈ A of a labeled
graph is defined by the formula
g(Γ ) := s · b 1 (Γ ) +
v∈Vert(Γ )
(v),
where b 1 (Γ ) is the first Betti number of the geometric realization of Γ , i.e., the
number of independent circuits of Γ .
An isomorphism of φ
φ : (Γ 0 , , 0 ) → (Γ 1 , , 1 ) of labeled graphs is an isomorphism
φ : Γ 0
∼ =
−→ Γ 1 of the underlying graphs compatible with the labelings. The grafting
extends to labeled graphs in the straightforward manner, namely
Γ 1 a ◦ b Γ 2 = (Γ 1 a ◦ b Γ 2 ; 1 a ◦ b 2 ),
(6.65)
where Γ 1 a ◦ b Γ 2 is as in Definition 6.9 and
(( 1 a ◦ b 2 )| Vert(Γ i ) := i for i = 1, 2.
One clearly has
g(Γ 1 a ◦ b Γ 2 ) = g(Γ 1 ) + g(Γ 2 ).
Suppose that Γ = (Γ, ,) is a labeled graph with Leg(Γ ) = S {u, v}. We define
the contracted labeled graph ◦ uv Γ = (◦ uv Γ, ◦ uv ) as follows. The graph ◦ uv Γ has
the same set of flags and its partition as Γ , in particular, Vert(◦ uv Γ ) = Vert(Γ ). The
involution ◦ uv (σ ) of Flag(◦ uv Γ ) agrees with the involution of Γ on Flag(Γ )\{u, v}
while ◦ uv (σ )(u) = v. Informally, ◦ uv Γ is obtained from Γ by connecting the free
ends of the legs u and v, creating a loop, as expressed in the schematic picture
Γ
u
v