Section5.6Presenting Spaces and the 2-way Street Theorem
Subsection5.6.1CW complexes
Definition5.6.1.
Let X(0) be a set of points with the discrete topology. For each n \geq 1, form the space X(n) from X(n-1) by: For each \alpha in an index set Jn, let D\alpha n be a copy of the closed unit disk Dn in \R^n (with Euclidean topology), and let φ\alpha : S\alpha n-1 \to X(n-1) be a continuous function. Let Zn := X(n-1) ∐ (∐\alpha \in Jn D\alpha n) have the disjoint union topology, and let \sim be the minimal equivalence relation on this space such that d \sim φ\alpha (d) for all d \in S\alpha n-1 and \alpha \in Jn. Define X(n) be the quotient space and let qn be the quotient map. Let X := \cup n \in \N X(n) (where each X(n) is identified with its image in X(n+1)). If there is no n such that X = X(n), then a subset A of X is defined to be open in X if and only if A \cap X(n) is open in X(n) for all n. The space X is a CW complex, and the topology on X is the CW topology or weak topology. The subspace X(n) of X is the n-skeleton of X. The procedure of building X(n) from X(n-1) is called gluing on n-disks. For each n \geq 1 and each \alpha \in Jn, let Φ\alpha : D\alpha n \to X be the composition Φ\alpha := iX(n)X \circ qn \circ iD\alpha nZn (where iX(n)X: X(n) \to X and iD\alpha nZn: D\alpha n \to Zn are the inclusion maps). For n = 0, let J0 be a set in bijection with X(0), and for each \alpha \in J0, let D\alpha 0 = B_\alpha 0 :=\{∗\} be a singleton space and let Φ\alpha : D\alpha 0 \to X be defined by Φ\alpha (p) := v\alpha, the vertex in X corresponding to \alpha. For all n \geq 0; and \alpha \in Jn, the map Φ\alpha is the characteristic map, or gluing map, or attaching map associated to \alpha. The image e\alpha n := Φ\alpha (B_\alpha n) is the open n-cell of X corresponding to \alpha. If there is an n \in \N such that there is an open n-cell but there are no open k-cells for any k > n, then X = X(n) is n-dimensional. If there is no such n, then X is infinite dimensional.
Proposition5.6.2.
Let X be a CW complex, let n \geq 1, and let \alpha \in Jn.
The map qn|X(n-1) is an embedding of X(n-1) into X(n).
As a set, X(n) is the disjoint union X(n-1) ∐ (∐\alpha \in Jn e\alpha n).
Φ\alpha |S\alpha n-1X(n-1) = φ\alpha.
The open n-cell e\alpha n is an open set of X and Φ\alpha |B_\alpha ne\alpha n: B_\alpha n \to e\alpha n is a homeomorphism.
Example5.6.3.
Examples
Theorem5.6.4.
Let X be a CW complex and let A be a subset of X. The set A is open in X if and only if for all n \geq 1 and all \alpha \in Jn, Φ\alpha -1(A) is open in D\alpha n.
Proposition5.6.5.
Let X be a CW complex.
A subset B of X is closed in X if and only if B \cap X(n) is closed in X(n) for all n.
If n \geq 0 and \beta \in Jn, then \text{Cl}_X(e\beta n) has nonempty intersection with at most finitely many open cell of X.
A subset B of X is closed in X if and only if B \cap \text{Cl}_X(e\beta n) is closed in \text{Cl}_X(e\beta n) for all open cells e\beta n.
Remark5.6.6.
To prove properties of a CW complex X, a standard method is to use induction (on n).
Lemma5.6.7.
A graph is a 1-dimensional CW complex.
Definition5.6.8.
For a CW complex X, the closed n-cell associated to an open n-cell e\beta n is \text{Cl}_X(e\beta n). A vertex, or 0-cell, of X is an element of X(0). An edge of X is a closed 1-cell, and a face of X is a closed 2-cell.
Theorem5.6.9.
Let X be a CW complex with associated index sets Jn for the n-cells and characterstic maps Φ\alpha for each \alpha \in Jn and each n, and let Y be a topological space.
CW-CFBT = "CW-Continuous Function Building Thm".
Suppose that for all n \geq 0 and \alpha \in Jn, g_\alpha : Dn \to Y is a continuous function. Suppose also that for all \alpha \in Jm and \beta \in Jn with \alpha \ne \beta, whenever p \in D\alpha m, q \in D\beta n, and Φ\alpha (p) = Φ\beta (q), then g_\alpha (p) = g\beta (q). Then the collection\{g_\alpha \} induces a unique continuous function g: X \to Y with g \circ Φ\alpha = g_\alpha for all n \geq 0 and \alpha \in Jn.
CW-CFCT = "CW-Continuous Function Checking Thm".
If f:X \to Y is a function, then f is continuous iff every composition f \circ Φ\alpha of f with an attaching map is continuous.
Theorem5.6.10.CW-HBT = "CW-Homeomorphism Building Thm".
Let X be a compact CW complex with associated index sets Jn for the n-cells and characterstic maps Φ\alpha for each \alpha \in Jn and each n, and let Y be a T_2 topological space. Suppose that for all n \geq 0 and \alpha \in Jn, g_\alpha : Dn \to Y is a continuous function such that g_\alpha |Bn is injective. Suppose also that for all \alpha \in Jm and \beta \in Jn with \alpha \ne \beta, (i) whenever p \in D\alpha m, q \in D\beta n, and Φ\alpha (p) = Φ\beta (q), then g_\alpha (p) = g\beta (q), and (ii) g_\alpha (B_\alpha m) \cap g\beta (B\beta n) = \es. If Y = \cup n \geq 0, \alpha \in Jn g_\alpha (Bn), then the collection\{g_\alpha \} induces a unique homeomorphism g: X \to Y with g \circ Φ\alpha = g_\alpha for all n \geq 0 and \alpha \in Jn.
