Lecture 4

Graph of Spaces.

Definition 1. A graph of spaces X is a topological space with a structure dictated by a graph Γ and certain data.

1) There is a topological space X_{v} for each vertex v in Γ. X_{v} is called a vertex space of X.

2) There is a topological space X_{e} for each edge e of Γ. X_{e} or X_{e}×I is edge space of X. (I = [−1, 1])

3)For each edge e attached to vertices i(e), τ(e) there are continuous (“attaching”) maps Φ_{e−} : X_{e} → X_{i(e)}, Φ_{e+} : X_{e} → X_{τ(e)}.

From this data we can build the space X with quotient topology.

X = ( ⨆_{vertices ; v} X) ⊔ ( ⨆_{\begin{matrix}edges \\ e \end{matrix}} X_{e} × I)/ \begin{matrix}Φ_{e−}(a) ∼ (a, −1) ∀a ∈ X_{e} \\ Φ_{e+}(a) ∼ (a, +1) ∀a ∈ X_{e}. \end{matrix}

Example (see figure).

¡figure¿

Theorem 1. Assume all attaching maps of edge spaces in our graph of spaces are π_{1}-injective (f : A → B s.t. f_{∗} : π_{1}A → π_{1}B is a monomorphism).

Then π_{1}X “splits” as a graph of groups in the obvious way: use the same graph Γ, G_{v} ≅ π_{1}X_{v}, G_{e} ≅ π_{1}X_{e}, ϕ_{e_{+}} = (Φ_{e_{+}})_{∗} : π_{1}X_{e} → π_{1}X_{τ(e)}, ϕ_{e_{−}} = (Φ_{e_{−}})_{∗} : π_{1}X_{e} → π_{1}X_{i(e)}.

(We can assume all spaces have a base point, and all the maps are base point preserving. In fact, it doesn’t matter.)

Then π_{1}X ≅ π_{1}associated graph of groups.

Example (see figure “pair of pants”).

¡figure¿

Example (see figure “BS(n, m)”).

¡figure¿

Reminder. How to compute π_{1}Γ. Example (see figure “house”).

¡figure¿

Pick a base point p in Γ. Pick a maximal tree T ⊂ Γ. Then there is a one-to-one correspondence “generators of π_{1}Γ ↔ edges Γ \ T ”.

For each oriented edge e in Γ \ T put \overrightarrow{p i(e)}e\overleftarrow{p τ(e)} = σ_{e}. (Let p ∈ T be a base point. For each v ∈ T define \overrightarrow{p v} to be the shortest edge path from p to v in T.)

Then, in our example, π_{1}(Γ) = ⟨σ_{a}, σ_{b}, σ_{c}⟩.

Arbitrary Group a Fundamental Group. Every group is a π_{1}X for some topological space X.

Theorem 2. Let G be a group, let P = ⟨g_{1}, … |R_{1}, …⟩ be its presentation, let X be the standard 2-complex of P. Then G = π_{1}X.

(BTW, proof of the theorem is Van Kampen’s theorem.)

In particular, let M be the multiplication table presentation. Let X_{G} be the standard 2-complex of M. (This is called canonical graph of spaces.)

Now, a monomorphism H ⊂ G corresponds to a π_{1}-injective embedding X_{H} ⊂ X_{G}.

Using this we see that every graph of groups is induced (arises) from some graph of spaces (in fact, many graphs of spaces).

Example (see figure “non-canonical graph of spaces”).