Section6.2Building Covering Spaces Using Group Actions
Subsection6.2.1Covering space group actions
Definition6.2.1.
Given a group \(G\) and a topological space \(Y\text{,}\) an action of \(G\) on \(Y\) (or "action of \(G\) on \(Y\) by homeomorphisms") is a homomorphism \(ρ:G \to Homeo(Y)\text{,}\) where \(Homeo(Y)\) is the group of homeomorphisms \(:Y \to Y\) with group operation given by composition. For each \(g \in G\) and \(y \in Y\text{,}\) denote \(g · y := (ρ(g))(y)\text{.}\)
Definition6.2.2.
Given an action of a group \(G\) on a space \(Y\text{,}\) let \(\sim G\) be the equivalence relation on \(Y\) defined by \(y \sim G y'\) iff there is a \(g \in G\) with \(g · y = y'\) for all \(y,y' \in Y\text{.}\) For each \(y \in Y\text{,}\) the equivalence class \([y] = [y]G = G · y\) is called the orbit of \(y\text{.}\) The quotient space \(Y/\sim\) is called the orbit space of the group action and denoted \(Y/G\text{.}\)
Definition6.2.3.
A covering space action of a group \(G\) on a space \(Y\) is an action satisfying: For each \(y \in Y\) there is an open set \(Uy\) in \(Y\) containing \(y\) such that whenever \(g,g' \in G\) and \((g · Uy) \cap (g' · Uy) \ne \es\) then \(g = g'\text{.}\)
Theorem6.2.4.
If \(G\) has a covering space action on \(Y\text{,}\) then:
The quotient \(p:Y \to Y/G\) is a covering space.
If \(Y\) is PC and LPC and \(y_0 \in Y\text{,}\) then \(p∗(\pi_1(Y,y_0)) ⊴ \pi_1(Y/G,[y_0])\) and \(G \cong \pi_1(Y/G,[y_0]) / p∗(\pi_1(Y,y_0))\text{.}\)
Corollary6.2.5.
If \(Y\) is an SC CW complex and \(G\) acts on \(Y\) with a covering space action, then \(\pi_1(Y/G) \cong G\text{.}\)
Subsection6.2.2Building SC covering spaces from group presentations
Definition6.2.6.
Let \(\langle A | R \rangle\) be a presentation of a group \(G\text{.}\) The Cayley complex of this presentation is the CW complex \(X̃\) constructed by: \(X̃(0) = G\text{.}\) The set of edges is in bijection with \(G \times A\text{;}\) for each \(g \in G\) and \(a \in A\text{,}\) the attaching map of the edge eg, \(a\) is \(φg,a(-1) = g\) and \(φg,a(1) = ga\text{.}\) The set of faces is in bijection with \(G \times R\text{;}\) for each \(g \in G\) and \(r \in R\text{,}\) the attaching map \(φg,r\) of the face \(fg,r\) satisfies \(φg,r \circ ω := \text{ edge path in }X̃(1)\) starting at \(g\) labeled by \(r\text{.}\) The \(1\)-skeleton \(X̃(1)\) is the Cayley graph of \(G\) with respect to \(A\text{.}\)
Theorem6.2.7.
Let \(\langle A | R \rangle\) be a presentation of a group \(G\text{,}\) let \(X̃\) be the Cayley complex, and let \(X\) be the presentation complex. Then
the action of \(G\) on \(X̃\text{,}\) given by \(g · h := (gh), g · eh,a := e(gh),a\text{,}\) and \(g · fh,r := f(gh),r\) for all \(h \in G\text{,}\)\(a\in A\text{,}\) and \(r \in R\text{,}\) is a covering space action;
\(X̃\) is a SC CW complex (and hence PC, LPC, and SLSC), and hence \(\pi_1(X̃/G) \cong G\text{;}\) and
\(X̃/G \cong X\text{.}\)
Example6.2.8.
Examples
Subsection6.2.3Existence of covering spaces
Theorem6.2.9.