Interactions with homeomorphism invariants.
Theorem5.6.11.
CW complexes are T_2.
Theorem5.6.12.
A CW complex X is compact if and only if X has only finitely many cells.
Theorem5.6.13.
A CW complex X is PC if and only if the 1-skeleton X(1) is PC.
Definition5.6.14.
A space X locally deforms to each point if each point p of X is contained in an open subset Up of X such that there is a deformation retraction from the subspace Up onto\{p\}.
Theorem5.6.15.
Every CW complex locally deforms to each point.
Corollary5.6.16.
If X and Y are PC CW complexes, then X ∨ Y is a CW complex and \pi_1(X ∨ Y) \cong \pi_1(X) ∗ \pi_1(Y).
Definition5.6.17.
A topological space X is defined to be locally path-connected, or LPC, if for every point p \in X and every open set U of X containing p, there is an open set V of X such that p \in V \sse U and (as a subspace of X) V is path-connected.
Corollary5.6.18.
CW complexes are LPC.
Definition5.6.19.
A topological space X is simply-connected, or SC, if X is PC and \pi_1(X) = 1.
Definition5.6.20.
A space X is semi-locally simply connected, or SLSC if for each point p of X there is an open set Up of X containing p such that the inclusion map Up \to X induces the constant homomorphism \pi_1(Up,p) \to \pi_1(X,p) to the identity element of \pi_1(X,p).
Corollary5.6.21.
CW complexes are SLSC.
Interactions with constructions:.
Definition5.6.22.
Let X be a CW complex. A subcomplex of X is a closed subspace A in X satisfying the property that whenever e\alpha is an open cell of X and e\alpha \cap A is nonempty, then \text{Cl}_X(e\alpha ) \sse A.
Theorem5.6.23.
If A is a subcomplex of a CW complex X, then A is a CW complex.
Theorem5.6.24.
If X and Y are compact CW complexes, then X \times Y is also a compact CW complex.
Subsection5.6.2Fundamental groups of CW complexes
Theorem5.6.25.
Let X be a 2-dimensional PC CW complex and let p \in X be the basepoint. The inclusion map X(1) \to X(2) induces a surjective homomorphism of fundamental groups whose kernel is generated by loops based at p in bijection with the 2-cells of X. More precisely: For each face f = e\alpha 2 of X define \theta \alpha : I \to X by \theta \alpha := iX(1)X \circ φ\alpha \circ ω, where iX(1)X: X(1) \to X is the inclusion map and ω:I \to S\alpha 1 is defined by ω(t) := (cos(2\pi t),sin(2\pi t)) for all t \in I. Let γ\alpha :I \to X be a path from p to \theta \alpha (0), and let wf := γ\alpha ∗ \theta \alpha ∗ γ\alpha rev. Then \pi_1(X,p) \cong \pi_1(X(1),p) / \langle{[wf]\}\rangle^N.
Theorem5.6.26.
For any PC CW complex X, \pi_1(X) \cong \pi_1(X(2)).
Theorem5.6.27.Algorithm to present \pi_1(CW).
Let X be a PC CW complex and let p \in X(0) be the basepoint. Let ω:I \to S\alpha 1 be defined by ω(t) := (cos(2\pi t),sin(2\pi t)) and let ρ:I \to D1 be defined by ρ(t) := 2t - 1 for all t \in I. Suppose that for each face (2-cell) f = e\alpha 2 of X, the map sf: I \to X defined by sf := iX(1)X \circ φ\alpha \circ ω satisfies sf(0) \in X(0) and sf follows an edge path in X(1). Then a presentation for \pi_1(X,p) can be computed as follows: Step 0: Choose a maximal tree T in the 1-skeleton X(1). For each vertex v of X, let tv: I \to X be the (shortest) path in T from p to v. Step 1: For each edge e = e\alpha e1 of X(1) outside of T, let ae be the loop in X at p defined by ae' := tΦ\alpha e(-1) ∗ be ∗ tΦ\alpha e(1)rev, where be := Φ\alpha e \circ ρ is the path in X from Φ\alpha e(-1) to Φ\alpha e(1) that traverses the edge e. Let ae := [ae']. Then \pi_1(X(1),p) = F({ae | e is an edge of X outside of T\}). Step 2: For each face f = e\alpha f2 of X(2), let rf'' := tsf(0) ∗ sf ∗ tsf(1)rev. By inserting and/or deleting path products with paths of the form e ∗ erev and erev ∗ e for edges e in T into rf'', create a path rf' that is a path product of paths of the form ae' or (ae')rev for edges e outside of T; that is, [rf'] =\pi_1(X(1),p) rf for some word rf = ae1j1 ··· aemjm with m \geq 0 and each ji \in\{± 1\}. Then \pi_1(X,p) = \pi_1(X(2),p) = \langle\{ae | e is an edge of X outside of T\} |\{rf = 1 | f is a face of X\} \rangle.
Definition5.6.28.
The presentation complex associated to a presentation \langle A | R \rangle of a group G is a CW complex with one vertex v, an edge ea for each a \in A (with attaching maps gluing both endpoints of ea to v), and a face fr for each r \in R with attaching map determined by following the edges according to the word r.
Theorem5.6.29."2-Way Street Thm".
For every group G, there is a 2-dimensional PC CW complex X with \pi_1(X) \cong G. Moreover, if \langle A | R \rangle is a presentation of G and Y is the associated presentation complex, then \pi_1(Y) \cong G.
