3. GRAPHS AND CW-COMPLEXES. Graphs.
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1 Graphs. 3. GRAPHS AND CW-COMPLEXES Definition. A directed graph X consists of two disjoint sets, V (X) and E(X) (the set of vertices and edges of X, resp.), together with two mappings o, t : E(X) V (X). Vertices o(e), t(e) are the endpoints of e, Definition. A graph is a directed graph X together with a mapping E(X) E(X), e ē, satisfying: e ē, ē = e, o(ē) = t(e) and (consequently) t(ē) = o(e), for all e E(X). An unoriented edge of X is a pair {e,ē}, where e E(X). Depict vertices by small circles, and edges e as line segments joining their endpoints, with an arrow pointing from o(e) to t(e). Unoriented edges are depicted as line segments without an arrow. e ē {e, ē} Note: o(e) = t(e) is allowed. Definition. An orientation of a graph X is a set containing exactly one edge from each unoriented edge {e,ē}. Definition. Let X and Y be graphs. A graph map from X to Y is a mapping f : V (X) E(X) V (Y ) E(Y ) which maps vertices to vertices and edges to edges, such that, for all edges e V (X), f (o(e)) = o( f (e)), f (t(e)) = t( f (e)) and f (ē) = f (e). Graph map f is called an isomorphism if it is bijective. A graph X is a subgraph of a graph Y if V (X) V (Y ), E(X) E(Y ) and the inclusion map V (X) E(X) V (Y ) E(Y ) is a graph map, i.e. if e E(X) then o(e), t(e) and ē have the same meaning in Y as they do in X. Paths. Let L n (n 0) be the graph with vertex set {0,1,...,n} and edge set {(0,1),(1,2),...,(n 1,n)} {(1,0),(2,1),...,(n,n 1)} with o(i,i + 1) = i, t(i,i + 1) = i + 1, (i,i + 1) = (i + 1,i). L n : n 1 n Definition. A path of length n in a graph X is a graph map p : L n X. Denote p(0) by o(p) and p(n) by t(p) and call these the endpoints of p. 1
2 [Say that p joins o(p) to t(p), and that it is a path from o(p) to t(p).] A path p of length n > 0 can be viewed as a sequence of edges (e 1,...,e n ), where t(e i ) = o(e i+1 ) for 1 i n 1, and o(p) = o(e 1 ), t(p) = t(e n ) (e i = p(i 1,i)). Also, for every vertex v of X, there is a trivial path, denoted by 1 v, with o(1 v ) = t(1 v ) = v, of length 0. Definition. A graph X is connected if, given x, y V (X), there exists a path in X from x to y. A path (e 1,...,e n ) is reduced if e i ē i+1 for 1 i n 1 (trivial paths are reduced). A path p is closed if o(p) = t(p). A circuit is a closed reduced path of positive length, (e 1,...,e n ) such that o(e i ) o(e j ) for 1 i, j n with i j. Definition. A tree is a connected graph having no circuits. Lemma 3.1. A graph X is a tree given vertices u, v of X, there is a unique reduced path which joins u to v. Proof. Omitted. A subtree of a graph is a subgraph which is a tree. A maximal tree in a graph X is a subtree of X which is not properly contained in any other subtree of X. Lemma 3.2. Let X be a connected graph. Then a subtree T of X is maximal if and only if V (T ) = V (X). Further, X has a maximal subtree. Proof. Omitted. Fundamental group. Given two paths p = (e 1,...,e n ) and q = ( f 1,..., f m ), with t(p) = o(q), the product pq is defined to be (e 1,...,e n, f 1,..., f m ), a path from o(p) to t(q). Note: if q is trivial, pq = p, and if p is trivial then pq = q. If p = (e 1,...,e n ) is a path, p means the path (ē n,...,ē 1 ) ( 1 v = 1 v ). Note: for any path p, p = p, o( p) = t(p), t( p) = o(p). Let p, q be paths; p q means one is obtained from the other by deleting a pair of edges of the form eē. Definition. Paths p and q are freely equivalent, written p q, if there is a sequence of paths p = p 1, p 2,... p n = q, where p i p i+1 for 1 i n 1. Equivalence relation on the set of all paths; denote equivalence class of p by [p]. Note: freely equivalent paths have the same endpoints. 2