Let \(X\) be a PC, LPC, SLSC space and \(x_0 \in X\text{.}\) Then:
(SC Covering Space Thm): There is a SC covering space \(p:(X̃,x̃_0) \to (X,x_0)\text{,}\) and there is covering space group action of \(G := \pi_1(X,x_0)\) on \(X̃\) that induces \(p\text{;}\) hence \((X̃/G,[x̃_0]) \cong (X,x_0)\text{.}\) Moreover, if \(X\) is a CW complex, then \(X̃\) is a CW complex, and for each open \(n\)-cell \sigma of \(X\) there are \(|G|\) open \(n\)-cells of \(X̃\) mapping via \(p\) to \(\sigma\) such that the action of each \(g \in G\) on \(X̃\) permutes these \(n\)-cells.
(Covering Space Existence Thm): For each subgroup \(H < \pi_1(X,x_0)\text{,}\) there is a covering space \(rH: (X̃/H,[x̃_0]H) \to (X,x_0)\) such that \(rH \circ qH = p\text{,}\) where \(qH: X̃ \to X̃/H\) is the quotient map, and \(H = rH∗(\pi_1(X̃/H,[x̃_0]H))\text{.}\)
Example6.2.10.
Examples
Corollary6.2.11.
If \((X,x_0)\) is a PC, LPC, SLSC space or PC CW complex, \(H < \pi_1(X,x_0)\text{,}\) and \(rH:(X̃/H,[x̃_0]H) \to (X,x_0)\) is the covering space of Thm 6.60(b), then the number of sheets of this covering space is the index of \(H\) in \(\pi_1(X,x_0)\text{.}\)
Subsection6.2.4Building covering spaces for subgroups from group presentations
Theorem6.2.12.
Let \(\langle A | R \rangle\) be a presentation of a group \(G\) and let \(X\) be the presentation complex with vertex \(x_0\text{.}\) Let \(H < G\text{.}\)
(Top-down Method): Let \(X̃\) be the Cayley complex of the presentation with quotient map \(p:X̃ \to X\text{.}\) Then \(H\) has a covering space action on \(X̃\text{,}\) with quotient map \(qH:X̃ \to X̃/H\text{,}\) and the covering space \(rH:X̃/H \to X\) defined by \(rH([s]H) := p(s)\) for all \([s]H \in X̃/H\) satisfies \(Im rH∗ = H\text{.}\)
(Bottom-up Method): Let \(Z\) be the CW complex constructed by: \(Z(0) := H\G\) (the set of right cosets of \(H\) in \(G\)). The set of edges of \(Z\) is in bijection with \(H\G \times A\text{;}\) for each \(Hg \in H\G\) and \(a \in A\text{,}\) the attaching map of the edge \(eHg\text{,}\)\(a\) is \(φHg,a(-1) = Hg and φHg,a(1) = Hga\text{.}\) The set of faces of \(Z\) is in bijection with \(H\G \times R\text{;}\) for each \(Hg \in H\G\) and \(r \in R\text{,}\) the attaching map \(φHg,r\) of the face \(fHg,r\) satisfies \(φHg,r \circ ω :=\) edge path in \(Z(1)\) starting at \(Hg\) labeled by \(r\text{.}\) Then \(r:Z \to X\) such that \(r(Hg) = x_0, r(eHg,a) = \)the edge of \(X\) corresponding to \(a\text{,}\) and \(r(fHg,r) =\) the face of \(X\) corresponding to \(r\) (for all \(Hg, a\text{,}\) and \(r\)) is a covering space and \(Im r∗ = H\text{.}\)
Subsection6.2.5Applications to group theory
Theorem6.2.13.
Every subgroup of a free group is a free group.
Theorem6.2.14.
Let \(H\) be a finite index subgroup of a group \(G\text{.}\)
If \(G\) is a finitely generated group, then \(H\) is also finitely generated.
If \(G\) is a finitely presented group, then \(H\) is also finitely presented.