3 Lemma 3.3. Any path is equivalent to a unique reduced path. Proof. Just like the proof of Lemma 1.2. Lemma 3.4. (1) If p p, q q and t(p) = o(q) then pq p q ; (2) If p q then p q. Proof. Exercise. Let v 0 be a vertex of the connected graph X. Let π 1 (X,v 0 ) be the set of equivalence classes of paths starting and ending at v 0. If [p], [q] π 1 (X,v 0 ), then by Lemma 3.4(1), can define their product by [p][q] = [pq]. This makes π 1 (X,v 0 ) into a group. Identity elt is [1 v0 ], and [p] 1 is [ p]. Definition. The group π 1 (X,v 0 ) is called the fundamental group of X at v 0. Now choose a maximal tree T of X. For v V (X), let p v be the reduced path in T from v 0 to v. For e E(X), let w e = p o(e) ep t(e). Then [w e ] π 1 (X,v 0 ), and [w e ] = 1 if e E(T ). If e E(T ) then w e is reduced. Further, wē = p o(ē) ē p t(ē) = p t(e) ē p o(e) = w e hence [w e ] 1 = [wē]. Suppose p = e 1...e n is a path from v 0 to v 0 in X. Then w e1...w en p, so [p] = [w e1 ]...[w en ]. Let A be an orientation of X. Follows: π 1 (X,v 0 ) is generated by U = {[w e ] e A \ E(T )}. Theorem 3.5. In fact, π 1 (X,v 0 ) is free with basis U. Proof. Omitted. Cayley Graphs. Let G be a gp, α : X G a mapping such that α(x) generates G. Form a directed graph: set of vertices is G; set of edges is G X; edge (g,x) has label x. o(g,x) = g, t(g,x) = gα(x). For every edge e = (g,x) add an edge ē, with label x 1, defining o(ē) = gα(x), t(ē) = g. Get a graph, the Cayley graph of G wrt α, denoted Γ(G,α). Usually, α is suppressed and graph is denoted by Γ(G,X). CW-complexes. Let E n = {x R n x 1}, U n = {x R n x < 1}, S n 1 = E n U n. Let A be a Hausdorff space, Λ a set; for λ Λ, let E λ be a copy of E n, so containing a copy S λ, U λ of S n 1, U n. Let f λ : S λ A be a cts map, for 3
4 λ Λ. Let Z = A λ Λ E λ, and let X be the quotient space Z/, where is the equiv. rel. gend by z f λ (z), z S λ, λ Λ. Let q : Z X be the quotient map: q maps A homeomorphically onto a closed subspace of X, which is identified with A via q. Also, X is Hausdorff. Definition. Let Y be a Hausdorff space having A as a subspace. Then Y is obtained from A by adjoining n-cells if a homeom. h : X Y, for some X constructed as above, with h A = id A. Put p = h q, let c λ = p(e λ ) (the n-cells), let p λ = p Eλ (the characteristic maps), and let f λ = p λ Sλ (the attaching maps). Facts: (i) A is closed in Y. (ii) p λ Uλ is a homeom. onto an open subset c λ of Y whose closure in Y is c λ. (iii)the c λ are the path components of Y A. (iv) A subset B of Y is closed iff B A is closed in A and B c λ is closed in c λ, for all λ Λ. Definition. A top. space X is a CW-complex if subspaces X 0 X 1 X 2... with X = X n n 0 such that each X n is closed, X 0 is discrete, X n is obtained from X n 1 (n 1) by adjoining n-cells and a subset Y of X is closed iff Y X n is closed in X n, for all n 0. Note: E 1 = [ 1,1], U 1 = ( 1,1). Let X be a 1-dimensional CW-complex (X = X 1 ), let c be a 1-cell. If g 1, g 2 are homeoms. U 1 c, define g 1 g 2 to mean g 1 1 g 2 is strictly increasing. (Equiv. rel. on set of such homeoms. with two equivalence classes, [g] and [g ], where g (t) = g( t).) An oriented 1-cell is a pair (c,[g]) where c is a 1-cell and [g] is one of the equivalence classes. Define (c,[g]) = (c,[g ]). If ϕ : [ 1,1] c is the char. map, put g = ϕ ( 1,1). Then define o(c,[g]) = ϕ( 1), t(c,[g]) = ϕ(1). This defines a graph Γ, with V (Γ) = X 0, E(Γ) the set of oriented 1-cells. If c is a 1-cell with char. map ϕ and g = ϕ (0,1), e = (c,[g]), then there is a path α e in X from o(e) to t(e), α e = ϕ h, where h : [0,1] [ 1,1] is h(t) = 2t 1. Define αē to be ϕ h (where ϕ (t) = ϕ( t)), so αē(t) = α e (1 t), hence α e αē is homotopic to the constant path at o(e). If p = (e 1,...,e m ) is a path in Γ, define θ(p) = homotopy class of α e1...α em (bracketing is unimportant) and let θ(1 v ) =homotopy class of const. path at v, v V (Γ). 4
5 Follows: if p q then θ(p) = θ(q). Hence, if v V (Γ), an induced map ψ : π 1 (Γ,v) π 1 (X,v), [p] θ(p). Theorem 3.6. The map ψ : π 1 (Γ,v) π 1 (X,v), is a group isomorphism, so π 1 (X,v) is a free group. Proof. See Massey s book, Thm 5.1, Ch.6 (needs Seifert-van Kampen Theorem). 2-complexes. Let X be obtained from A by attaching 2-cells. Let {c λ λ Λ} be the set of 2-cells, with char. maps p λ. Assume A path connected. Let g : [0,1] S 1 be the map t e 2πit. Then α λ := p λ g is a closed path in A. Choose v A and a path β λ in A from v to α λ (0). Then β λ α λ β 1 λ is a closed path at v, so its homotopy class w λ is in π 1 (A,v). Theorem 3.7. The homomorphism π 1 (A,v) π 1 (X,v) induced by the inclusion map is onto, and the kernel is the normal subgroup of π 1 (A,v) generated by {w λ λ Λ}. Proof. See Massey, Thm 2.1, Ch.7 (another application of Seifert-van Kampen). Let X R be a group presn. Let K 0 = {v}, and let K 1 be obtained from K 0 by adjoining a set of 1-cells {c x x X}. (Only one choice for attaching maps.) [ Bouquet of circles joined at a single point.] Let p x be the char. map, g x its restriction to ( 1,1), e x = (c x,g x ) corr. oriented edge, and define e x 1 = ē x. For oriented edge e y, y X ±1, let α y = α ey be the corr. path in K 1 as above. (preceding Thm 3.6). [If y = x ±1, x X, α y goes once round c x in a preferred direction.] Let F be the free gp on X. By Thms 3.5 and 3.6, x homotopy class of α x extends to an isomorphism Φ : F π 1 (K 1,x). If r R, say r = y 1...y k, y i X ±1, define α r = α y1...α yk. Then α r induces β r : S 1 K 1 with α r = β r g (g : t e 2πit ). Adjoin 2-cells {c r r R} to K 1 with β r as attaching map for c r, to obtain a CW-cx K = K(X R). By Thm 3.7, π 1 (K,v) = π 1 (K 1,v)/N, N being the normal subgp gend by {w r r R}, where w r is the htopy class of α r, ie w r = Φ(r). Hence π 1 (K,x) = F/N, where N is the normal subgp of F gend by R, so π 1 (K,x) has presn X R. 5
6 6 Examples. (1) x,y xyx 1 y 1 ; K the torus S 1 S 1. c x c y (2) x x 2 ; K is the real projective plane RP 2, obtained by identifying antipodal points of S 2. (3) x,y xyx 1 y ; K is the Klein bottle. Presn transforms to a,b a 2 = b 2 by Tietze transfs.
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